<table><caption>**Table 14**

Spectral Fitting Results for LAT-detected Millisecond Pulsars</caption><thead><tr><th>PSR<sup>a</sup></th><th>Energy Flux<br>(erg cm<sup>-2</sup> s<sup>-1</sup>)<br>(×10<sup>-12</sup>)</th><th>Γ<sub>100</sub></th><th>E<sub>p</sub><br>(GeV)</th><th>d<sub>p</sub></th><th>b<sub>free</sub></th><th>Luminosity<br>(10<sup>33</sup> erg s<sup>-1</sup>)</th><th>Efficiency<sup>b</sup><br>(%)</th></tr></thead><tbody><tr><td>J0023+0923</td><td>7.8 ± 0.6</td><td>...</td><td>1.28 ± 0.15</td><td>0.54 ± 0.11</td><td>...</td><td>3.03 ± 0.21<sup>+2.0</sup><sub>-1.00</sub></td><td>19 ± 1<sup>+10</sup><sub>-6</sub></td></tr><tr><td>J0030+0451</td><td>60.0 ± 1.1</td><td>0.88 ± 0.15</td><td>1.42 ± 0.03</td><td>0.58 ± 0.03</td><td>0.53 ± 0.11</td><td>0.743 ± 0.013 ± 0.026</td><td>21.3 ± 0.4 ± 0.8</td></tr><tr><td>J0034-0534</td><td>20.0 ± 0.6</td><td>1.5 ± 0.16</td><td>0.99 ± 0.09</td><td>0.32 ± 0.03</td><td>0.65 ± 0.23</td><td>4.38 ± 0.14<sup>+4.2</sup><sub>-2.8</sub></td><td>14.7 ± 0.5 ± 10</td></tr><tr><td>J0101-6422</td><td>13.0 ± 0.4</td><td>0.87 ± 0.31</td><td>1.34 ± 0.07</td><td>0.53 ± 0.05</td><td>0.43 ± 0.20</td><td>1.562 ± 0.053<sup>+1.5</sup><sub>-1.00</sub></td><td>13.3 ± 0.4 ± 10</td></tr><tr><td>J0102+4839</td><td>15.0 ± 0.7</td><td>1.2 ± 0.33</td><td>1.70 ± 0.17</td><td>0.41 ± 0.05</td><td>0.48 ± 0.25</td><td>9.59 ± 0.48<sup>+9.2</sup><sub>-6.1</sub></td><td>55 ± 3 ± 50</td></tr><tr><td>J0154+1833 †</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J0218+4232</td><td>48.0 ± 1.2</td><td>1.9 ± 0.07</td><td>0.33 ± 0.08</td><td>0.26 ± 0.02</td><td>0.76 ± 0.17</td><td>57.4 ± 1.4<sup>+35</sup><sub>-20</sub></td><td>23.5 ± 0.6<sup>+10</sup><sub>-8</sub></td></tr><tr><td>J0248+4230</td><td>1.9 ± 0.3</td><td>...</td><td>1.80 ± 0.26</td><td>1.39 ± 0.41</td><td>...</td><td>0.79 ± 0.11<sup>+0.76</sup><sub>-0.51</sub></td><td>2.1 ± 0.3 ± 2</td></tr><tr><td>J0251+2606</td><td>4.9 ± 0.4</td><td>...</td><td>1.57 ± 0.18</td><td>0.78 ± 0.17</td><td>...</td><td>0.799 ± 0.070<sup>+0.77</sup><sub>-0.51</sub></td><td>4.4 ± 0.4 ± 4</td></tr><tr><td>J0307+7443</td><td>16.0 ± 0.6</td><td>1.2 ± 0.22</td><td>1.63 ± 0.07</td><td>0.99 ± 0.08</td><td>1.12 ± 0.28</td><td>0.293 ± 0.011<sup>+0.28</sup><sub>-0.19</sub></td><td>1.35 ± 0.05 ± 1</td></tr><tr><td>J0312-0921</td><td>5.5 ± 0.4</td><td>...</td><td>1.53 ± 0.13</td><td>0.95 ± 0.18</td><td>...</td><td>0.443 ± 0.034<sup>+0.43</sup><sub>-0.28</sub></td><td>2.9 ± 0.2 ± 3</td></tr><tr><td>J0318+0253</td><td>6.3 ± 0.5</td><td>...</td><td>1.89 ± 0.18</td><td>0.89 ± 0.16</td><td>...</td><td>1.33 ± 0.10<sup>+1.3</sup><sub>-0.85</sub></td><td>27 ± 2 ± 30</td></tr><tr><td>J0340+4130</td><td>19.0 ± 0.7</td><td>0.76 ± 0.35</td><td>2.91 ± 0.14</td><td>0.56 ± 0.05</td><td>0.55 ± 0.23</td><td>5.94 ± 0.22<sup>+5.7</sup><sub>-3.8</sub></td><td>77 ± 3 ± 70</td></tr><tr><td>J0418+6635</td><td>10.0 ± 0.7</td><td>1.3 ± 0.41</td><td>1.80 ± 0.23</td><td>0.42 ± 0.07</td><td>0.55 ± 0.34</td><td>6.02 ± 0.40<sup>+5.8</sup><sub>-3.9</sub></td><td>28 ± 2 ± 30</td></tr><tr><td>J0437-4715</td><td>18.0 ± 0.5</td><td>1.6 ± 0.11</td><td>0.70 ± 0.03</td><td>0.74 ± 0.06</td><td>1.37 ± 0.24</td><td>0.052 ± 0.001 ± 0.006</td><td>0.44 ± 0.01 ± 0.06</td></tr><tr><td>J0533+6759</td><td>9.3 ± 0.5</td><td>1.2 ± 0.27</td><td>2.29 ± 0.17</td><td>0.53 ± 0.07</td><td>0.80 ± 0.30</td><td>6.38 ± 0.32<sup>+6.1</sup><sub>-4.1</sub></td><td>108 ± 5 ± 100</td></tr><tr><td>J0605+3757</td><td>6.4 ± 0.5</td><td>...</td><td>1.95 ± 0.13</td><td>1.39 ± 0.23</td><td>...</td><td>0.036 ± 0.003<sup>+0.034</sup><sub>-0.023</sub></td><td>0.39 ± 0.03 ± 0.4</td></tr><tr><td>J0610-2100</td><td>7.2 ± 0.5</td><td>...</td><td>1.34 ± 0.12</td><td>0.79 ± 0.12</td><td>...</td><td>4.19 ± 0.27 ± 2.2</td><td>50 ± 3 ± 40</td></tr><tr><td>J0613-0200</td><td>38.0 ± 1.1</td><td>1.6 ± 0.16</td><td>1.16 ± 0.07</td><td>0.43 ± 0.03</td><td>0.93 ± 0.27</td><td>2.777 ± 0.078 ± 0.51</td><td>21.1 ± 0.6 ± 4</td></tr><tr><td>J0614-3329</td><td>110.0 ± 1.4</td><td>0.86 ± 0.09</td><td>2.84 ± 0.06</td><td>0.33 ± 0.01</td><td>0.37 ± 0.07</td><td>5.443 ± 0.069 ± 1.6</td><td>24.7 ± 0.3 ± 8</td></tr><tr><td>J0621+2514</td><td>4.1 ± 0.4</td><td>...</td><td>2.69 ± 0.24</td><td>1.83 ± 0.42</td><td>...</td><td>1.31 ± 0.14<sup>+1.3</sup><sub>-0.84</sub></td><td>2.7 ± 0.3 ± 3</td></tr><tr><td>J0636+5128 †</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J0653+4706</td><td>2.0 ± 0.3</td><td>...</td><td>1.24 ± 0.15</td><td>1.82 ± 0.58</td><td>...</td><td>0.195 ± 0.029<sup>+0.19</sup><sub>-0.12</sub></td><td>2.6 ± 0.4 ± 2</td></tr><tr><td>J0737-3039A †</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J0740+6620</td><td>2.8 ± 0.3</td><td>...</td><td>1.88 ± 0.29</td><td>0.59 ± 0.17</td><td>...</td><td>0.448 ± 0.051 ± 0.12</td><td>2.2 ± 0.3 ± 0.7</td></tr><tr><td>J0751+1807</td><td>10.0 ± 0.5</td><td>0.07 ± 1.2</td><td>2.31 ± 0.12</td><td>1.22 ± 0.15</td><td>0.50 ± 0.53</td><td>1.483 ± 0.081 ± 0.49</td><td>20 ± 1 ± 8</td></tr><tr><td>J0931-1902</td><td>1.4 ± 0.2</td><td>...</td><td>1.89 ± 0.19</td><td>2.89 ± 1.16</td><td>...</td><td>2.39 ± 0.35<sup>+2.3</sup><sub>-1.5</sub></td><td>170 ± 20 ± 200</td></tr><tr><td>J0952-0607</td><td>2.4 ± 0.3</td><td>...</td><td>1.83 ± 0.23</td><td>1.29 ± 0.38</td><td>...</td><td>11.0 ± 1.5 ± 1.4</td><td>17 ± 2 ± 2</td></tr><tr><td>J0955-6150</td><td>7.7 ± 0.8</td><td>...</td><td>0.84 ± 0.28</td><td>0.49 ± 0.13</td><td>...</td><td>4.36 ± 0.45<sup>+4.2</sup><sub>-2.8</sub></td><td>6.2 ± 0.6 ± 6</td></tr><tr><td>J1012-4235</td><td>5.6 ± 0.5</td><td>...</td><td>1.30 ± 0.40</td><td>0.40 ± 0.11</td><td>...</td><td>0.093 ± 0.009<sup>+0.089</sup><sub>-0.060</sub></td><td>1.1 ± 0.1 ± 1</td></tr><tr><td>J1024-0719</td><td>4.3 ± 0.4</td><td>...</td><td>1.66 ± 0.24</td><td>0.65 ± 0.16</td><td>...</td><td>0.767 ± 0.075 ± 0.19</td><td>14 ± 1 ± 4</td></tr><tr><td>J1035-6720</td><td>19.0 ± 0.7</td><td>0.55 ± 0.44</td><td>1.65 ± 0.08</td><td>0.87 ± 0.08</td><td>0.46 ± 0.22</td><td>4.81 ± 0.19<sup>+4.6</sup><sub>-3.1</sub></td><td>6.2 ± 0.2 ± 6</td></tr><tr><td>J1036-8317</td><td>4.1 ± 0.4</td><td>...</td><td>2.15 ± 0.25</td><td>0.80 ± 0.19</td><td>...</td><td>2.29 ± 0.23<sup>+1.6</sup><sub>-0.54</sub></td><td>7.5 ± 0.7<sup>+5</sup><sub>-2</sub></td></tr><tr><td>J1048+2339</td><td>4.9 ± 0.5</td><td>...</td><td>0.89 ± 0.33</td><td>0.31 ± 0.09</td><td>...</td><td>1.021 ± 0.096<sup>+1.0</sup><sub>-0.32</sub></td><td>8.7 ± 0.8<sup>+9</sup><sub>-3</sub></td></tr><tr><td>J1124-3653</td><td>12.0 ± 0.6</td><td>1.4 ± 0.08</td><td>1.96 ± 0.15</td><td>0.51 ± 0.07</td><td>1.01 ± 0.17</td><td>1.465 ± 0.071<sup>+1.4</sup><sub>-0.94</sub></td><td>8.6 ± 0.4 ± 8</td></tr><tr><td>J1125-5825</td><td>6.5 ± 0.7</td><td>...</td><td>3.02 ± 0.36</td><td>0.93 ± 0.21</td><td>...</td><td>2.35 ± 0.24<sup>+2.3</sup><sub>-1.5</sub></td><td>2.9 ± 0.3 ± 3</td></tr><tr><td>J1125-6014</td><td>3.1 ± 0.9</td><td>...</td><td>1.87 ± 0.51</td><td>1.52 ± 0.69</td><td>...</td><td>0.361 ± 0.099<sup>+0.35</sup><sub>-0.23</sub></td><td>4 ± 1 ± 4</td></tr><tr><td>J1137+7528</td><td>0.6 ± 0.1</td><td>...</td><td>3.20 ± 0.64</td><td>1.72 ± 0.78</td><td>...</td><td>1.06 ± 0.24<sup>+1.0</sup><sub>-0.68</sub></td><td>13 ± 3 ± 10</td></tr><tr><td>J1142+0119</td><td>6.4 ± 0.5</td><td>...</td><td>2.86 ± 0.30</td><td>0.46 ± 0.11</td><td>...</td><td>3.63 ± 0.28<sup>+3.5</sup><sub>-2.5</sub></td><td>80 ± 6 ± 80</td></tr><tr><td>J1207-5050</td><td>4.8 ± 0.5</td><td>...</td><td>1.75 ± 0.20</td><td>1.26 ± 0.28</td><td>...</td><td>0.920 ± 0.099<sup>+0.88</sup><sub>-0.59</sub></td><td>43 ± 5 ± 40</td></tr><tr><td>J1221-0633</td><td>5.8 ± 0.5</td><td>...</td><td>1.48 ± 0.22</td><td>0.48 ± 0.11</td><td>...</td><td>1.090 ± 0.092<sup>+1.0</sup><sub>-0.70</sub></td><td>1.8 ± 0.2 ± 2</td></tr><tr><td>J1227-4853</td><td>19.0 ± 1.6</td><td>1.7 ± 0.45</td><td>0.66 ± 0.35</td><td>0.34 ± 0.07</td><td>0.59 ± 0.53</td><td>6.98 ± 0.60<sup>+4.5</sup><sub>-1.5</sub></td><td>6.4 ± 0.5<sup>+1</sup><sub>-1</sub></td></tr><tr><td>J1231-1411</td><td>100.0 ± 1.3</td><td>0.69 ± 0.11</td><td>2.02 ± 0.03</td><td>0.57 ± 0.02</td><td>0.53 ± 0.08</td><td>2.133 ± 0.027 ± 0.89</td><td>11.9 ± 0.2 ± 6</td></tr><tr><td>J1301+0833</td><td>7.7 ± 0.5</td><td>...</td><td>1.30 ± 0.13</td><td>0.59 ± 0.10</td><td>...</td><td>2.88 ± 0.18 ± 0.35</td><td>4.3 ± 0.3 ± 0.5</td></tr><tr><td>J1302-3258</td><td>11.0 ± 0.5</td><td>0.78 ± 0.43</td><td>2.54 ± 0.14</td><td>0.84 ± 0.10</td><td>0.86 ± 0.36</td><td>2.67 ± 0.13<sup>+2.6</sup><sub>-1.7</sub></td><td>55 ± 3 ± 50</td></tr><tr><td>J1311-3430</td><td>61.0 ± 1.2</td><td>1.6 ± 0.10</td><td>0.85 ± 0.07</td><td>0.24 ± 0.02</td><td>0.54 ± 0.15</td><td>18.11 ± 0.37<sup>+29</sup><sub>-7.1</sub></td><td>37.0 ± 0.7<sup>+60</sup><sub>-10</sub></td></tr><tr><td>J1312+0051</td><td>15.0 ± 0.7</td><td>1.1 ± 0.27</td><td>1.46 ± 0.08</td><td>0.84 ± 0.09</td><td>0.90 ± 0.30</td><td>3.78 ± 0.17<sup>+3.6</sup><sub>-2.4</sub></td><td>41 ± 2 ± 40</td></tr><tr><td>J1327-0755 †</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J1335-5656</td><td>8.2 ± 0.9</td><td>...</td><td>2.09 ± 0.35</td><td>0.52 ± 0.12</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J1400-1431</td><td>6.4 ± 0.5</td><td>...</td><td>1.43 ± 0.14</td><td>0.74 ± 0.13</td><td>...</td><td>0.059 ± 0.004<sup>+0.064</sup><sub>-0.025</sub></td><td>0.61 ± 0.05<sup>+0.7</sup><sub>-0.3</sub></td></tr><tr><td>J1431-4715</td><td>4.7 ± 0.7</td><td>...</td><td>0.44 ± 0.24</td><td>0.80 ± 0.30</td><td>...</td><td>1.74 ± 0.26<sup>+1.2</sup><sub>-0.39</sub></td><td>2.5 ± 0.4<sup>+2</sup><sub>-0.6</sub></td></tr><tr><td>J1446-4701</td><td>7.7 ± 0.7</td><td>...</td><td>1.38 ± 0.29</td><td>0.47 ± 0.11</td><td>...</td><td>2.26 ± 0.20<sup>+2.2</sup><sub>-1.4</sub></td><td>6.2 ± 0.6 ± 6</td></tr><tr><td>J1455-3330</td><td>1.1 ± 0.3</td><td>...</td><td>1.47 ± 0.38</td><td>1.34 ± 0.66</td><td>...</td><td>0.064 ± 0.016<sup>+0.061</sup><sub>-0.041</sub></td><td>3.4 ± 0.9 ± 3</td></tr><tr><td>J1513-2550</td><td>7.6 ± 0.6</td><td>...</td><td>1.05 ± 0.20</td><td>0.55 ± 0.12</td><td>...</td><td>14.3 ± 1.2<sup>+14</sup><sub>-9.1</sub></td><td>16 ± 1 ± 20</td></tr><tr><td>J1514-4946</td><td>42.0 ± 1.2</td><td>1.3 ± 0.16</td><td>2.55 ± 0.11</td><td>0.47 ± 0.04</td><td>0.84 ± 0.19</td><td>4.13 ± 0.12<sup>+4.0</sup><sub>-2.6</sub></td><td>25.9 ± 0.7 ± 20</td></tr><tr><td>J1536-4948</td><td>80.0 ± 1.9</td><td>1.4 ± 0.12</td><td>2.14 ± 0.11</td><td>0.29 ± 0.02</td><td>0.56 ± 0.14</td><td>9.14 ± 0.21<sup>+8.8</sup><sub>-5.8</sub></td><td>31.9 ± 0.7 ± 30</td></tr><tr><td>J1543-5149</td><td>18.0 ± 2.2</td><td>...</td><td>0.64 ± 0.29</td><td>0.58 ± 0.13</td><td>...</td><td>2.78 ± 0.35<sup>+2.7</sup><sub>-1.8</sub></td><td>3.8 ± 0.5 ± 4</td></tr><tr><td>J1544+4937</td><td>2.4 ± 0.3</td><td>...</td><td>2.16 ± 0.39</td><td>0.45 ± 0.14</td><td>...</td><td>2.53 ± 0.31<sup>+4.5</sup><sub>-1.4</sub></td><td>23 ± 3<sup>+40</sup><sub>-10</sub></td></tr></tbody></table>
<table><caption>**Table 14**
(Continued)</caption><thead><tr><th>PSR<sup>a</sup></th><th>Energy Flux<br>(erg cm<sup>-2</sup> s<sup>-1</sup>)<br>(×10<sup>-12</sup>)</th><th>Γ<sub>100</sub></th><th>E<sub>p</sub><br>(GeV)</th><th>d<sub>p</sub></th><th>b<sub>free</sub></th><th>Luminosity<br>(10<sup>33</sup> erg s<sup>-1</sup>)</th><th>Efficiency<sup>b</sup><br>(%)</th></tr></thead><tbody><tr><td>J1552+5437</td><td>2.7 ± 0.3</td><td>...</td><td>1.71 ± 0.31</td><td>0.50 ± 0.14</td><td>...</td><td>2.28 ± 0.24<sup>+2.2</sup><sub>-1.5</sub></td><td>29 ± 3 ± 30</td></tr><tr><td>J1555-2908</td><td>4.7 ± 0.6</td><td>...</td><td>0.32 ± 0.36</td><td>0.43 ± 0.21</td><td>...</td><td>14.5 ± 1.9 ± 3.7</td><td>4.7 ± 0.6 ± 1.0</td></tr><tr><td>J1600-3053</td><td>7.7 ± 0.5</td><td>...</td><td>2.74 ± 0.23</td><td>0.82 ± 0.14</td><td>...</td><td>3.42 ± 0.24<sup>+1.4</sup><sub>-0.87</sub></td><td>42 ± 3<sup>+20</sup><sub>-10</sub></td></tr><tr><td>J1614-2230</td><td>26.0 ± 0.8</td><td>-0.16 ± 0.56</td><td>2.26 ± 0.08</td><td>0.91 ± 0.07</td><td>0.30 ± 0.18</td><td>1.517 ± 0.050<sup>+1.6</sup><sub>-1.0</sub></td><td>12.5 ± 0.4<sup>+10</sup><sub>-8</sub></td></tr><tr><td>J1622-0315</td><td>7.2 ± 0.7</td><td>...</td><td>1.51 ± 0.28</td><td>0.48 ± 0.11</td><td>...</td><td>1.31 ± 0.12<sup>+1.4</sup><sub>-0.36</sub></td><td>17 ± 1<sup>+20</sup><sub>-4</sub></td></tr><tr><td>J1625-0021</td><td>21.0 ± 0.7</td><td>1.1 ± 0.18</td><td>2.06 ± 0.08</td><td>0.86 ± 0.05</td><td>1.27 ± 0.29</td><td>2.240 ± 0.073<sup>+2.2</sup><sub>-1.4</sub></td><td>6.1 ± 0.2 ± 6</td></tr><tr><td>J1628-3205</td><td>11.0 ± 0.9</td><td>...</td><td>1.07 ± 0.15</td><td>0.81 ± 0.14</td><td>...</td><td>1.92 ± 0.16<sup>+2.5</sup><sub>-0.57</sub></td><td>14 ± 1<sup>+20</sup><sub>-4</sub></td></tr><tr><td>J1630+3734</td><td>6.0 ± 0.5</td><td>...</td><td>1.80 ± 0.15</td><td>0.97 ± 0.18</td><td>...</td><td>1.006 ± 0.080<sup>+0.97</sup><sub>-0.94</sub></td><td>8.7 ± 0.7 ± 8</td></tr><tr><td>J1640+2224</td><td>2.3 ± 0.3</td><td>...</td><td>2.09 ± 0.35</td><td>0.88 ± 0.27</td><td>...</td><td>0.621 ± 0.091 ± 0.11</td><td>18 ± 3 ± 4</td></tr><tr><td>J1641+8049</td><td>2.0 ± 0.3</td><td>...</td><td>1.29 ± 0.24</td><td>1.02 ± 0.44</td><td>...</td><td>2.19 ± 0.34<sup>+2.1</sup><sub>-1.4</sub></td><td>4.7 ± 0.7 ± 4</td></tr><tr><td>J1649-3012</td><td>9.0 ± 0.8</td><td>...</td><td>1.71 ± 0.23</td><td>0.71 ± 0.14</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J1653-0158</td><td>34.0 ± 1.0</td><td>1.6 ± 0.15</td><td>0.83 ± 0.07</td><td>0.40 ± 0.04</td><td>0.73 ± 0.22</td><td>2.249 ± 0.067<sup>+1.9</sup><sub>-0.35</sub></td><td>18.1 ± 0.5<sup>+200</sup><sub>-3</sub></td></tr><tr><td>J1658-5324</td><td>21.0 ± 1.1</td><td>0.69 ± 0.61</td><td>0.82 ± 0.08</td><td>0.89 ± 0.11</td><td>0.32 ± 0.30</td><td>1.92 ± 0.11<sup>+1.3</sup><sub>-1.2</sub></td><td>6.3 ± 0.3 ± 6</td></tr><tr><td>J1713+0747</td><td>7.3 ± 0.6</td><td>...</td><td>2.13 ± 0.20</td><td>0.77 ± 0.14</td><td>...</td><td>1.50 ± 0.11 ± 0.079</td><td>42 ± 3 ± 2</td></tr><tr><td>J1730-2304</td><td>8.6 ± 1.7</td><td>...</td><td>0.63 ± 0.18</td><td>1.38 ± 0.53</td><td>...</td><td>0.395 ± 0.077<sup>+0.31</sup><sub>-0.12</sub></td><td>27 ± 5<sup>+10</sup><sub>-8</sub></td></tr><tr><td>J1732-5049</td><td>5.6 ± 0.7</td><td>...</td><td>1.34 ± 0.30</td><td>0.69 ± 0.19</td><td>...</td><td>2.37 ± 0.29<sup>+2.3</sup><sub>-1.5</sub></td><td>63 ± 8 ± 60</td></tr><tr><td>J1741+1351</td><td>3.6 ± 0.5</td><td>...</td><td>1.90 ± 0.30</td><td>0.74 ± 0.23</td><td>...</td><td>2.03 ± 0.27<sup>+1.4</sup><sub>-0.69</sub></td><td>9 ± 1<sup>+6</sup><sub>-3</sub></td></tr><tr><td>J1744-1134</td><td>37.0 ± 1.4</td><td>0.47 ± 0.47</td><td>1.02 ± 0.05</td><td>1.04 ± 0.08</td><td>0.48 ± 0.24</td><td>0.692 ± 0.026<sup>+0.39</sup><sub>-0.038</sub></td><td>13.3 ± 0.5<sup>+8</sup><sub>-0.7</sub></td></tr><tr><td>J1744-7619</td><td>20.0 ± 0.6</td><td>1.0 ± 0.22</td><td>1.78 ± 0.06</td><td>0.87 ± 0.07</td><td>1.04 ± 0.25</td><td>...</td><td>...</td></tr><tr><td>J1745+1017</td><td>7.6 ± 0.6</td><td>...</td><td>1.80 ± 0.19</td><td>0.80 ± 0.14</td><td>...</td><td>1.34 ± 0.11<sup>+1.3</sup><sub>-0.86</sub></td><td>21 ± 2 ± 20</td></tr><tr><td>J1747-4036</td><td>13.0 ± 1.1</td><td>...</td><td>0.82 ± 0.18</td><td>0.63 ± 0.12</td><td>...</td><td>81.8 ± 6.6<sup>+7.9</sup><sub>-52</sub></td><td>70 ± 6 ± 70</td></tr><tr><td>J1805+0615</td><td>5.3 ± 0.5</td><td>...</td><td>2.05 ± 0.25</td><td>0.90 ± 0.20</td><td>...</td><td>9.51 ± 0.98<sup>+9.1</sup><sub>-6.1</sub></td><td>10 ± 1 ± 10</td></tr><tr><td>J1810+1744</td><td>23.0 ± 0.9</td><td>1.2 ± 0.36</td><td>0.50 ± 0.11</td><td>0.38 ± 0.05</td><td>0.13 ± 0.23</td><td>9.73 ± 0.39<sup>+15</sup><sub>-3.6</sub></td><td>25 ± 1<sup>+40</sup><sub>-9</sub></td></tr><tr><td>J1811-2405</td><td>14.0 ± 3.5</td><td>...</td><td>0.49 ± 1.25</td><td>0.28 ± 0.14</td><td>...</td><td>5.5 ± 1.4<sup>+5.3</sup><sub>-3.6</sub></td><td>20 ± 5 ± 20</td></tr><tr><td>J1816+4510</td><td>11.0 ± 0.5</td><td>0.98 ± 0.38</td><td>1.76 ± 0.13</td><td>0.49 ± 0.07</td><td>0.46 ± 0.27</td><td>14.88 ± 0.70<sup>+12</sup><sub>-3.9</sub></td><td>28 ± 1<sup>+20</sup><sub>-7</sub></td></tr><tr><td>J1823-3021A</td><td>14.0 ± 0.9</td><td>...</td><td>1.84 ± 0.24</td><td>0.49 ± 0.09</td><td>...</td><td>98.2 ± 6.5 ± 10</td><td>11.9 ± 0.8 ± 1</td></tr><tr><td>J1824+1014</td><td>6.2 ± 0.6</td><td>...</td><td>2.16 ± 0.35</td><td>0.51 ± 0.13</td><td>...</td><td>6.23 ± 0.64<sup>+6.0</sup><sub>-4.0</sub></td><td>190 ± 20 ± 200</td></tr><tr><td>J1824-2452A</td><td>22.0 ± 1.6</td><td>0.29 ± 1.00</td><td>0.95 ± 0.16</td><td>0.57 ± 0.09</td><td>-0.06 ± 0.33</td><td>81.1 ± 5.7 ± 8.6</td><td>3.6 ± 0.3 ± 0.4</td></tr><tr><td>J1827-0849</td><td>30.0 ± 3.6</td><td>...</td><td>0.10 ± 0.51</td><td>0.24 ± 0.08</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J1832-0836 †</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J1833-3840</td><td>2.8 ± 0.4</td><td>...</td><td>1.25 ± 0.29</td><td>1.01 ± 0.34</td><td>...</td><td>7.3 ± 1.2<sup>+7.0</sup><sub>-4.7</sub></td><td>7 ± 1 ± 6</td></tr><tr><td>J1835-3259B</td><td>5.3 ± 0.7</td><td>...</td><td>...</td><td>...</td><td>...</td><td>45.0 ± 6.0 ± 7.9</td><td>16 ± 2 ± 4</td></tr><tr><td>J1843-1113</td><td>23.0 ± 2.2</td><td>...</td><td>0.21 ± 0.20</td><td>0.44 ± 0.10</td><td>...</td><td>4.33 ± 0.41<sup>+4.1</sup><sub>-2.8</sub></td><td>7.2 ± 0.7 ± 7</td></tr><tr><td>J1855-1436</td><td>5.1 ± 0.8</td><td>...</td><td>2.35 ± 0.79</td><td>0.39 ± 0.14</td><td>...</td><td>15.9 ± 2.5<sup>+15</sup><sub>-10</sub></td><td>170 ± 30 ± 200</td></tr><tr><td>J1858-2216</td><td>11.0 ± 0.7</td><td>0.10 ± 0.94</td><td>1.81 ± 0.12</td><td>1.07 ± 0.16</td><td>0.38 ± 0.38</td><td>1.160 ± 0.073<sup>+1.1</sup><sub>-0.74</sub></td><td>10.3 ± 0.7 ± 10</td></tr><tr><td>J1901-0125</td><td>19.0 ± 1.7</td><td>...</td><td>1.14 ± 0.25</td><td>0.62 ± 0.12</td><td>...</td><td>13.0 ± 1.1<sup>+12</sup><sub>-8.3</sub></td><td>20 ± 2 ± 20</td></tr><tr><td>J1902-5105</td><td>24.0 ± 1.0</td><td>0.88 ± 0.47</td><td>0.55 ± 0.12</td><td>0.37 ± 0.05</td><td>-0.06 ± 0.24</td><td>7.62 ± 0.32<sup>+7.3</sup><sub>-4.9</sub></td><td>11.1 ± 0.5 ± 10</td></tr><tr><td>J1903-7051</td><td>5.3 ± 0.4</td><td>...</td><td>2.57 ± 0.21</td><td>0.92 ± 0.16</td><td>...</td><td>0.546 ± 0.040<sup>+0.52</sup><sub>-0.35</sub></td><td>6.2 ± 0.5 ± 6</td></tr><tr><td>J1908+2105</td><td>4.9 ± 0.8</td><td>...</td><td>1.77 ± 0.66</td><td>0.47 ± 0.18</td><td>...</td><td>3.95 ± 0.64<sup>+3.8</sup><sub>-2.5</sub></td><td>12 ± 2 ± 10</td></tr><tr><td>J1909-3744 †</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J1921+0137</td><td>11.0 ± 1.0</td><td>...</td><td>1.32 ± 0.32</td><td>0.50 ± 0.11</td><td>...</td><td>33.5 ± 3.1<sup>+32</sup><sub>-21</sub></td><td>70 ± 7 ± 70</td></tr><tr><td>J1921+1929</td><td>3.5 ± 0.8</td><td>...</td><td>2.55 ± 0.68</td><td>1.13 ± 0.49</td><td>...</td><td>2.48 ± 0.60<sup>+2.4</sup><sub>-1.6</sub></td><td>3.0 ± 0.7 ± 3</td></tr><tr><td>J1939+2134</td><td>21.0 ± 2.7</td><td>...</td><td>0.68 ± 0.28</td><td>0.67 ± 0.16</td><td>...</td><td>23.3 ± 3.0<sup>+8.3</sup><sub>-5.4</sub></td><td>2.1 ± 0.3<sup>+0.8</sup><sub>-0.5</sub></td></tr><tr><td>J1946+3417 †</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J1946-5403</td><td>9.8 ± 0.5</td><td>1.4 ± 0.20</td><td>1.27 ± 0.08</td><td>1.01 ± 0.12</td><td>1.38 ± 0.38</td><td>1.555 ± 0.076<sup>+1.5</sup><sub>-1.00</sub></td><td>29 ± 1 ± 30</td></tr><tr><td>J1959+2048</td><td>16.0 ± 0.9</td><td>0.19 ± 1.1</td><td>1.04 ± 0.10</td><td>0.82 ± 0.11</td><td>0.11 ± 0.41</td><td>3.02 ± 0.18<sup>+4.6</sup><sub>-1.1</sub></td><td>1.9 ± 0.1<sup>+3</sup><sub>-0.7</sub></td></tr><tr><td>J2006+0148</td><td>3.0 ± 0.4</td><td>...</td><td>3.37 ± 0.64</td><td>0.65 ± 0.22</td><td>...</td><td>2.15 ± 0.28<sup>+2.1</sup><sub>-1.4</sub></td><td>17 ± 2 ± 20</td></tr><tr><td>J2017+0603</td><td>36.0 ± 1.0</td><td>1.1 ± 0.16</td><td>3.17 ± 0.12</td><td>0.58 ± 0.04</td><td>0.94 ± 0.19</td><td>8.34 ± 0.24<sup>+6.6</sup><sub>-1.7</sub></td><td>64 ± 2<sup>+50</sup><sub>-10</sub></td></tr><tr><td>J2017-1614</td><td>6.5 ± 0.6</td><td>...</td><td>1.60 ± 0.31</td><td>0.47 ± 0.12</td><td>...</td><td>1.61 ± 0.15<sup>+1.5</sup><sub>-1.0</sub></td><td>21 ± 2 ± 20</td></tr><tr><td>J2034+3632</td><td>12.0 ± 0.8</td><td>...</td><td>3.16 ± 0.18</td><td>1.38 ± 0.19</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J2039-3616</td><td>3.8 ± 0.5</td><td>...</td><td>1.18 ± 0.25</td><td>0.69 ± 0.21</td><td>...</td><td>1.33 ± 0.16<sup>+1.3</sup><sub>-0.85</sub></td><td>14 ± 2 ± 10</td></tr><tr><td>J2039-5617</td><td>15.0 ± 0.6</td><td>1.8 ± 0.08</td><td>1.47 ± 0.17</td><td>0.30 ± 0.04</td><td>1.67 ± 0.43</td><td>5.21 ± 0.22<sup>+4.0</sup><sub>-1.3</sub></td><td>17.4 ± 0.7<sup>+10</sup><sub>-4</sub></td></tr><tr><td>J2042+0246</td><td>5.8 ± 0.6</td><td>...</td><td>0.84 ± 0.09</td><td>1.36 ± 0.27</td><td>...</td><td>0.286 ± 0.029<sup>+0.27</sup><sub>-0.18</sub></td><td>4.8 ± 0.5 ± 5</td></tr><tr><td>J2043+1711</td><td>28.0 ± 0.8</td><td>1.3 ± 0.17</td><td>1.87 ± 0.10</td><td>0.41 ± 0.03</td><td>0.62 ± 0.19</td><td>6.57 ± 0.19 ± 1.1</td><td>43 ± 1 ± 10</td></tr><tr><td>J2047+1053</td><td>4.3 ± 0.6</td><td>...</td><td>2.75 ± 0.69</td><td>0.36 ± 0.14</td><td>...</td><td>3.98 ± 0.52<sup>+3.8</sup><sub>-2.5</sub></td><td>38 ± 5 ± 40</td></tr><tr><td>J2051-0827</td><td>2.5 ± 0.3</td><td>...</td><td>2.33 ± 0.29</td><td>1.27 ± 0.36</td><td>...</td><td>0.646 ± 0.081<sup>+0.62</sup><sub>-0.41</sub></td><td>12 ± 1 ± 10</td></tr><tr><td>J2052+1219</td><td>4.6 ± 0.6</td><td>...</td><td>0.45 ± 0.62</td><td>0.21 ± 0.09</td><td>...</td><td>17.0 ± 2.1 ± 3.2</td><td>50 ± 6 ± 9</td></tr></tbody></table>
<table><caption>**Table 14**
(Continued)</caption><thead><tr><th>PSR<sup>a</sup></th><th>Energy Flux<br>(erg cm<sup>-2</sup> s<sup>-1</sup>)<br>(×10<sup>-12</sup>)</th><th>Γ<sub>100</sub></th><th>E<sub>p</sub><br>(GeV)</th><th>d<sub>p</sub></th><th>b<sub>free</sub></th><th>Luminosity<br>(10<sup>33</sup> erg s<sup>-1</sup>)</th><th>Efficiency<sup>b</sup><br>(%)</th></tr></thead><tbody><tr><td>J1552+5437</td><td>2.7 ± 0.3</td><td>...</td><td>1.71 ± 0.31</td><td>0.50 ± 0.14</td><td>...</td><td>2.28 ± 0.24<sup>+2.2</sup><sub>-1.5</sub></td><td>29 ± 3 ± 30</td></tr><tr><td>J1555-2908</td><td>4.7 ± 0.6</td><td>...</td><td>0.32 ± 0.36</td><td>0.43 ± 0.21</td><td>...</td><td>14.5 ± 1.9 ± 3.7</td><td>4.7 ± 0.6 ± 1.0</td></tr><tr><td>J1600-3053</td><td>7.7 ± 0.5</td><td>...</td><td>2.74 ± 0.23</td><td>0.82 ± 0.14</td><td>...</td><td>3.42 ± 0.24<sup>+1.4</sup><sub>-0.87</sub></td><td>42 ± 3<sup>+20</sup><sub>-10</sub></td></tr><tr><td>J1614-2230</td><td>26.0 ± 0.8</td><td>-0.16 ± 0.56</td><td>2.26 ± 0.08</td><td>0.91 ± 0.07</td><td>0.30 ± 0.18</td><td>1.517 ± 0.050<sup>+1.6</sup><sub>-1.0</sub></td><td>12.5 ± 0.4<sup>+10</sup><sub>-8</sub></td></tr><tr><td>J1622-0315</td><td>7.2 ± 0.7</td><td>...</td><td>1.51 ± 0.28</td><td>0.48 ± 0.11</td><td>...</td><td>1.31 ± 0.12<sup>+1.4</sup><sub>-0.36</sub></td><td>17 ± 1<sup>+20</sup><sub>-4</sub></td></tr><tr><td>J1625-0021</td><td>21.0 ± 0.7</td><td>1.1 ± 0.18</td><td>2.06 ± 0.08</td><td>0.86 ± 0.05</td><td>1.27 ± 0.29</td><td>2.240 ± 0.073<sup>+2.2</sup><sub>-1.4</sub></td><td>6.1 ± 0.2 ± 6</td></tr><tr><td>J1628-3205</td><td>11.0 ± 0.9</td><td>...</td><td>1.07 ± 0.15</td><td>0.81 ± 0.14</td><td>...</td><td>1.92 ± 0.16<sup>+2.5</sup><sub>-0.57</sub></td><td>14 ± 1<sup>+20</sup><sub>-4</sub></td></tr><tr><td>J1630+3734</td><td>6.0 ± 0.5</td><td>...</td><td>1.80 ± 0.15</td><td>0.97 ± 0.18</td><td>...</td><td>1.006 ± 0.080<sup>+0.97</sup><sub>-0.94</sub></td><td>8.7 ± 0.7 ± 8</td></tr><tr><td>J1640+2224</td><td>2.3 ± 0.3</td><td>...</td><td>2.09 ± 0.35</td><td>0.88 ± 0.27</td><td>...</td><td>0.621 ± 0.091 ± 0.11</td><td>18 ± 3 ± 4</td></tr><tr><td>J1641+8049</td><td>2.0 ± 0.3</td><td>...</td><td>1.29 ± 0.24</td><td>1.02 ± 0.44</td><td>...</td><td>2.19 ± 0.34<sup>+2.1</sup><sub>-1.4</sub></td><td>4.7 ± 0.7 ± 4</td></tr><tr><td>J1649-3012</td><td>9.0 ± 0.8</td><td>...</td><td>1.71 ± 0.23</td><td>0.71 ± 0.14</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J1653-0158</td><td>34.0 ± 1.0</td><td>1.6 ± 0.15</td><td>0.83 ± 0.07</td><td>0.40 ± 0.04</td><td>0.73 ± 0.22</td><td>2.249 ± 0.067<sup>+1.9</sup><sub>-0.35</sub></td><td>18.1 ± 0.5<sup>+200</sup><sub>-3</sub></td></tr><tr><td>J1658-5324</td><td>21.0 ± 1.1</td><td>0.69 ± 0.61</td><td>0.82 ± 0.08</td><td>0.89 ± 0.11</td><td>0.32 ± 0.30</td><td>1.92 ± 0.11<sup>+1.3</sup><sub>-1.2</sub></td><td>6.3 ± 0.3 ± 6</td></tr><tr><td>J1713+0747</td><td>7.3 ± 0.6</td><td>...</td><td>2.13 ± 0.20</td><td>0.77 ± 0.14</td><td>...</td><td>1.50 ± 0.11 ± 0.079</td><td>42 ± 3 ± 2</td></tr><tr><td>J1730-2304</td><td>8.6 ± 1.7</td><td>...</td><td>0.63 ± 0.18</td><td>1.38 ± 0.53</td><td>...</td><td>0.395 ± 0.077<sup>+0.31</sup><sub>-0.12</sub></td><td>27 ± 5<sup>+10</sup><sub>-8</sub></td></tr><tr><td>J1732-5049</td><td>5.6 ± 0.7</td><td>...</td><td>1.34 ± 0.30</td><td>0.69 ± 0.19</td><td>...</td><td>2.37 ± 0.29<sup>+2.3</sup><sub>-1.5</sub></td><td>63 ± 8 ± 60</td></tr><tr><td>J1741+1351</td><td>3.6 ± 0.5</td><td>...</td><td>1.90 ± 0.30</td><td>0.74 ± 0.23</td><td>...</td><td>2.03 ± 0.27<sup>+1.4</sup><sub>-0.69</sub></td><td>9 ± 1<sup>+6</sup><sub>-3</sub></td></tr><tr><td>J1744-1134</td><td>37.0 ± 1.4</td><td>0.47 ± 0.47</td><td>1.02 ± 0.05</td><td>1.04 ± 0.08</td><td>0.48 ± 0.24</td><td>0.692 ± 0.026<sup>+0.39</sup><sub>-0.038</sub></td><td>13.3 ± 0.5<sup>+8</sup><sub>-0.7</sub></td></tr><tr><td>J1744-7619</td><td>20.0 ± 0.6</td><td>1.0 ± 0.22</td><td>1.78 ± 0.06</td><td>0.87 ± 0.07</td><td>1.04 ± 0.25</td><td>...</td><td>...</td></tr><tr><td>J1745+1017</td><td>7.6 ± 0.6</td><td>...</td><td>1.80 ± 0.19</td><td>0.80 ± 0.14</td><td>...</td><td>1.34 ± 0.11<sup>+1.3</sup><sub>-0.86</sub></td><td>21 ± 2 ± 20</td></tr><tr><td>J1747-4036</td><td>13.0 ± 1.1</td><td>...</td><td>0.82 ± 0.18</td><td>0.63 ± 0.12</td><td>...</td><td>81.8 ± 6.6<sup>+7.9</sup><sub>-52</sub></td><td>70 ± 6 ± 70</td></tr><tr><td>J1805+0615</td><td>5.3 ± 0.5</td><td>...</td><td>2.05 ± 0.25</td><td>0.90 ± 0.20</td><td>...</td><td>9.51 ± 0.98<sup>+9.1</sup><sub>-6.1</sub></td><td>10 ± 1 ± 10</td></tr><tr><td>J1810+1744</td><td>23.0 ± 0.9</td><td>1.2 ± 0.36</td><td>0.50 ± 0.11</td><td>0.38 ± 0.05</td><td>0.13 ± 0.23</td><td>9.73 ± 0.39<sup>+15</sup><sub>-3.6</sub></td><td>25 ± 1<sup>+40</sup><sub>-9</sub></td></tr><tr><td>J1811-2405</td><td>14.0 ± 3.5</td><td>...</td><td>0.49 ± 1.25</td><td>0.28 ± 0.14</td><td>...</td><td>5.5 ± 1.4<sup>+5.3</sup><sub>-3.6</sub></td><td>20 ± 5 ± 20</td></tr><tr><td>J1816+4510</td><td>11.0 ± 0.5</td><td>0.98 ± 0.38</td><td>1.76 ± 0.13</td><td>0.49 ± 0.07</td><td>0.46 ± 0.27</td><td>14.88 ± 0.70<sup>+12</sup><sub>-3.9</sub></td><td>28 ± 1<sup>+20</sup><sub>-7</sub></td></tr><tr><td>J1823-3021A</td><td>14.0 ± 0.9</td><td>...</td><td>1.84 ± 0.24</td><td>0.49 ± 0.09</td><td>...</td><td>98.2 ± 6.5 ± 10</td><td>11.9 ± 0.8 ± 1</td></tr><tr><td>J1824+1014</td><td>6.2 ± 0.6</td><td>...</td><td>2.16 ± 0.35</td><td>0.51 ± 0.13</td><td>...</td><td>6.23 ± 0.64<sup>+6.0</sup><sub>-4.0</sub></td><td>190 ± 20 ± 200</td></tr><tr><td>J1824-2452A</td><td>22.0 ± 1.6</td><td>0.29 ± 1.00</td><td>0.95 ± 0.16</td><td>0.57 ± 0.09</td><td>-0.06 ± 0.33</td><td>81.1 ± 5.7 ± 8.6</td><td>3.6 ± 0.3 ± 0.4</td></tr><tr><td>J1827-0849</td><td>30.0 ± 3.6</td><td>...</td><td>0.10 ± 0.51</td><td>0.24 ± 0.08</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J1832-0836 †</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J1833-3840</td><td>2.8 ± 0.4</td><td>...</td><td>1.25 ± 0.29</td><td>1.01 ± 0.34</td><td>...</td><td>7.3 ± 1.2<sup>+7.0</sup><sub>-4.7</sub></td><td>7 ± 1 ± 6</td></tr><tr><td>J1835-3259B</td><td>5.3 ± 0.7</td><td>...</td><td>...</td><td>...</td><td>...</td><td>45.0 ± 6.0 ± 7.9</td><td>16 ± 2 ± 4</td></tr><tr><td>J1843-1113</td><td>23.0 ± 2.2</td><td>...</td><td>0.21 ± 0.20</td><td>0.44 ± 0.10</td><td>...</td><td>4.33 ± 0.41<sup>+4.1</sup><sub>-2.8</sub></td><td>7.2 ± 0.7 ± 7</td></tr><tr><td>J1855-1436</td><td>5.1 ± 0.8</td><td>...</td><td>2.35 ± 0.79</td><td>0.39 ± 0.14</td><td>...</td><td>15.9 ± 2.5<sup>+15</sup><sub>-10</sub></td><td>170 ± 30 ± 200</td></tr><tr><td>J1858-2216</td><td>11.0 ± 0.7</td><td>0.10 ± 0.94</td><td>1.81 ± 0.12</td><td>1.07 ± 0.16</td><td>0.38 ± 0.38</td><td>1.160 ± 0.073<sup>+1.1</sup><sub>-0.74</sub></td><td>10.3 ± 0.7 ± 10</td></tr><tr><td>J1901-0125</td><td>19.0 ± 1.7</td><td>...</td><td>1.14 ± 0.25</td><td>0.62 ± 0.12</td><td>...</td><td>13.0 ± 1.1<sup>+12</sup><sub>-8.3</sub></td><td>20 ± 2 ± 20</td></tr><tr><td>J1902-5105</td><td>24.0 ± 1.0</td><td>0.88 ± 0.47</td><td>0.55 ± 0.12</td><td>0.37 ± 0.05</td><td>-0.06 ± 0.24</td><td>7.62 ± 0.32<sup>+7.3</sup><sub>-4.9</sub></td><td>11.1 ± 0.5 ± 10</td></tr><tr><td>J1903-7051</td><td>5.3 ± 0.4</td><td>...</td><td>2.57 ± 0.21</td><td>0.92 ± 0.16</td><td>...</td><td>0.546 ± 0.040<sup>+0.52</sup><sub>-0.35</sub></td><td>6.2 ± 0.5 ± 6</td></tr><tr><td>J1908+2105</td><td>4.9 ± 0.8</td><td>...</td><td>1.77 ± 0.66</td><td>0.47 ± 0.18</td><td>...</td><td>3.95 ± 0.64<sup>+3.8</sup><sub>-2.5</sub></td><td>12 ± 2 ± 10</td></tr><tr><td>J1909-3744 †</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J1921+0137</td><td>11.0 ± 1.0</td><td>...</td><td>1.32 ± 0.32</td><td>0.50 ± 0.11</td><td>...</td><td>33.5 ± 3.1<sup>+32</sup><sub>-21</sub></td><td>70 ± 7 ± 70</td></tr><tr><td>J1921+1929</td><td>3.5 ± 0.8</td><td>...</td><td>2.55 ± 0.68</td><td>1.13 ± 0.49</td><td>...</td><td>2.48 ± 0.60<sup>+2.4</sup><sub>-1.6</sub></td><td>3.0 ± 0.7 ± 3</td></tr><tr><td>J1939+2134</td><td>21.0 ± 2.7</td><td>...</td><td>0.68 ± 0.28</td><td>0.67 ± 0.16</td><td>...</td><td>23.3 ± 3.0<sup>+8.3</sup><sub>-5.4</sub></td><td>2.1 ± 0.3<sup>+0.8</sup><sub>-0.5</sub></td></tr><tr><td>J1946+3417 †</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J1946-5403</td><td>9.8 ± 0.5</td><td>1.4 ± 0.20</td><td>1.27 ± 0.08</td><td>1.01 ± 0.12</td><td>1.38 ± 0.38</td><td>1.555 ± 0.076<sup>+1.5</sup><sub>-1.00</sub></td><td>29 ± 1 ± 30</td></tr><tr><td>J1959+2048</td><td>16.0 ± 0.9</td><td>0.19 ± 1.1</td><td>1.04 ± 0.10</td><td>0.82 ± 0.11</td><td>0.11 ± 0.41</td><td>3.02 ± 0.18<sup>+4.6</sup><sub>-1.1</sub></td><td>1.9 ± 0.1<sup>+3</sup><sub>-0.7</sub></td></tr><tr><td>J2006+0148</td><td>3.0 ± 0.4</td><td>...</td><td>3.37 ± 0.64</td><td>0.65 ± 0.22</td><td>...</td><td>2.15 ± 0.28<sup>+2.1</sup><sub>-1.4</sub></td><td>17 ± 2 ± 20</td></tr><tr><td>J2017+0603</td><td>36.0 ± 1.0</td><td>1.1 ± 0.16</td><td>3.17 ± 0.12</td><td>0.58 ± 0.04</td><td>0.94 ± 0.19</td><td>8.34 ± 0.24<sup>+6.6</sup><sub>-1.7</sub></td><td>64 ± 2<sup>+50</sup><sub>-10</sub></td></tr><tr><td>J2017-1614</td><td>6.5 ± 0.6</td><td>...</td><td>1.60 ± 0.31</td><td>0.47 ± 0.12</td><td>...</td><td>1.61 ± 0.15<sup>+1.5</sup><sub>-1.0</sub></td><td>21 ± 2 ± 20</td></tr><tr><td>J2034+3632</td><td>12.0 ± 0.8</td><td>...</td><td>3.16 ± 0.18</td><td>1.38 ± 0.19</td><td>...</td><td>...</td><td>...</td></tr><tr><td>J2039-3616</td><td>3.8 ± 0.5</td><td>...</td><td>1.18 ± 0.25</td><td>0.69 ± 0.21</td><td>...</td><td>1.33 ± 0.16<sup>+1.3</sup><sub>-0.85</sub></td><td>14 ± 2 ± 10</td></tr><tr><td>J2039-5617</td><td>15.0 ± 0.6</td><td>1.8 ± 0.08</td><td>1.47 ± 0.17</td><td>0.30 ± 0.04</td><td>1.67 ± 0.43</td><td>5.21 ± 0.22<sup>+4.0</sup><sub>-1.3</sub></td><td>17.4 ± 0.7<sup>+10</sup><sub>-4</sub></td></tr><tr><td>J2042+0246</td><td>5.8 ± 0.6</td><td>...</td><td>0.84 ± 0.09</td><td>1.36 ± 0.27</td><td>...</td><td>0.286 ± 0.029<sup>+0.27</sup><sub>-0.18</sub></td><td>4.8 ± 0.5 ± 5</td></tr><tr><td>J2043+1711</td><td>28.0 ± 0.8</td><td>1.3 ± 0.17</td><td>1.87 ± 0.10</td><td>0.41 ± 0.03</td><td>0.62 ± 0.19</td><td>6.57 ± 0.19 ± 1.1</td><td>43 ± 1 ± 10</td></tr><tr><td>J2047+1053</td><td>4.3 ± 0.6</td><td>...</td><td>2.75 ± 0.69</td><td>0.36 ± 0.14</td><td>...</td><td>3.98 ± 0.52<sup>+3.8</sup><sub>-2.5</sub></td><td>38 ± 5 ± 40</td></tr><tr><td>J2051-0827</td><td>2.5 ± 0.3</td><td>...</td><td>2.33 ± 0.29</td><td>1.27 ± 0.36</td><td>...</td><td>0.646 ± 0.081<sup>+0.62</sup><sub>-0.41</sub></td><td>12 ± 1 ± 10</td></tr><tr><td>J2052+1219</td><td>4.6 ± 0.6</td><td>...</td><td>0.45 ± 0.62</td><td>0.21 ± 0.09</td><td>...</td><td>17.0 ± 2.1 ± 3.2</td><td>50 ± 6 ± 9</td></tr></tbody></table>
<table><caption>**Table 14**
(Continued)</caption><thead><tr><th>PSR<sup>a</sup></th><th>Energy Flux<br>(erg cm<sup>-2</sup> s<sup>-1</sup>)<br>(×10<sup>-12</sup>)</th><th>Γ<sub>100</sub></th><th>E<sub>p</sub><br>(GeV)</th><th>d<sub>p</sub></th><th>b<sub>free</sub></th><th>Luminosity<br>(10<sup>33</sup> erg s<sup>-1</sup>)</th><th>Efficiency<sup>b</sup><br>(%)</th></tr></thead><tbody><tr><td>J2115+5448</td><td>7.0 ± 0.7</td><td>...</td><td>3.06 ± 0.30</td><td>1.31 ± 0.27</td><td>...</td><td>8.10 ± 0.80<sup>+7.8</sup><sub>-5.2</sub></td><td>4.8 ± 0.5 ± 5</td></tr><tr><td>J2124-3358</td><td>39.0 ± 0.8</td><td>0.97 ± 0.12</td><td>1.95 ± 0.05</td><td>0.77 ± 0.04</td><td>1.09 ± 0.16</td><td>0.780 ± 0.016<sup>+0.43</sup><sub>-0.18</sub></td><td>11.5 ± 0.2<sup>+6</sup><sub>-3</sub></td></tr><tr><td>J2129-0429</td><td>6.8 ± 0.5</td><td>...</td><td>1.66 ± 0.17</td><td>0.58 ± 0.11</td><td>...</td><td>2.73 ± 0.19 ± 0.32</td><td>9.3 ± 0.6 ± 1</td></tr><tr><td>J2205+6012</td><td>3.7 ± 1.0</td><td>...</td><td>2.40 ± 0.52</td><td>2.90 ± 1.67</td><td>...</td><td>5.6 ± 1.6<sup>+5.4</sup><sub>-3.6</sub></td><td>10 ± 3 ± 10</td></tr><tr><td>J2214+3000</td><td>33.0 ± 0.7</td><td>1.0 ± 0.15</td><td>1.76 ± 0.05</td><td>0.68 ± 0.04</td><td>0.92 ± 0.17</td><td>1.406 ± 0.032<sup>+1.8</sup><sub>-1.1</sub></td><td>7.3 ± 0.2<sup>+10</sup><sub>-6</sub></td></tr><tr><td>J2215+5135</td><td>18.0 ± 0.8</td><td>1.4 ± 0.30</td><td>1.45 ± 0.17</td><td>0.40 ± 0.05</td><td>0.52 ± 0.28</td><td>15.26 ± 0.69<sup>+21</sup><sub>-4.9</sub></td><td>24 ± 1<sup>+30</sup><sub>-8</sub></td></tr><tr><td>J2234+0944</td><td>10.0 ± 0.6</td><td>0.74 ± 0.65</td><td>1.87 ± 0.18</td><td>0.59 ± 0.10</td><td>0.36 ± 0.34</td><td>2.45 ± 0.15<sup>+5.0</sup><sub>-1.2</sub></td><td>14.7 ± 0.9<sup>+30</sup><sub>-8</sub></td></tr><tr><td>J2241-5236</td><td>25.0 ± 1.1</td><td>1.3 ± 0.11</td><td>1.90 ± 0.08</td><td>0.95 ± 0.07</td><td>1.35 ± 0.22</td><td>3.25 ± 0.14 ± 0.25</td><td>12.5 ± 0.5 ± 1</td></tr><tr><td>J2256-1024</td><td>8.2 ± 0.5</td><td>1.7 ± 0.16</td><td>1.41 ± 0.16</td><td>0.43 ± 0.08</td><td>1.44 ± 0.52</td><td>4.25 ± 0.26<sup>+4.7</sup><sub>-1.8</sub></td><td>11.4 ± 0.7<sup>+10</sup><sub>-5</sub></td></tr><tr><td>J2302+4442</td><td>39.0 ± 0.9</td><td>1.1 ± 0.12</td><td>2.36 ± 0.07</td><td>0.56 ± 0.03</td><td>0.85 ± 0.14</td><td>3.472 ± 0.076<sup>+3.3</sup><sub>-2.2</sub></td><td>89 ± 2 ± 90</td></tr><tr><td>J2310-0555</td><td>5.6 ± 0.4</td><td>...</td><td>1.74 ± 0.15</td><td>0.78 ± 0.14</td><td>...</td><td>1.62 ± 0.12<sup>+1.6</sup><sub>-1.0</sub></td><td>15 ± 1 ± 10</td></tr><tr><td>J2317+1439</td><td>0.6 ± 0.2</td><td>...</td><td>1.58 ± 0.21</td><td>5.78 ± 3.76</td><td>...</td><td>0.213 ± 0.057<sup>+0.070</sup><sub>-0.047</sub></td><td>9 ± 2 ± 3</td></tr><tr><td>J2339-0533</td><td>29.0 ± 0.8</td><td>1.3 ± 0.12</td><td>2.40 ± 0.10</td><td>0.37 ± 0.03</td><td>0.86 ± 0.19</td><td>5.90 ± 0.15<sup>+3.6</sup><sub>-1.4</sub></td><td>25.4 ± 0.7<sup>+20</sup><sub>-6</sub></td></tr></tbody></table>

**Notes.** Unbinned maximum likelihood spectral fit results for the LAT millisecond (MSP) gamma-ray pulsars, using the PLEC4 model (Equation (15) in Section 6). Column 2 lists the phase-averaged integral energy flux in the 0.1–300 GeV energy band, $G_{100}$. Column 3 lists $\Gamma_{100}$, the photon index at 100 MeV (Equation (23)), reliable only for fits where the *b* parameter is left free, and not shown when fixed to *b* = 2/3. Columns 4 and 5 list the energy $E_p$ of the maximum of the SED (Equation (21)), and the SED curvature $d_p$ at the maximum (Equation (22)). Column 6 lists the fitted value of *b* for pulsars with TS > 1000 (fixed to *b* = 2/3 otherwise). Tabulated values are from the *b*-free fit when available, and the *b* = 2/3 fit otherwise. Columns 7 and 8 give the total gamma-ray luminosity $L_\gamma$ in the 0.1–100 GeV energy band, and the gamma-ray conversion efficiency $\eta \equiv L_\gamma/\dot{E}$, assuming $f_{\Omega} = 1$ as described in Section 6.7. $\dot{E}$ is not Doppler corrected. The first uncertainties in $L_\gamma$ and $\eta$ come from the statistical uncertainties in the spectral fit, whereas the second are due to the distance uncertainty. The strong dependence of these quantities on distance (see Table 7) and beaming factor means that these values should be considered with care.

<sup>a</sup> A dagger (†) means the point source has TS < 25; hence, we show no spectral fit results.

<sup>b</sup> Overestimated distances or the assumed beaming factor, $f_{\Omega} = 1$, can result in an efficiency > 100%.

which is closely related to a monoenergetic curvature emission or synchrotron radiation spectrum, but with the cutoff strength modulated by the parameter *b*. Blends of particle energies and averaging over pulse phases both tend to broaden the peak, yielding “subexponential” shapes with $b < 1$ (Abdo et al. 2010g). One innovation since 2PC, adopted beginning with the 8 yr 4FGL catalog and justified therein, is to fix $b = 2/3$ instead of $b = 1$ for pulsars with lower TS, as this choice better matches the observed spectra.

A good spectral model should have the flexibility to capture the features evoked above, but in most cases, measuring them all is not possible. The LAT sensitivity peaks at ~1 GeV and the SED slope at 100 MeV, $2 - \Gamma_{100}$, can only be measured for the brightest pulsars. Furthermore, it is only in the cases of sharply peaked spectra that quickly transition to a low-energy power law that $\Gamma_{100}$ may be a good proxy for the asymptotic low-energy slope $\Gamma_0$. Above 1 GeV, pulsar spectra cut off quickly, making it difficult to measure the cutoff shape and strength. We show below that for most pulsars, the only reliable features that can be measured are the position and width of the SED peak. Fortunately, it is these features that strongly correlate with underlying pulsar properties like $\dot{E}$, so they are likely to be the most important in capturing information about the pulsar emission mechanism.

The PLEC formula suffers from large covariance between the model parameters (Abdollahi et al. 2022), magnifying the effect of statistical uncertainty. To reduce these correlations, the spectral analysis of 4FGL-DR3 adopted the PLEC4 model (PLSuperExpCutoff4 in the Fermitools,<fnref n="121" /> specifically

designed to minimize the covariance between the model parameters at a reference energy, $E_0$:

$$ \frac{dN}{dE} = N_0 \left(\frac{E}{E_0}\right)^{-\Gamma+\frac{d}{b}} \exp \left[ \frac{d}{b^2} \left(1 - \left(\frac{E}{E_0}\right)^b\right) \right] \quad (15) $$

$$ \begin{aligned} & \equiv N_0 x^{-\Gamma+\frac{d}{b}} \exp [d/b^2(1-x^b)], \\ & x = \frac{E}{E_0}. \end{aligned} \quad (16) $$

For PLEC4, $N_0$ is the flux density at $E_0$. The PLEC4 spectral shape is exactly the same as for the PLEC model, as can be seen by identifying

$$ \begin{aligned} \text{PLEC} &\Leftrightarrow \text{PLEC4} \\ E_c &= E_0 \left(\frac{b^2}{d}\right)^{\frac{1}{b}} \end{aligned} \quad (17) $$

$$ \Gamma_0 = \Gamma - \frac{d}{b} \quad (18) $$

$$ N_0 = N_0 \exp\left(\frac{d}{b^2}\right). \quad (19) $$

In the fits presented below, the reference energies $E_0$ are confined to a relatively narrow energy range with a typical value of 1.6 GeV.

The PLEC4 model parameters $\Gamma$ and $d$ are related to the logarithmic derivatives of the model. The gamma-ray

<footnote n="121">¹²¹ https://fermi.gsfc.nasa.gov/ssc/data/analysis/software/. The catalog FITS files tabulate negative $-\Gamma < 0$ values.</footnote>
<table><caption>**Table 14**
(Continued)</caption><thead><tr><th>PSR<sup>a</sup></th><th>Energy Flux<br>(erg cm<sup>-2</sup> s<sup>-1</sup>)<br>(×10<sup>-12</sup>)</th><th>Γ<sub>100</sub></th><th>E<sub>p</sub><br>(GeV)</th><th>d<sub>p</sub></th><th>b<sub>free</sub></th><th>Luminosity<br>(10<sup>33</sup> erg s<sup>-1</sup>)</th><th>Efficiency<sup>b</sup><br>(%)</th></tr></thead><tbody><tr><td>J2115+5448</td><td>7.0 ± 0.7</td><td>...</td><td>3.06 ± 0.30</td><td>1.31 ± 0.27</td><td>...</td><td>8.10 ± 0.80<sup>+7.8</sup><sub>-5.2</sub></td><td>4.8 ± 0.5 ± 5</td></tr><tr><td>J2124–3358</td><td>39.0 ± 0.8</td><td>0.97 ± 0.12</td><td>1.95 ± 0.05</td><td>0.77 ± 0.04</td><td>1.09 ± 0.16</td><td>0.780 ± 0.016<sup>+0.43</sup><sub>-0.18</sub></td><td>11.5 ± 0.2<sup>+6</sup><sub>-3</sub></td></tr><tr><td>J2129–0429</td><td>6.8 ± 0.5</td><td>...</td><td>1.66 ± 0.17</td><td>0.58 ± 0.11</td><td>...</td><td>2.73 ± 0.19 ± 0.32</td><td>9.3 ± 0.6 ± 1</td></tr><tr><td>J2205+6012</td><td>3.7 ± 1.0</td><td>...</td><td>2.40 ± 0.52</td><td>2.90 ± 1.67</td><td>...</td><td>5.6 ± 1.6<sup>+5.4</sup><sub>-3.6</sub></td><td>10 ± 3 ± 10</td></tr><tr><td>J2214+3000</td><td>33.0 ± 0.7</td><td>1.0 ± 0.15</td><td>1.76 ± 0.05</td><td>0.68 ± 0.04</td><td>0.92 ± 0.17</td><td>1.406 ± 0.032<sup>+1.8</sup><sub>-1.1</sub></td><td>7.3 ± 0.2<sup>+10</sup><sub>-6</sub></td></tr><tr><td>J2215+5135</td><td>18.0 ± 0.8</td><td>1.4 ± 0.30</td><td>1.45 ± 0.17</td><td>0.40 ± 0.05</td><td>0.52 ± 0.28</td><td>15.26 ± 0.69<sup>+21</sup><sub>-4.9</sub></td><td>24 ± 1<sup>+30</sup><sub>-8</sub></td></tr><tr><td>J2234+0944</td><td>10.0 ± 0.6</td><td>0.74 ± 0.65</td><td>1.87 ± 0.18</td><td>0.59 ± 0.10</td><td>0.36 ± 0.34</td><td>2.45 ± 0.15<sup>+5.0</sup><sub>-1.2</sub></td><td>14.7 ± 0.9<sup>+30</sup><sub>-8</sub></td></tr><tr><td>J2241–5236</td><td>25.0 ± 1.1</td><td>1.3 ± 0.11</td><td>1.90 ± 0.08</td><td>0.95 ± 0.07</td><td>1.35 ± 0.22</td><td>3.25 ± 0.14 ± 0.25</td><td>12.5 ± 0.5 ± 1</td></tr><tr><td>J2256–1024</td><td>8.2 ± 0.5</td><td>1.7 ± 0.16</td><td>1.41 ± 0.16</td><td>0.43 ± 0.08</td><td>1.44 ± 0.52</td><td>4.25 ± 0.26<sup>+4.7</sup><sub>-1.8</sub></td><td>11.4 ± 0.7<sup>+10</sup><sub>-5</sub></td></tr><tr><td>J2302+4442</td><td>39.0 ± 0.9</td><td>1.1 ± 0.12</td><td>2.36 ± 0.07</td><td>0.56 ± 0.03</td><td>0.85 ± 0.14</td><td>3.472 ± 0.076<sup>+3.3</sup><sub>-2.2</sub></td><td>89 ± 2 ± 90</td></tr><tr><td>J2310–0555</td><td>5.6 ± 0.4</td><td>...</td><td>1.74 ± 0.15</td><td>0.78 ± 0.14</td><td>...</td><td>1.62 ± 0.12<sup>+1.6</sup><sub>-1.0</sub></td><td>15 ± 1 ± 10</td></tr><tr><td>J2317+1439</td><td>0.6 ± 0.2</td><td>...</td><td>1.58 ± 0.21</td><td>5.78 ± 3.76</td><td>...</td><td>0.213 ± 0.057<sup>+0.070</sup><sub>-0.047</sub></td><td>9 ± 2 ± 3</td></tr><tr><td>J2339–0533</td><td>29.0 ± 0.8</td><td>1.3 ± 0.12</td><td>2.40 ± 0.10</td><td>0.37 ± 0.03</td><td>0.86 ± 0.19</td><td>5.90 ± 0.15<sup>+3.6</sup><sub>-1.4</sub></td><td>25.4 ± 0.7<sup>+20</sup><sub>-6</sub></td></tr></tbody></table>

**Notes.** Unbinned maximum likelihood spectral fit results for the LAT millisecond (MSP) gamma-ray pulsars, using the PLEC4 model (Equation (15) in Section 6). Column 2 lists the phase-averaged integral energy flux in the 0.1–300 GeV energy band, $G_{100}$. Column 3 lists $\Gamma_{100}$, the photon index at 100 MeV (Equation (23)), reliable only for fits where the *b* parameter is left free, and not shown when fixed to *b* = 2/3. Columns 4 and 5 list the energy $E_p$ of the maximum of the SED (Equation (21)), and the SED curvature $d_p$ at the maximum (Equation (22)). Column 6 lists the fitted value of *b* for pulsars with TS > 1000 (fixed to *b* = 2/3 otherwise). Tabulated values are from the *b*-free fit when available, and the *b* = 2/3 fit otherwise. Columns 7 and 8 give the total gamma-ray luminosity $L_\gamma$ in the 0.1–100 GeV energy band, and the gamma-ray conversion efficiency $\eta \equiv L_\gamma/\dot{E}$, assuming $f_{\Omega} = 1$ as described in Section 6.7. $\dot{E}$ is not Doppler corrected. The first uncertainties in $L_\gamma$ and $\eta$ come from the statistical uncertainties in the spectral fit, whereas the second are due to the distance uncertainty. The strong dependence of these quantities on distance (see Table 7) and beaming factor means that these values should be considered with care.

a A dagger (†) means the point source has TS < 25; hence, we show no spectral fit results.

b Overestimated distances or the assumed beaming factor, $f_{\Omega} = 1$, can result in an efficiency > 100%.

which is closely related to a monoenergetic curvature emission or synchrotron radiation spectrum, but with the cutoff strength modulated by the parameter *b*. Blends of particle energies and averaging over pulse phases both tend to broaden the peak, yielding “subexponential” shapes with $b < 1$ (Abdo et al. 2010g). One innovation since 2PC, adopted beginning with the 8 yr 4FGL catalog and justified therein, is to fix $b = 2/3$ instead of $b = 1$ for pulsars with lower TS, as this choice better matches the observed spectra.

A good spectral model should have the flexibility to capture the features evoked above, but in most cases, measuring them all is not possible. The LAT sensitivity peaks at ~1 GeV and the SED slope at 100 MeV, $2 - \Gamma_{100}$, can only be measured for the brightest pulsars. Furthermore, it is only in the cases of sharply peaked spectra that quickly transition to a low-energy power law that $\Gamma_{100}$ may be a good proxy for the asymptotic low-energy slope $\Gamma_0$. Above 1 GeV, pulsar spectra cut off quickly, making it difficult to measure the cutoff shape and strength. We show below that for most pulsars, the only reliable features that can be measured are the position and width of the SED peak. Fortunately, it is these features that strongly correlate with underlying pulsar properties like $\dot{E}$, so they are likely to be the most important in capturing information about the pulsar emission mechanism.

The PLEC formula suffers from large covariance between the model parameters (Abdollahi et al. 2022), magnifying the effect of statistical uncertainty. To reduce these correlations, the spectral analysis of 4FGL-DR3 adopted the PLEC4 model (PLSuperExpCutoff4 in the Fermitools,<fnref n="121" /> specifically

designed to minimize the covariance between the model parameters at a reference energy, $E_0$:

$$ \frac{dN}{dE} = N_0 \left(\frac{E}{E_0}\right)^{-\Gamma + \frac{d}{b}} \exp \left[ \frac{d}{b^2} \left(1 - \left(\frac{E}{E_0}\right)^b\right) \right] \quad (15) $$

$$ \begin{aligned} & \equiv N_0 x^{-\Gamma + \frac{d}{b}} \exp[d/b^2(1 - x^b)], \\ & x = \frac{E}{E_0}. \end{aligned} \quad (16) $$

For PLEC4, $N_0$ is the flux density at $E_0$. The PLEC4 spectral shape is exactly the same as for the PLEC model, as can be seen by identifying

$$ \begin{aligned} \text{PLEC} &\Leftrightarrow \text{PLEC4} \\ E_c &= E_0 \left(\frac{b^2}{d}\right)^{\frac{1}{b}} \end{aligned} \quad (17) $$

$$ \Gamma_0 = \Gamma - \frac{d}{b} \quad (18) $$

$$ N_0 = N_0 \exp\left(\frac{d}{b^2}\right). \quad (19) $$

In the fits presented below, the reference energies $E_0$ are confined to a relatively narrow energy range with a typical value of 1.6 GeV.

The PLEC4 model parameters $\Gamma$ and $d$ are related to the logarithmic derivatives of the model. The gamma-ray

<footnote n="121">¹²¹ https://fermi.gsfc.nasa.gov/ssc/data/analysis/software/. The catalog FITS files tabulate negative $-\Gamma < 0$ values.</footnote>
<figure><img src="image_16.png" /><figcaption>**Figure 15.** Realizations of the PLEC4 model for three values of the peak curvature, $d_p = 0.2, 0.6, \text{ and } 1.3$, and the asymmetry parameter $b = 0.3, 0.6, \text{ and } 1.3$. We set $E_p = E_0 = 1$ GeV, which uniquely determines $\Gamma$ and $\Gamma_0$. The SEDs are normalized to 1 at $E_p$ but offset by 50% for clarity. Each curve is labeled with the tuple $(d, b, \Gamma_0)$. The shaded regions show the difference between the SED and the asymptotic power law, normalized to 0.1 GeV.</figcaption></figure>

convention defines the photon spectral index as

$$ \Gamma_E = -\frac{\partial \log N}{\partial \log E} \\ = \Gamma - \frac{d}{b}(1 - x^b) = \Gamma_0 + \frac{d}{b}x^b, \qquad (20) $$

from which it clear that $\Gamma$ is the photon index at the reference energy ($x=1$), and $\Gamma_0$ is the asymptotic spectral index ($x \to 0$). The spectral curvature can be defined as $-\frac{\partial^2 \log N}{\partial \log E^2} = dx^b$, so $d$ is the SED curvature at $E_0$.

Figure 15 shows some example realizations of the PLEC4 model and in particular highlights the role of $d$ in producing narrow (large $d$) or broad (small $d$) peaks. $b$ controls the overall shape of the spectrum, with larger values of $b$ producing greater asymmetry. For $b \ll 1$, the spectrum becomes more symmetric, with both a slower cutoff and a slower convergence to a power law at low energies. $b > 1$ produces asymmetric spectra with sharp, “superexponential” cutoffs and power-law behavior at low energy. When $b < 0$, the cutoff and power-law regimes are reflected. For many of these examples, the transition to power-law behavior has not yet happened at 100 MeV. In this example, the local spectral index at 100 MeV agrees with the asymptotic index to within 0.1 for $b = 0.6$, but for $b = 0.3$ it differs by 0.3 ($d = 0.2$), 0.9 ($d = 0.6$), and 1.9 ($d = 1.3$).

The energy at which the SED peaks (if the peak exists, is nonzero, and is finite) is

$$ x_p = \frac{E_p}{E_0} = \left[ 1 + \frac{b}{d}(2 - \Gamma) \right]^{\frac{1}{b}} \\ = \left[ \frac{b}{d}(2 - \Gamma_0) \right]^{\frac{1}{b}}. \qquad (21) $$

Note that $b < 0$ can yield a valid $E_p$ if $\Gamma_0 \ge 2$. $E_p$ is a key physical property of interest, and we can obtain a second key

property, the width of the SED at its peak, by evaluating the curvature at $x_p$:

$$ dx_p^b \equiv d_p = d + b(2 - \Gamma) = b(2 - \Gamma_0). \qquad (22) $$

For reference, the maximum expected spectral curvature, obtained for monoenergetic synchrotron radiation, is $d_p = 4/3$.

We expect these quantities associated with the SED peak to be well determined and relatively independent because $E_p$ is generally close to $E_0$. However, Equation (22) indicates that we **cannot** measure the peak curvature and the asymptotic index independently when $b$ is fixed, and since we expect $d_p$ to be the better-determined quantity, we cannot use the PLEC4 model to extrapolate the asymptotic spectral index, $\Gamma_0$. $\Gamma_{100}$ could be used as a proxy for $\Gamma_0$ (when $E_p \gg 100$ MeV and the peak is not too broad), but it too suffers from bias: for a typical $E_0 = 1.6$ GeV and with $b = 2/3$,

$$ \Gamma_{100} = \Gamma_0 + 0.23d = \Gamma - 1.26d. \qquad (23) $$

This formulation all but maximizes the correlation between the model parameters $\Gamma$ and $d$, undoing the benefits of PLEC4, and like $\Gamma_0$, $\Gamma_{100}$ will potentially be biased if $b = 2/3$ does not describe the true SED.

Thus, to use the PLEC4 model to estimate reliable physical properties, we need to answer two primary questions:

1. For bright pulsars where $b$ can be measured, does the PLEC4 model allow for robust estimation of other physical parameters of interest, like $\Gamma_{100}$? That is, can PLEC4 reproduce the true shapes of pulsar SEDs?

2. For fainter pulsars, for which $b$ must be fixed to a canonical value, does the reduced parameter space introduce a bias on physical parameters?
<figure><img src="image_17.png" /><figcaption>**Figure 15.** Realizations of the PLEC4 model for three values of the peak curvature, $d_p = 0.2, 0.6, \text{ and } 1.3$, and the asymmetry parameter $b = 0.3, 0.6, \text{ and } 1.3$. We set $E_p = E_0 = 1$ GeV, which uniquely determines $\Gamma$ and $\Gamma_0$. The SEDs are normalized to 1 at $E_p$ but offset by 50% for clarity. Each curve is labeled with the tuple $(d, b, \Gamma_0)$. The shaded regions show the difference between the SED and the asymptotic power law, normalized to 0.1 GeV.</figcaption></figure>

convention defines the photon spectral index as

$$ \Gamma_E = - \frac{\partial \log N}{\partial \log E} \\ = \Gamma - \frac{d}{b}(1 - x^b) = \Gamma_0 + \frac{d}{b}x^b, \qquad (20) $$

from which it clear that $\Gamma$ is the photon index at the reference energy ($x=1$), and $\Gamma_0$ is the asymptotic spectral index ($x \to 0$). The spectral curvature can be defined as $-\frac{\partial^2 \log N}{\partial \log E^2} = dx^b$, so $d$ is the SED curvature at $E_0$.

Figure 15 shows some example realizations of the PLEC4 model and in particular highlights the role of $d$ in producing narrow (large $d$) or broad (small $d$) peaks. $b$ controls the overall shape of the spectrum, with larger values of $b$ producing greater asymmetry. For $b \ll 1$, the spectrum becomes more symmetric, with both a slower cutoff and a slower convergence to a power law at low energies. $b > 1$ produces asymmetric spectra with sharp, “superexponential” cutoffs and power-law behavior at low energy. When $b < 0$, the cutoff and power-law regimes are reflected. For many of these examples, the transition to power-law behavior has not yet happened at 100 MeV. In this example, the local spectral index at 100 MeV agrees with the asymptotic index to within 0.1 for $b = 0.6$, but for $b = 0.3$ it differs by 0.3 ($d = 0.2$), 0.9 ($d = 0.6$), and 1.9 ($d = 1.3$).

The energy at which the SED peaks (if the peak exists, is nonzero, and is finite) is

$$ x_p = \frac{E_p}{E_0} = \left[ 1 + \frac{b}{d}(2 - \Gamma) \right]^{\frac{1}{b}} \\ = \left[ \frac{b}{d}(2 - \Gamma_0) \right]^{\frac{1}{b}}. \qquad (21) $$

Note that $b < 0$ can yield a valid $E_p$ if $\Gamma_0 \ge 2$. $E_p$ is a key physical property of interest, and we can obtain a second key

property, the width of the SED at its peak, by evaluating the curvature at $x_p$:

$$ dx_p^b \equiv d_p = d + b(2 - \Gamma) = b(2 - \Gamma_0). \qquad (22) $$

For reference, the maximum expected spectral curvature, obtained for monoenergetic synchrotron radiation, is $d_p = 4/3$.

We expect these quantities associated with the SED peak to be well determined and relatively independent because $E_p$ is generally close to $E_0$. However, Equation (22) indicates that we **cannot** measure the peak curvature and the asymptotic index independently when $b$ is fixed, and since we expect $d_p$ to be the better-determined quantity, we cannot use the PLEC4 model to extrapolate the asymptotic spectral index, $\Gamma_0$. $\Gamma_{100}$ could be used as a proxy for $\Gamma_0$ (when $E_p \gg 100$ MeV and the peak is not too broad), but it too suffers from bias: for a typical $E_0 = 1.6$ GeV and with $b = 2/3$,

$$ \Gamma_{100} = \Gamma_0 + 0.23d = \Gamma - 1.26d. \qquad (23) $$

This formulation all but maximizes the correlation between the model parameters $\Gamma$ and $d$, undoing the benefits of PLEC4, and like $\Gamma_0$, $\Gamma_{100}$ will potentially be biased if $b = 2/3$ does not describe the true SED.

Thus, to use the PLEC4 model to estimate reliable physical properties, we need to answer two primary questions:

1. For bright pulsars where $b$ can be measured, does the PLEC4 model allow for robust estimation of other physical parameters of interest, like $\Gamma_{100}$? That is, can PLEC4 reproduce the true shapes of pulsar SEDs?

2. For fainter pulsars, for which $b$ must be fixed to a canonical value, does the reduced parameter space introduce a bias on physical parameters?
<figure><img src="image_17.png" /><figcaption>**Figure 16.** The best-fit values of *b*, with $1\sigma$ uncertainties, shown as a function of $\dot{E}$. Maximum likelihood linear models are shown for each class and for the combined population. The centroid indicates the model fit, while the distance between the dashed lines indicates the intrinsic scatter ($\pm\sigma_i$, see the main text). The dark shaded region indicates the same quantities for the full population with measured values of *b*.</figcaption></figure>

## 6.2. Spectral Model Fits

We now address these questions using phase-averaged spectral fits. All spectral results in this section rely on the 4FGL-DR3 spectral fitting methodology. Some results are incorporated directly from that catalog, and we have extended these results using the DR3 model as a baseline. In particular, we have

1. performed spectral fits in which *b* was free to vary for 119 pulsars with $\text{TS} > 1000$ (see 28 pulsars with $\text{TS} > 10,000$ in 4FGL-DR3).

2. fit the 28 *b*-free pulsars of 4FGL with a $b=2/3$ model for comparison.

3. re-fit PSRs J0922+0638, J1757-2421, J1913+1011, and J2205+6012 with a PLEC4 model, as opposed to the power law used in 4FGL-DR3.

4. added a power-law component to the spectral models for pulsars with TeV counterparts (PSRs J0007+7303, J0205+6449, J1016-5857, J1119-6127, J1714-3830, J1833-1034, J1907+0602, and J2032+4127) to encapsulate potential emission from a PWN and re-optimized the PLEC4 model. Crab, Vela, and a half-dozen other pulsars were already colocated with TeV power-law sources in 4FGL.

5. directly measured the low-energy spectral indices with a power-law fit of the 50-300 MeV band, for the 42 pulsars with $\text{TS} > 25$ in that band and $\text{TS} > 1000$ overall.

Of the 255 pulsars, 251 produce a spectral fit with a well-defined $E_p$.

Here we quickly note a possible point of confusion. The reference energy $E_0$ in Equation (15) is used in fitting models and evaluating covariance between parameters. From these results, a “pivot energy” can be determined at which the uncertainty on the flux density normalization $N_0$ is minimized. The results presented here require the covariance matrices from the likelihood fits and so use the reference energy, and thus

parameter values will generally differ from those in the FITS files accompanying 4FGL-DR3, which have been scaled to the pivot energy. A script provided with the supplementary material shows how to correctly unpack the parameters and covariance from the electronic catalog.

### 6.2.1. Distribution of *b*

First, we consider the distribution of best-fit *b* values, plotted against $\dot{E}$ in Figure 16. It is clear that *b* does not take on a constant value, so we used maximum likelihood to fit a linear model $b = b_0 + m_b \log_{10}(\dot{E}/10^{36} \text{ erg s}^{-1})$ to each class and to the full population. We assumed that each measured value follows a Gaussian distribution with the measurement uncertainty, and we also included an intrinsic scatter, $\sigma_i$, which is added in quadrature to the measurement error.

We excluded the Crab pulsar from the fits. In Figures 17–21 its well-measured parameters generally deviate from those of most other pulsars. With its high $\dot{E}$, including it changes the fit results substantially, such that they no longer describe most other gamma-ray pulsars well. If the Crab is atypical only in having both high $\dot{E}$ and good statistics, then perhaps our simple linear fit breaks down for high $\dot{E}$. But the Crab is also atypical in being a rare pulsar with much stronger flux density in the MeV range than in the GeV range, as we further discuss in Section 8.1.

The results, depicted in Figure 16, are (1) there is strong evidence (log likelihood improves by >40) of intrinsic scatter, i.e., $\sigma_i > 0$, and (2) there is no evidence (log likelihood improves only by a few) for a different linear relation among the classes. Therefore, we report a single model for all pulsars, finding $b_0 = 0.42$, $m_B = -0.15$, and $\sigma_i = 0.15$.

These values are intended as a guide only: the simple model does not fully capture the observed variations, and the fitting approximates the measurement uncertainty on *b* as a Gaussian rather than using the full likelihood surface. Nonetheless, it is clear that many pulsars fit with a $b=2/3$ model will suffer
<figure><img src="image_18.png" /><figcaption>**Figure 16.** The best-fit values of *b*, with $1\sigma$ uncertainties, shown as a function of $\dot{E}$. Maximum likelihood linear models are shown for each class and for the combined population. The centroid indicates the model fit, while the distance between the dashed lines indicates the intrinsic scatter ($\pm\sigma_i$, see the main text). The dark shaded region indicates the same quantities for the full population with measured values of *b*.</figcaption></figure>

## 6.2. Spectral Model Fits

We now address these questions using phase-averaged spectral fits. All spectral results in this section rely on the 4FGL-DR3 spectral fitting methodology. Some results are incorporated directly from that catalog, and we have extended these results using the DR3 model as a baseline. In particular, we have

1. performed spectral fits in which *b* was free to vary for 119 pulsars with $\text{TS} > 1000$ (see 28 pulsars with $\text{TS} > 10,000$ in 4FGL-DR3).

2. fit the 28 *b*-free pulsars of 4FGL with a $b=2/3$ model for comparison.

3. re-fit PSRs J0922+0638, J1757-2421, J1913+1011, and J2205+6012 with a PLEC4 model, as opposed to the power law used in 4FGL-DR3.

4. added a power-law component to the spectral models for pulsars with TeV counterparts (PSRs J0007+7303, J0205+6449, J1016-5857, J1119-6127, J1714-3830, J1833-1034, J1907+0602, and J2032+4127) to encapsulate potential emission from a PWN and re-optimized the PLEC4 model. Crab, Vela, and a half-dozen other pulsars were already colocated with TeV power-law sources in 4FGL.

5. directly measured the low-energy spectral indices with a power-law fit of the 50-300 MeV band, for the 42 pulsars with $\text{TS} > 25$ in that band and $\text{TS} > 1000$ overall.

Of the 255 pulsars, 251 produce a spectral fit with a well-defined $E_p$.

Here we quickly note a possible point of confusion. The reference energy $E_0$ in Equation (15) is used in fitting models and evaluating covariance between parameters. From these results, a “pivot energy” can be determined at which the uncertainty on the flux density normalization $N_0$ is minimized. The results presented here require the covariance matrices from the likelihood fits and so use the reference energy, and thus

parameter values will generally differ from those in the FITS files accompanying 4FGL-DR3, which have been scaled to the pivot energy. A script provided with the supplementary material shows how to correctly unpack the parameters and covariance from the electronic catalog.

### 6.2.1. Distribution of *b*

First, we consider the distribution of best-fit *b* values, plotted against $\dot{E}$ in Figure 16. It is clear that *b* does not take on a constant value, so we used maximum likelihood to fit a linear model $b = b_0 + m_b \log_{10}(\dot{E}/10^{36} \text{ erg s}^{-1})$ to each class and to the full population. We assumed that each measured value follows a Gaussian distribution with the measurement uncertainty, and we also included an intrinsic scatter, $\sigma_i$, which is added in quadrature to the measurement error.

We excluded the Crab pulsar from the fits. In Figures 17–21 its well-measured parameters generally deviate from those of most other pulsars. With its high $\dot{E}$, including it changes the fit results substantially, such that they no longer describe most other gamma-ray pulsars well. If the Crab is atypical only in having both high $\dot{E}$ and good statistics, then perhaps our simple linear fit breaks down for high $\dot{E}$. But the Crab is also atypical in being a rare pulsar with much stronger flux density in the MeV range than in the GeV range, as we further discuss in Section 8.1.

The results, depicted in Figure 16, are (1) there is strong evidence (log likelihood improves by >40) of intrinsic scatter, i.e., $\sigma_i > 0$, and (2) there is no evidence (log likelihood improves only by a few) for a different linear relation among the classes. Therefore, we report a single model for all pulsars, finding $b_0 = 0.42$, $m_B = -0.15$, and $\sigma_i = 0.15$.

These values are intended as a guide only: the simple model does not fully capture the observed variations, and the fitting approximates the measurement uncertainty on *b* as a Gaussian rather than using the full likelihood surface. Nonetheless, it is clear that many pulsars fit with a $b=2/3$ model will suffer
<figure><img src="image_18.png" /><figcaption>**Figure 17.** The comparison between spectral indices measured directly with a low-energy fit (y-axis) and the predicted local index at 100 MeV from b-free models.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Figure 18.** Dependence of the spectral peak curvature $d_p$ on $\dot{E}$, showing a well-defined trend. The horizontal dashed line at 4/3 indicates the expected peak curvature for a monoenergetic curvature radiation spectrum. The left-hand panel shows results for all 251 pulsars (i.e., mixes b-free and b = 2/3 fits), while the right-hand panel shows only the 116 pulsars with a b-free fit (and a well-defined peak energy). The solid line indicates a best-fit relation to the merged sample (left-hand panel).</figcaption></figure>

appreciable biases. In particular, estimates of $\Gamma_0$ and $\Gamma_{100}$ for MSPs (with $\dot{E} \sim 10^{34}$ erg s$^{-1}$) will tend to be too soft, and those of RL pulsars (with $\dot{E} > 10^{36}$ erg s$^{-1}$) too hard. We analyze these biases more quantitatively below. With reference to Figure 15, we also see that the distributions of b indicate that MSP spectra have more closely to the power-law with exponential cutoff model, while young pulsars, particularly high-$\dot{E}$ RL pulsars, have substantial curvature over the full LAT band and more gradual high-energy cutoffs. This is

suggestive of MSPs possessing an intrinsically narrower radiating particle distribution than do their RL and RQ counterparts; how this might connect to the smaller magnetospheres of MSPs remains to be understood.

In Appendix C, we study the change in model parameters between the 119 pulsars that have b-free and b = 2/3 fits available. In particular, we find that d and $\Gamma$ agree well between the models, while $\Gamma_{100}$ is irrecoverably biased. The physical properties $d_p$ and $E_p$ are robustly measured in both b-free and
<figure><img src="image_19.png" /><figcaption>**Figure 17.** The comparison between spectral indices measured directly with a low-energy fit (y-axis) and the predicted local index at 100 MeV from b-free models.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Figure 18.** Dependence of the spectral peak curvature $d_p$ on $\dot{E}$, showing a well-defined trend. The horizontal dashed line at 4/3 indicates the expected peak curvature for a monoenergetic curvature radiation spectrum. The left-hand panel shows results for all 251 pulsars (i.e., mixes b-free and b = 2/3 fits), while the right-hand panel shows only the 116 pulsars with a b-free fit (and a well-defined peak energy). The solid line indicates a best-fit relation to the merged sample (left-hand panel).</figcaption></figure>

appreciable biases. In particular, estimates of $\Gamma_0$ and $\Gamma_{100}$ for MSPs (with $\dot{E} \sim 10^{34}$ erg s$^{-1}$) will tend to be too soft, and those of RL pulsars (with $\dot{E} > 10^{36}$ erg s$^{-1}$) too hard. We analyze these biases more quantitatively below. With reference to Figure 15, we also see that the distributions of b indicate that MSP spectra have more closely to the power-law with exponential cutoff model, while young pulsars, particularly high-$\dot{E}$ RL pulsars, have substantial curvature over the full LAT band and more gradual high-energy cutoffs. This is

suggestive of MSPs possessing an intrinsically narrower radiating particle distribution than do their RL and RQ counterparts; how this might connect to the smaller magnetospheres of MSPs remains to be understood.

In Appendix C, we study the change in model parameters between the 119 pulsars that have b-free and b = 2/3 fits available. In particular, we find that d and $\Gamma$ agree well between the models, while $\Gamma_{100}$ is irrecoverably biased. The physical properties $d_p$ and $E_p$ are robustly measured in both b-free and
<figure><img src="image_20.png" /><figcaption>**Figure 19.** Dependence of the spectral peak energy on $\dot{E}$, with low-$\dot{E}$ pulsars having a peak energy roughly twice that of high-$\dot{E}$ pulsars. The samples and best-fit model are the same as in the left panel of Figure 18.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Figure 20.** Dependence of the spectral curvature $d_p$ on the peak energy, $E_p$. Panel selections are the same as in Figure 18.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Figure 21.** Dependence of the low-energy spectral index $\Gamma_{100}$ on $\dot{E}$. The left panel shows fits in which every pulsar has $b = 2/3$, revealing a strong but erroneous trend. The right panel shows only the 119 unbiased $b$-free fits. The horizontal dashed line indicates the monoenergetic limit, $2/3$.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>**Figure 19.** Dependence of the spectral peak energy on $\dot{E}$, with low-$\dot{E}$ pulsars having a peak energy roughly twice that of high-$\dot{E}$ pulsars. The samples and best-fit model are the same as in the left panel of Figure 18.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Figure 20.** Dependence of the spectral curvature $d_p$ on the peak energy, $E_p$. Panel selections are the same as in Figure 18.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Figure 21.** Dependence of the low-energy spectral index $\Gamma_{100}$ on $\dot{E}$. The left panel shows fits in which every pulsar has $b = 2/3$, revealing a strong but erroneous trend. The right panel shows only the 119 unbiased $b$-free fits. The horizontal dashed line indicates the monoenergetic limit, $2/3$.</figcaption></figure>
<figure><img src="image_23.png" /></figure>

**Figure 23.** Gamma-ray luminosity $L_{\gamma} = 4\pi f_{\gamma}d^2G_{100}$ in the 0.1–100 GeV energy band vs. spindown power $\dot{E}$. The vertical error bars from the statistical uncertainty on the energy flux $G_{100}$ are colored. The vertical error bars due to the distance uncertainties are black-dashed, and generally larger. Doppler corrections (Section 4.3) have been applied to MSPs with known proper motions, leading to visible horizontal error bars in many cases, and when the corrections yield negative $\dot{E}$, we place the limit at $\dot{E} > 2 \times 10^{32} \text{ erg s}^{-1}$ for clarity. Luminosity upper limits use distance upper limits, while $\dot{E}$ limits result from insignificant proper-motion measurements or undetermined distances. We use a light color shade to plot these limits in order to de-emphasize them, and we do not include them in the tallies in the legend. The upper diagonal line indicates 100% conversion of spindown power into gamma-ray flux: pulsars above this line may have smaller distance *d*, and/or the assumed beam correction $f_{\Omega} \equiv 1$ is wrong. The lower diagonal line indicates the heuristic luminosity $L_{\gamma}^{h} \propto \sqrt{\dot{E}}$, to guide the eye. For the Crab and PSR J0540−6919 in the Large Magellanic Cloud, at far right, the upper points include the X-ray energy flux.

that the gamma pulse farther from the magnetic pole generally has the harder spectrum (Figure 13).

## 6.4. Summary of Results

We summarize our main results here:

1. The broadband shape, governed by the parameter *b*, evolves with $\dot{E}$, yielding slowly decaying spectra for high-$\dot{E}$ pulsars and sharply cutoff spectra for low-$\dot{E}$ pulsars.

2. Pulsar spectral energy densities ($E^2dN/dE$) are most effectively characterized by the curvature at the spectral energy density peak, $d_p$, which is inversely proportional to the spectral energy density peak width. $d_p$ anticorrelates with $\dot{E}$ with low (17%) scatter; see Figure 18.

3. The energy at which the spectral energy density peaks, $E_p$, also varies inversely with $\dot{E}$ with somewhat larger (30%) scatter; see Figure 19.

4. Consequently, the spectra of MSPs are more sharply peaked than those of young pulsars, and reach the highest peak energies.

5. The asymptotic ($E \to 0$) spectral index, $\Gamma_0$ lies out of the LAT band. The spectral index near 100 MeV, $\Gamma_{100}$, is close to $\Gamma_0$ when $E_p \gg 100$ MeV, and it can be estimated for 116 of the brightest pulsars (Figure 21). These indices appear to take on an almost universal value of about 1.2, though low-$\dot{E}$ pulsars, particularly MSPs, may have harder spectra that approach the “monoenergetic limit” of 2/3.

6. Thus, for our LAT pulsars, nearly all of the rotational energy that is converted to electromagnetic radiation is concentrated in a single decade of the spectrum.

7. The spectral energy density peak energy $E_p$ seems to be highest for pulsars with pulse phase separations $\Delta < 0.4$ (Figure 22). This, coupled with the observation that the second pulse tends to have a harder spectrum than the first (Figure 13), is new empirical evidence that the same underlying magnetosphere configurations affect both profile and spectral shapes.

## 6.5. Notes on Individual Pulsars

Twenty gamma-ray pulsars in Tables 13 and 14 are flagged with † symbols, meaning that no 4FGL-DR3 source with Test Statistic > 25 is colocated with the pulsar. The pulsed signature is detectable in these cases, in spite of a low gamma-ray flux or large local background.

Guillemot et al. (2013) provided spectral parameters for the double pulsar PSR J0737−3039A via a dedicated analysis. PSR J1513−5908 (B1509−58) is brightest in the MeV domain, with a spectral cutoff below 100 MeV, measured by Kuiper et al. (2018). The colocated extended source 4FGL J1514.2−5909e corresponds to the PWN MSH 15−52, and its spectral parameters do not apply to the pulsar. Similarly, PSR J1023+0038 is a transitional pulsar, and the 4FGL-DR3 spectral results are dominated by the data obtained after it transitioned to its accreting state, where the (unpulsed) gamma-ray flux is greatly enhanced.

PSR J1748-2815 was colocated with a TS ~ 300 source in the LAT 8 yr catalog. Prior to 4FGL-DR1, the LAT team
<figure><img src="image_24.png" /></figure>

**Figure 23.** Gamma-ray luminosity $L_{\gamma} = 4\pi f_{\gamma}d^2G_{100}$ in the 0.1–100 GeV energy band vs. spindown power $\dot{E}$. The vertical error bars from the statistical uncertainty on the energy flux $G_{100}$ are colored. The vertical error bars due to the distance uncertainties are black-dashed, and generally larger. Doppler corrections (Section 4.3) have been applied to MSPs with known proper motions, leading to visible horizontal error bars in many cases, and when the corrections yield negative $\dot{E}$, we place the limit at $\dot{E} > 2 \times 10^{32} \text{ erg s}^{-1}$ for clarity. Luminosity upper limits use distance upper limits, while $\dot{E}$ limits result from insignificant proper-motion measurements or undetermined distances. We use a light color shade to plot these limits in order to de-emphasize them, and we do not include them in the tallies in the legend. The upper diagonal line indicates 100% conversion of spindown power into gamma-ray flux: pulsars above this line may have smaller distance *d*, and/or the assumed beam correction $f_{\Omega} \equiv 1$ is wrong. The lower diagonal line indicates the heuristic luminosity $L_{\gamma}^{h} \propto \sqrt{\dot{E}}$, to guide the eye. For the Crab and PSR J0540−6919 in the Large Magellanic Cloud, at far right, the upper points include the X-ray energy flux.

that the gamma pulse farther from the magnetic pole generally
has the harder spectrum (Figure 13).

6.4. Summary of Results

We summarize our main results here:

1. The broadband shape, governed by the parameter *b*, evolves with *E*, yielding slowly decaying spectra for high-*E* pulsars and sharply cutoff spectra for low-*E* pulsars.

2. Pulsar spectral energy densities ($E^2dN/dE$) are most effectively characterized by the curvature at the spectral energy density peak, $d_p$, which is inversely proportional to the spectral energy density peak width. $d_p$ anticorrelates with $\dot{E}$ with low (17%) scatter; see Figure 18.

3. The energy at which the spectral energy density peaks, $E_p$, also varies inversely with $\dot{E}$ with somewhat larger (30%) scatter; see Figure 19.

4. Consequently, the spectra of MSPs are more sharply peaked than those of young pulsars, and reach the highest peak energies.

5. The asymptotic ($E \to 0$) spectral index, $\Gamma_0$ lies out of the LAT band. The spectral index near 100 MeV, $\Gamma_{100}$, is close to $\Gamma_0$ when $E_p \gg 100$ MeV, and it can be estimated for 116 of the brightest pulsars (Figure 21). These indices appear to take on an almost universal value of about 1.2, though low-$\dot{E}$ pulsars, particularly MSPs, may have harder spectra that approach the “monoenergetic limit” of 2/3.

6. Thus, for our LAT pulsars, nearly all of the rotational energy that is converted to electromagnetic radiation is concentrated in a single decade of the spectrum.

7. The spectral energy density peak energy $E_p$ seems to be highest for pulsars with pulse phase separations $\Delta < 0.4$ (Figure 22). This, coupled with the observation that the second pulse tends to have a harder spectrum than the first (Figure 13), is new empirical evidence that the same underlying magnetosphere configurations affect both profile and spectral shapes.

6.5. Notes on Individual Pulsars

Twenty gamma-ray pulsars in Tables 13 and 14 are flagged with † symbols, meaning that no 4FGL-DR3 source with Test Statistic > 25 is colocated with the pulsar. The pulsed signature is detectable in these cases, in spite of a low gamma-ray flux or large local background.

Guillemot et al. (2013) provided spectral parameters for the double pulsar PSR J0737−3039A via a dedicated analysis. PSR J1513−5908 (B1509−58) is brightest in the MeV domain, with a spectral cutoff below 100 MeV, measured by Kuiper et al. (2018). The colocated extended source 4FGL J1514.2−5909e corresponds to the PWN MSH 15−52, and its spectral parameters do not apply to the pulsar. Similarly, PSR J1023+0038 is a transitional pulsar, and the 4FGL-DR3 spectral results are dominated by the data obtained after it transitioned to its accreting state, where the (unpulsed) gamma-ray flux is greatly enhanced.

PSR J1748-2815 was colocated with a TS ~ 300 source in
the LAT 8 yr catalog. Prior to 4FGL-DR1, the LAT team
<table><caption>**Table 15**
Candidate “Spider” and Transitional MSPs (tMSPs) Discovered in Deep Searches of Previously Unidentified LAT Catalog Sources</caption><thead><tr><th>LAT Name</th><th>Type</th><th>E<sub>peak</sub></th><th>Class, Assn Name</th><th>References</th></tr></thead><tbody><tr><td>J0212.1+5321</td><td>RB</td><td>1.20</td><td>bin 1SXPS J021210.6+532136</td><td>Li et al. (2016), Linares et al. (2017)</td></tr><tr><td>J0336.0+7502</td><td>BW</td><td>1.71</td><td>lmb 4FGL J0336.0+7502</td><td>Li et al. (2021)</td></tr><tr><td>J0407.7-5702</td><td>DStMSP</td><td>PL</td><td>(—)</td><td>Miller et al. (2020)</td></tr><tr><td>J0427.8-6704</td><td>DStMSP</td><td>PL</td><td>LMB 1SXPS J042749.2-670434</td><td>Strader et al. (2016)</td></tr><tr><td>J0523.3-2527</td><td>RB</td><td>1.77</td><td>bin CRTS J052316.9-252737</td><td>Strader et al. (2014)</td></tr><tr><td>J0540.0-7552</td><td>DStMSP</td><td>0.59</td><td>(—)</td><td>Strader et al. (2021)</td></tr><tr><td>J0846.0+2820</td><td>RB</td><td></td><td></td><td>Swihart et al. (2017)</td></tr><tr><td>J0935.3+0901</td><td>RB</td><td>0.50</td><td>(—)</td><td>Wang et al. (2020)</td></tr><tr><td>J0940.3-7610</td><td>RB</td><td>1.13</td><td>lmb 2SXPS J094023.5-761001</td><td>Swihart et al. (2021)</td></tr><tr><td>J1120.0-2204</td><td>RB+</td><td>1.0</td><td>(—)</td><td>Swihart et al. (2022a)</td></tr><tr><td>J1408.6-2917</td><td>BW</td><td>0.18</td><td>bin 4FGL J1408.6-2917</td><td>Swihart et al. (2022b)</td></tr><tr><td>J1544.5-1126</td><td>DStMSP</td><td>0.36</td><td>lmb 1RXS J154439.4-112820</td><td>Britt et al. (2017)</td></tr><tr><td>J1702.7-5655</td><td>RB</td><td>0.54</td><td>(—)</td><td>Corbet et al. (2022)</td></tr></tbody></table>

**Note.** “BW” means black widow, “RB” means redback, and “DStMSP” means that a “disk state” is presumed to have obscured pulsed detections to date. “PL” in place of the SED peak energy (in GeV) indicates that the best spectral fit uses a power law instead of log parabola. Two tMSPs seen to pulse in gamma-rays are PSRs J1023+0038 and J1227-4853. All LAT names are 4FGL except for 2FGL J0846.0+2820, detected only in 2009.

adopted a new model for the diffuse Galactic gamma-ray emission that is brighter around the Galactic center, and the source disappeared, although its pulsations remain significant with the model weights used here.

For PSR J1531-5610, 2PC mistakenly reported a luminosity $L_γ$ over 10 times smaller than the value in Table 13. An outlier in 2PC, with corrected $L_γ$, its efficiency $\epsilon = L_γ/\dot{E}$ becomes quite typical.

## 6.6. Searches for Pulsar Flux Variability

In 4FGL-DR3, fluxes for sources with Variability Index < 24.7 are considered stable. Three pulsars exceed this threshold: PSR J2021+4026, cited in Section 3 with Variability Index = 285, and the two transitional MSPs J1023+0038 and J1227-4853 with Variability Index = 58 and 1470, respectively, which exhibit large flux changes when switching between accretion- and rotation-powered states, but appear to be stable emitters like all other gamma-ray MSPs when in the rotation-powered state. In contrast, $\dot{E}$ of PSR J0540-6910 in the Large Magellanic Cloud increased by one-third, with no corresponding change in the gamma-ray flux (Marshall et al. 2015), contrary to what the correlation between $L_γ$ and $\dot{E}$ seen in Figure 23 suggests (Variability Index = 11.1). A search for variability in our pulsar sample yielded no new cases.

The linear increase in pulsed significance (weighted H-test) with time of PSR J2256-1024 changes abruptly near MJD 56600. It has Variability Index = 18.3. A dedicated analysis confirms that the pulsar flux is constant, but that nearby 4FGL J2250.0+3825, associated with a blazar, has a very bright flare peaking at the same time. The softer significance slope is thus due to increased background rather than decreased signal. Exposure changes due to pointing changes also cause H-test versus MJD slope changes.

PSR J2116+1345 is an MSP discovered using the Arecibo radio telescope in a search of a pulsar-like LAT source. Gamma-ray pulsations were found using Arecibo and GMRT timing after we froze the list of pulsars reported in this catalog. Its corresponding source, 4FGL J2117.0+1344, has Variability Index = 16.2. Here too, pulsed significance is not linear with time. Analysis showed that the decreased rate of significance growth between MJD 56900 and 57900 is due to

the flaring activity of 4FGL J2035.4+1056 (PKS 2032+107), which is 10$^2.5$ away from the pulsar.

Clark et al. (2023b) discovered dips in the gamma-ray flux of “spider” MSPs at orbital phases when the companion star lies between Earth and the pulsar. The 4FGL-DR3 fluxes are integrated in both rotational and orbital phase; these eclipses hence do not affect Variability Index. Orbital modulations (see, for example, An et al. 2020, or the references cited in Table 15) similarly do not influence Variability Index.

## 6.7. Luminosity

Most of the low-frequency Poynting flux generated by the dipole radiation of the rotating neutron star converts to a wind of electrons and ions toward and/or beyond the light cylinder, carrying away angular momentum and energy, braking the rotation. About 10% of the known pulsar population—those in this catalog—convert much of their spindown power into intense gamma-ray beams. Pulsar energetics contribute a piece to the broader puzzle of Galactic ecology, as recently illustrated by the PeV energy detection by LHAASO of the Crab nebula (Cao et al. 2021) and by the Fermi observation of GeV-band halos around three bright HAWC sources (Di Mauro et al. 2021), and gamma-ray pulsars allow for the most direct look at the acceleration mechanisms.

Integrating the LAT spectral energy flux $E \frac{dN}{dE}$ above 100 MeV yields $G_{100}$, which is tabulated in 4FGL. The range of $G_{100}$ values for the gamma-ray pulsars is apparent in Figure 6. We can then calculate the luminosity

$$L_{\gamma} = 4\pi d^2 f_{\Omega} G_{100}, \qquad (24)$$

using the distance *d* to the pulsar, and the beaming fraction $f_Ω$.

Figure 23 shows that the luminosity (Equation (24)) is highly correlated with the available braking power. However, at a given $\dot{E}$, there are two decades of dispersion in the observed efficiency $\eta = L_γ/\dot{E}$, shown in Figure 24. Distance errors cause some of the dispersion. The total radiated power likely depends on the inclination $\alpha$ between the neutron star’s magnetic and rotation axes, adding more pulsar-to-pulsar variation. The factor $f_Ω$ in Equation (24) is the ratio of total beam power averaged over the whole sky to the power in the
<table><caption>**Table 15**
Candidate “Spider” and Transitional MSPs (tMSPs) Discovered in Deep Searches of Previously Unidentified LAT Catalog Sources</caption><thead><tr><th>LAT Name</th><th>Type</th><th>E<sub>peak</sub></th><th>Class, Assn Name</th><th>References</th></tr></thead><tbody><tr><td>J0212.1+5321</td><td>RB</td><td>1.20</td><td>bin 1SXPS J021210.6+532136</td><td>Li et al. (2016), Linares et al. (2017)</td></tr><tr><td>J0336.0+7502</td><td>BW</td><td>1.71</td><td>lmb 4FGL J0336.0+7502</td><td>Li et al. (2021)</td></tr><tr><td>J0407.7-5702</td><td>DStMSP</td><td>PL</td><td>(—)</td><td>Miller et al. (2020)</td></tr><tr><td>J0427.8-6704</td><td>DStMSP</td><td>PL</td><td>LMB 1SXPS J042749.2-670434</td><td>Strader et al. (2016)</td></tr><tr><td>J0523.3-2527</td><td>RB</td><td>1.77</td><td>bin CRTS J052316.9-252737</td><td>Strader et al. (2014)</td></tr><tr><td>J0540.0-7552</td><td>DStMSP</td><td>0.59</td><td>(—)</td><td>Strader et al. (2021)</td></tr><tr><td>J0846.0+2820</td><td>RB</td><td></td><td></td><td>Swihart et al. (2017)</td></tr><tr><td>J0935.3+0901</td><td>RB</td><td>0.50</td><td>(—)</td><td>Wang et al. (2020)</td></tr><tr><td>J0940.3-7610</td><td>RB</td><td>1.13</td><td>lmb 2SXPS J094023.5-761001</td><td>Swihart et al. (2021)</td></tr><tr><td>J1120.0-2204</td><td>RB+</td><td>1.0</td><td>(—)</td><td>Swihart et al. (2022a)</td></tr><tr><td>J1408.6-2917</td><td>BW</td><td>0.18</td><td>bin 4FGL J1408.6-2917</td><td>Swihart et al. (2022b)</td></tr><tr><td>J1544.5-1126</td><td>DStMSP</td><td>0.36</td><td>lmb 1RXS J154439.4-112820</td><td>Britt et al. (2017)</td></tr><tr><td>J1702.7-5655</td><td>RB</td><td>0.54</td><td>(—)</td><td>Corbet et al. (2022)</td></tr></tbody></table>

**Note.** “BW” means black widow, “RB” means redback, and “DStMSP” means that a “disk state” is presumed to have obscured pulsed detections to date. “PL” in place of the SED peak energy (in GeV) indicates that the best spectral fit uses a power law instead of log parabola. Two tMSPs seen to pulse in gamma-rays are PSRs J1023+0038 and J1227-4853. All LAT names are 4FGL except for 2FGL J0846.0+2820, detected only in 2009.

adopted a new model for the diffuse Galactic gamma-ray emission that is brighter around the Galactic center, and the source disappeared, although its pulsations remain significant with the model weights used here.

For PSR J1531-5610, 2PC mistakenly reported a luminosity $L_γ$ over 10 times smaller than the value in Table 13. An outlier in 2PC, with corrected $L_γ$, its efficiency $\epsilon = L_γ/\dot{E}$ becomes quite typical.

## 6.6. Searches for Pulsar Flux Variability

In 4FGL-DR3, fluxes for sources with Variability Index < 24.7 are considered stable. Three pulsars exceed this threshold: PSR J2021+4026, cited in Section 3 with Variability Index = 285, and the two transitional MSPs J1023+0038 and J1227-4853 with Variability Index = 58 and 1470, respectively, which exhibit large flux changes when switching between accretion- and rotation-powered states, but appear to be stable emitters like all other gamma-ray MSPs when in the rotation-powered state. In contrast, $\dot{E}$ of PSR J0540-6910 in the Large Magellanic Cloud increased by one-third, with no corresponding change in the gamma-ray flux (Marshall et al. 2015), contrary to what the correlation between $L_γ$ and $\dot{E}$ seen in Figure 23 suggests (Variability Index = 11.1). A search for variability in our pulsar sample yielded no new cases.

The linear increase in pulsed significance (weighted H-test) with time of PSR J2256-1024 changes abruptly near MJD 56600. It has Variability Index = 18.3. A dedicated analysis confirms that the pulsar flux is constant, but that nearby 4FGL J2250.0+3825, associated with a blazar, has a very bright flare peaking at the same time. The softer significance slope is thus due to increased background rather than decreased signal. Exposure changes due to pointing changes also cause H-test versus MJD slope changes.

PSR J2116+1345 is an MSP discovered using the Arecibo radio telescope in a search of a pulsar-like LAT source. Gamma-ray pulsations were found using Arecibo and GMRT timing after we froze the list of pulsars reported in this catalog. Its corresponding source, 4FGL J2117.0+1344, has Variability Index = 16.2. Here too, pulsed significance is not linear with time. Analysis showed that the decreased rate of significance growth between MJD 56900 and 57900 is due to

the flaring activity of 4FGL J2035.4+1056 (PKS 2032+107), which is 10°.5 away from the pulsar.

Clark et al. (2023b) discovered dips in the gamma-ray flux of “spider” MSPs at orbital phases when the companion star lies between Earth and the pulsar. The 4FGL-DR3 fluxes are integrated in both rotational and orbital phase; these eclipses hence do not affect Variability Index. Orbital modulations (see, for example, An et al. 2020, or the references cited in Table 15) similarly do not influence Variability Index.

## 6.7. Luminosity

Most of the low-frequency Poynting flux generated by the dipole radiation of the rotating neutron star converts to a wind of electrons and ions toward and/or beyond the light cylinder, carrying away angular momentum and energy, braking the rotation. About 10% of the known pulsar population—those in this catalog—convert much of their spindown power into intense gamma-ray beams. Pulsar energetics contribute a piece to the broader puzzle of Galactic ecology, as recently illustrated by the PeV energy detection by LHAASO of the Crab nebula (Cao et al. 2021) and by the Fermi observation of GeV-band halos around three bright HAWC sources (Di Mauro et al. 2021), and gamma-ray pulsars allow for the most direct look at the acceleration mechanisms.

Integrating the LAT spectral energy flux $E \frac{dN}{dE}$ above 100 MeV yields $G_{100}$, which is tabulated in 4FGL. The range of $G_{100}$ values for the gamma-ray pulsars is apparent in Figure 6. We can then calculate the luminosity

$$L_{\gamma} = 4\pi d^2 f_{\Omega} G_{100}, \qquad (24)$$

using the distance *d* to the pulsar, and the beaming fraction $f_Ω$.

Figure 23 shows that the luminosity (Equation (24)) is highly correlated with the available braking power. However, at a given $\dot{E}$, there are two decades of dispersion in the observed efficiency $\eta = L_γ/\dot{E}$, shown in Figure 24. Distance errors cause some of the dispersion. The total radiated power likely depends on the inclination $\alpha$ between the neutron star’s magnetic and rotation axes, adding more pulsar-to-pulsar variation. The factor $f_Ω$ in Equation (24) is the ratio of total beam power averaged over the whole sky to the power in the
<figure><img src="image_24.png" /><figcaption>**Figure 24.** Gamma-ray efficiency $\eta = L_{\gamma}/\dot{E}$ vs. spindown power $\dot{E}$. The error bars are the same as in Figure 23. The dashed diagonal shows $\eta \propto \dot{E}^{-1/2}$, corresponding to the heuristic luminosity $L_{\gamma}^{h}$, to guide the eye. The correlation is more striking than in 2PC.</figcaption></figure>

beam slice illuminating the Earth averaged in phase. It is defined in 2PC Equation (16), and by Romani & Watters (2010), and depends on both $\alpha$ and on the inclination $\zeta$ between the rotation axis and the line of sight. Different models predict different pulse profile shapes and thus $f_{\Omega}$, and in any case, observational difficulties cause large uncertainties in the $\alpha$, $\zeta$ model inputs. Lacking $f_{\Omega}(\alpha, \zeta)$, our choice of $f_{\Omega} = 1$ likely contributes to the observed $\eta$ dispersion. A key test of a successful theory of pulsar radiation is its predictions of $L_{\gamma}$ for given pulsars. Kalapotharakos et al. (2023) generally found $f_{\Omega} < 1$ for the LAT pulsar sample, which would make our $L_{\gamma}$ values overestimates.

The upper diagonal line in Figure 23 shows 100% efficiency, $L_{\gamma} = \dot{E}$, for converting spindown power into gamma-rays. Progress since 2PC in determining pulsar distances and MSP proper motions has eliminated pulsars more than $1\sigma$ above the line. Furthermore, $R_{NS} = 10$ km is an outdated convention, and $\dot{E} \propto M_{NS}R_{NS}^2$ is $\approx 2$ times larger than what we use, reducing all $\eta$ values. The neutron star mass values $M_{NS}$ in our sample probably have variance <20% (see, for example, Figure 5 in Shamohammadi et al. 2023), contributing imperceptibly to the luminosity dispersion.

The lower diagonal shows the “heuristic” gamma-ray pulsar luminosity,

$$L_{\gamma}^{h} = \epsilon_{\gamma} \dot{E} \approx \dot{E} \sqrt{\frac{10^{33} \text{ erg s}^{-1}}{\dot{E}}} \\ = 10^{33} \text{ erg s}^{-1} \sqrt{\frac{\dot{E}}{10^{33} \text{ erg s}^{-1}}}, \quad (25)$$

motivated by the idea that above some minimum open field-line voltage $V \approx \sqrt{10^{-5}\dot{E}}$ volts, electron-positron cascades occur in an acceleration region with a self-governing size that produces a gamma-ray efficiency $\epsilon_{\gamma} \propto 1/V$ (Arons 1996). Recent particle-in-cell simulations of pulsar magnetospheres also support the idea of a critical pair production rate below which pulsar magnetospheres become “dead” (e.g., Kalapotharakos et al. 2018; Philippov & Spitkovsky 2018; Chen et al. 2020).

The scaling of $L_{\gamma}$ with $\dot{E}$ depends on whether gamma-rays are radiated primarily via synchrotron or curvature processes (Kalapotharakos et al. 2019). We discuss our results in the context of these simulations further in Section 8.

## 7. The Pulsars Not Seen

Here we address known pulsars not seen with the LAT, in spite of characteristics shared with gamma-ray emitters. PSR J1846-0258 (Section 7.2) is a unique case of a pulsar seen in LAT data only below 100 MeV (Kuiper et al. 2018), not using the methods of this catalog. Finally, we present the LAT’s sensitivity for future pulsar discoveries.

### 7.1. Radio and X-Ray Pulsars

Figure 5 shows that about 5% and 35% of young pulsars and MSPs, respectively, are seen with the LAT in the $\dot{E} \approx 10^{33}$ erg s$^{-1}$ decade, climbing to over 60% and 80% at high $\dot{E}$. Removing pulsars discovered using LAT sources as seeds, the low $\dot{E}$ fractions decrease to about 3% and 25%. 2PC Table 13 listed 28 pulsars with $\dot{E} > 10^{36}$ erg s$^{-1}$ that were not seen at the time, of the 64 then known. Since 2PC, half of these 28 have been detected and are in this catalog, while the total number increased to 71. The majority of those not yet seen are young X-ray pulsars, discussed below.

With a distance estimate, we calculate the pulsar’s “heuristic” flux $G_h = L_{\gamma}^h/4\pi d^2$, which converts to the units of the spectral measurements using $G_{100}/G_h = 3.33 \times 10^{-27} = 10^{-26.48}$. Figure 5 shows the fraction of detected pulsars rising steadily with $G_h$ above the LAT detection threshold near $G_{100} \approx 10^{-12}$ erg cm$^{-2}$ s$^{-1}$ (detailed in Section 7.3). Here, removing pulsars discovered in searches of unidentified LAT sources moves the MSP points closer to those of the young pulsars. Smith et al. (2019) discussed whether the nondetection of lower $\dot{E}$ pulsars is due to a deathline, meaning that the emission mechanisms cease below a minimum braking power, or comes from the small number of pulsars close to Earth and the low detection fraction for low $G_h$.
<figure><img src="image_25.png" /><figcaption>**Figure 24.** Gamma-ray efficiency $\eta = L_{\gamma}/\dot{E}$ vs. spindown power $\dot{E}$. The error bars are the same as in Figure 23. The dashed diagonal shows $\eta \propto \dot{E}^{-1/2}$, corresponding to the heuristic luminosity $L_{\gamma}^{h}$, to guide the eye. The correlation is more striking than in 2PC.</figcaption></figure>

beam slice illuminating the Earth averaged in phase. It is defined in 2PC Equation (16), and by Romani & Watters (2010), and depends on both $\alpha$ and on the inclination $\zeta$ between the rotation axis and the line of sight. Different models predict different pulse profile shapes and thus $f_{\Omega}$, and in any case, observational difficulties cause large uncertainties in the $\alpha$, $\zeta$ model inputs. Lacking $f_{\Omega}(\alpha, \zeta)$, our choice of $f_{\Omega} = 1$ likely contributes to the observed $\eta$ dispersion. A key test of a successful theory of pulsar radiation is its predictions of $L_{\gamma}$ for given pulsars. Kalapotharakos et al. (2023) generally found $f_{\Omega} < 1$ for the LAT pulsar sample, which would make our $L_{\gamma}$ values overestimates.

The upper diagonal line in Figure 23 shows 100% efficiency, $L_{\gamma} = \dot{E}$, for converting spindown power into gamma-rays. Progress since 2PC in determining pulsar distances and MSP proper motions has eliminated pulsars more than $1\sigma$ above the line. Furthermore, $R_{NS} = 10$ km is an outdated convention, and $\dot{E} \propto M_{NS}R_{NS}^2$ is $\approx 2$ times larger than what we use, reducing all $\eta$ values. The neutron star mass values $M_{NS}$ in our sample probably have variance <20% (see, for example, Figure 5 in Shamohammadi et al. 2023), contributing imperceptibly to the luminosity dispersion.

The lower diagonal shows the “heuristic” gamma-ray pulsar luminosity,

$$L_{\gamma}^{h} = \epsilon_{\gamma} \dot{E} \approx \dot{E} \sqrt{\frac{10^{33} \text{ erg s}^{-1}}{\dot{E}}} \\ = 10^{33} \text{ erg s}^{-1} \sqrt{\frac{\dot{E}}{10^{33} \text{ erg s}^{-1}}}, \qquad (25)$$

motivated by the idea that above some minimum open field-line voltage $V \approx \sqrt{10^{-5}\dot{E}}$ volts, electron-positron cascades occur in an acceleration region with a self-governing size that produces a gamma-ray efficiency $\epsilon_{\gamma} \propto 1/V$ (Arons 1996). Recent particle-in-cell simulations of pulsar magnetospheres also support the idea of a critical pair production rate below which pulsar magnetospheres become “dead” (e.g., Kalapotharakos et al. 2018; Philippov & Spitkovsky 2018; Chen et al. 2020).

The scaling of $L_{\gamma}$ with $\dot{E}$ depends on whether gamma-rays are radiated primarily via synchrotron or curvature processes (Kalapotharakos et al. 2019). We discuss our results in the context of these simulations further in Section 8.

## 7. The Pulsars Not Seen

Here we address known pulsars not seen with the LAT, in spite of characteristics shared with gamma-ray emitters. PSR J1846-0258 (Section 7.2) is a unique case of a pulsar seen in LAT data only below 100 MeV (Kuiper et al. 2018), not using the methods of this catalog. Finally, we present the LAT’s sensitivity for future pulsar discoveries.

### 7.1. Radio and X-Ray Pulsars

Figure 5 shows that about 5% and 35% of young pulsars and MSPs, respectively, are seen with the LAT in the $\dot{E} \approx 10^{33}$ erg s$^{-1}$ decade, climbing to over 60% and 80% at high $\dot{E}$. Removing pulsars discovered using LAT sources as seeds, the low $\dot{E}$ fractions decrease to about 3% and 25%. 2PC Table 13 listed 28 pulsars with $\dot{E} > 10^{36}$ erg s$^{-1}$ that were not seen at the time, of the 64 then known. Since 2PC, half of these 28 have been detected and are in this catalog, while the total number increased to 71. The majority of those not yet seen are young X-ray pulsars, discussed below.

With a distance estimate, we calculate the pulsar’s “heuristic” flux $G_h = L_{\gamma}^h/4\pi d^2$, which converts to the units of the spectral measurements using $G_{100}/G_h = 3.33 \times 10^{-27} = 10^{-26.48}$. Figure 5 shows the fraction of detected pulsars rising steadily with $G_h$ above the LAT detection threshold near $G_{100} \approx 10^{-12}$ erg cm$^{-2}$ s$^{-1}$ (detailed in Section 7.3). Here, removing pulsars discovered in searches of unidentified LAT sources moves the MSP points closer to those of the young pulsars. Smith et al. (2019) discussed whether the nondetection of lower $\dot{E}$ pulsars is due to a deathline, meaning that the emission mechanisms cease below a minimum braking power, or comes from the small number of pulsars close to Earth and the low detection fraction for low $G_h$.
For the nondetections at high $\dot{E}$, for the young pulsars, Johnston et al. (2020) found that beam geometry and pulsar inclination dominate (the beam misses the Earth). Rookyard et al. (2017) observed that broad radio pulses are a good proxy for the inclinations unfavorable to gamma detection. An example is the young pulsar PSR J1302-6350 (B1259-63), with large $\sqrt{\dot{E}}/d^2$ and colocated with an LAT source (see Table 5), yet showing no gamma-ray pulsations (Abdo et al. 2011; Smith et al. 2019). GeV flares occur during periastron every 3.4 yr, when the neutron star moves out of the stellar disk of the companion star (Chang et al. 2018). Here, the absence of gamma-ray pulsations is likely due to the beam orientation. Some gamma-ray pulsars are detected at much lower efficiency than expected from the general trend (e.g., MSPs J0659+1414, J0940-5428 and J1740+1000 near $\dot{E} = 4, 23, \text{ and } 190 \times 10^{34} \text{ erg s}^{-1}$, respectively, in Figure 24). If these objects are typical gamma-ray pulsars, then Earth lies well outside of the standard strong beam, and a weak fringe of this beam or an alternate emission component may be responsible for such “subluminous” gamma-ray pulsars (Romani et al. 2011).

Different factors contribute to the larger fraction of MSP gamma-ray detections. For gamma-rays emitted in the inner magnetosphere, the small light cylinders broaden the beams. However, current sheet emission models predict similar beams for MSPs and young pulsars. MSPs have lower $\dot{E}$, and thus lower $L_\gamma$ than young pulsars (see Figure 23), but they are also generally closer to Earth. Figure 19 shows that $E_p$ is slightly higher for MSPs, facilitating detection, as was highlighted by Kalapotharakos et al. (2017) using 2PC data. The details of which MSPs are not seen by the LAT remain to be clarified.

Another category of as-yet unseen pulsars is those for which the radio rotation ephemerides do not suffice for multiyear gamma folding. An example is the 33 radio MSPs found in deep radio searches targeting unidentified LAT sources that have not yet been confirmed as gamma-ray pulsars (Tables 1 and 6). It also occurs for pulsars found in a survey, but never included in a long-term timing campaign: psrcat lists 278 pulsars with $30 < P < 800$ ms but without $\dot{P}$. Nearly half were recently discovered by the GPPS<fnref n="123" /> and CRAFT<fnref n="124" /> surveys, both on the FAST telescope (Han et al. 2021), with improved measurements ongoing. Some of the others have been added to the timing program for Fermi at the Parkes radio telescope, when their decl. and radio brightness allow. Of those found to have large $\dot{E}$, a fraction may be detected in LAT data in the coming years.

X-ray timing with RXTE enabled the gamma-ray discovery of PSR J0540-6919 in the Large Magellanic Cloud (Ackermann et al. 2015). Subsequent Swift timing revealed a rare 30% increase in $\dot{E}$ and a near-zero braking index (Marshall et al. 2016). XMM-Newton timing assisted the LAT pulsation searches for PSR J2022+3842 (Limyansky 2022).

Several of the high-$\dot{E}$ pulsars unseen in LAT data mentioned above are radio-quiet X-ray pulsars. Timing noise for such young pulsars is generally high, and while X-ray observations made over a few years with XMM or Chandra yield astrometry and the spindown rate, they cannot give the phase-connected ephemeris necessary for multiyear gamma-folding. Thus, for many years, only pulsars that have low timing noise and/or are bright enough for routine observations with RXTE or Swift

could be gamma-ray phase-folded with the same level of confidence as we have for our 1000 radio ephemerides.

NICER has provided new capabilities, allowing precision X-ray timing for more pulsars than previously possible. An example is the study of PSR J1813-1749, the third pulsar in 2PC Table 13, by Ho et al. (2020). The pulsar has no 4FGL counterpart and gamma folding yields a null result. Ho et al. (2022) presented NICER timing for five other high-$\dot{E}$ X-ray pulsars, unseen with the LAT.

X-ray and optical studies of unidentified 4FGL sources have led to the pulsar candidates listed in Table 15. Many of these have led to MSP discoveries in radio pulsation searches using the methods described in Section 3.3 or gamma-ray MSP discoveries in restricted blind searches (Section 3.4), once the orbital parameters are better constrained by further multi-wavelength studies. Transitional MSPs, of which PSR J1023+0038 is the prototype (see Stappers et al. 2014, and Appendix A), may “turn on” and show pulsations at some future date. In a similar vein, Hare et al. (2019) identified a point source in the heart of an X-ray PWN that merits pulsation searches.

## 7.2. PSR J1846-0258

PSR J1846-0258 is a young ($\tau \approx 723$ yr), high magnetic field ($5 \times 10^{13}$ G) pulsar located in the Kes 75 SNR (Gotthelf et al. 2000). Although radio-quiet (Archibald et al. 2008), it has been timed regularly in X-rays by RXTE, Swift, INTEGRAL, and NICER. PSR J1846-0258 behaves largely as an RPP, but has twice exhibited magnetar-like outbursts (Gavriil et al. 2008; Blumer et al. 2021). These outbursts have garnered interest in this source as a “magnetar-pulsar transitional object,” possibly able to shed light on the evolutionary relationship between RPPs and magnetars. Additionally, it is one of 18 “soft” gamma-ray pulsars (Kuiper & Hermsen 2015), with strong emission in the MeV range. The X-ray spectrum suggested that the LAT would be able to detect the pulsed high-energy tail, aiding research into the soft pulsar emission mechanism (Torres 2018).

PSR J1846-0258 is the only pulsar with a published gamma-ray detection (pulsed $4.2\sigma$, via unweighted $Z_{m=1}^2$, see Equation (2) in Section 3) that is not detected using the methods of this catalog (Kuiper et al. 2018). It has no 4FGL-DR3 counterpart source, and thus no $L_\gamma$ measurement. Following Kuiper et al.’s (2018) prescription, we confirmed this detection. It increases to $4.3\sigma$ using Pass 8 data. As discussed in Section 3, timing solutions may produce significance versus time curves that peak outside the ephemeris validity range. This is indeed the case with PSR J1846-0258, with a maximum H-Test ($Z_{m=1}^2$) of $4.4\sigma$ ($5.0\sigma$) occurring ~484 days after the end of the published 3295 day timing solution (including a 153 day loss of coherence). We also used NICER to construct an independent ~770 days timing solution outside the time range covered by Kuiper et al. (2018), which we applied to similarly filtered LAT data. This did not yield an LAT detection, which, given the 4× shorter span of the NICER data, is consistent with expectations from the previous detection (Limyansky 2022).

## 7.3. Flux Upper Limits and Sensitivity

Discovery of most of the pulsars in this list started with an initial phase-integrated source detection of the type applied to

<footnote n="123">¹²³ http://zmtt.bao.ac.cn/GPPS/GPPSnewPSR.html</footnote>

<footnote n="124">¹²⁴ http://groups.bao.ac.cn/ism/CRAFTS/</footnote>
For the nondetections at high $\dot{E}$, for the young pulsars, Johnston et al. (2020) found that beam geometry and pulsar inclination dominate (the beam misses the Earth). Rookyard et al. (2017) observed that broad radio pulses are a good proxy for the inclinations unfavorable to gamma detection. An example is the young pulsar PSR J1302-6350 (B1259-63), with large $\sqrt{\dot{E}}/d^2$ and colocated with an LAT source (see Table 5), yet showing no gamma-ray pulsations (Abdo et al. 2011; Smith et al. 2019). GeV flares occur during periastron every 3.4 yr, when the neutron star moves out of the stellar disk of the companion star (Chang et al. 2018). Here, the absence of gamma-ray pulsations is likely due to the beam orientation. Some gamma-ray pulsars are detected at much lower efficiency than expected from the general trend (e.g., MSPs J0659+1414, J0940-5428 and J1740+1000 near $\dot{E} = 4, 23, \text{ and } 190 \times 10^{34} \text{ erg s}^{-1}$, respectively, in Figure 24). If these objects are typical gamma-ray pulsars, then Earth lies well outside of the standard strong beam, and a weak fringe of this beam or an alternate emission component may be responsible for such “subluminous” gamma-ray pulsars (Romani et al. 2011).

Different factors contribute to the larger fraction of MSP gamma-ray detections. For gamma-rays emitted in the inner magnetosphere, the small light cylinders broaden the beams. However, current sheet emission models predict similar beams for MSPs and young pulsars. MSPs have lower $\dot{E}$, and thus lower $L_\gamma$ than young pulsars (see Figure 23), but they are also generally closer to Earth. Figure 19 shows that $E_p$ is slightly higher for MSPs, facilitating detection, as was highlighted by Kalapotharakos et al. (2017) using 2PC data. The details of which MSPs are not seen by the LAT remain to be clarified.

Another category of as-yet unseen pulsars is those for which the radio rotation ephemerides do not suffice for multiyear gamma folding. An example is the 33 radio MSPs found in deep radio searches targeting unidentified LAT sources that have not yet been confirmed as gamma-ray pulsars (Tables 1 and 6). It also occurs for pulsars found in a survey, but never included in a long-term timing campaign: psrcat lists 278 pulsars with $30 < P < 800$ ms but without $\dot{P}$. Nearly half were recently discovered by the GPPS<fnref n="123" /> and CRAFT<fnref n="124" /> surveys, both on the FAST telescope (Han et al. 2021), with improved measurements ongoing. Some of the others have been added to the timing program for Fermi at the Parkes radio telescope, when their decl. and radio brightness allow. Of those found to have large $\dot{E}$, a fraction may be detected in LAT data in the coming years.

X-ray timing with RXTE enabled the gamma-ray discovery of PSR J0540-6919 in the Large Magellanic Cloud (Ackermann et al. 2015). Subsequent Swift timing revealed a rare 30% increase in $\dot{E}$ and a near-zero braking index (Marshall et al. 2016). XMM-Newton timing assisted the LAT pulsation searches for PSR J2022+3842 (Limyansky 2022).

Several of the high-$\dot{E}$ pulsars unseen in LAT data mentioned above are radio-quiet X-ray pulsars. Timing noise for such young pulsars is generally high, and while X-ray observations made over a few years with XMM or Chandra yield astrometry and the spindown rate, they cannot give the phase-connected ephemeris necessary for multiyear gamma-folding. Thus, for many years, only pulsars that have low timing noise and/or are bright enough for routine observations with RXTE or Swift

could be gamma-ray phase-folded with the same level of confidence as we have for our 1000 radio ephemerides.

NICER has provided new capabilities, allowing precision X-ray timing for more pulsars than previously possible. An example is the study of PSR J1813-1749, the third pulsar in 2PC Table 13, by Ho et al. (2020). The pulsar has no 4FGL counterpart and gamma folding yields a null result. Ho et al. (2022) presented NICER timing for five other high-$\dot{E}$ X-ray pulsars, unseen with the LAT.

X-ray and optical studies of unidentified 4FGL sources have led to the pulsar candidates listed in Table 15. Many of these have led to MSP discoveries in radio pulsation searches using the methods described in Section 3.3 or gamma-ray MSP discoveries in restricted blind searches (Section 3.4), once the orbital parameters are better constrained by further multi-wavelength studies. Transitional MSPs, of which PSR J1023+0038 is the prototype (see Stappers et al. 2014, and Appendix A), may “turn on” and show pulsations at some future date. In a similar vein, Hare et al. (2019) identified a point source in the heart of an X-ray PWN that merits pulsation searches.

## 7.2. PSR J1846-0258

PSR J1846-0258 is a young ($\tau \approx 723$ yr), high magnetic field ($5 \times 10^{13}$ G) pulsar located in the Kes 75 SNR (Gotthelf et al. 2000). Although radio-quiet (Archibald et al. 2008), it has been timed regularly in X-rays by RXTE, Swift, INTEGRAL, and NICER. PSR J1846-0258 behaves largely as an RPP, but has twice exhibited magnetar-like outbursts (Gavriil et al. 2008; Blumer et al. 2021). These outbursts have garnered interest in this source as a “magnetar-pulsar transitional object,” possibly able to shed light on the evolutionary relationship between RPPs and magnetars. Additionally, it is one of 18 “soft” gamma-ray pulsars (Kuiper & Hermsen 2015), with strong emission in the MeV range. The X-ray spectrum suggested that the LAT would be able to detect the pulsed high-energy tail, aiding research into the soft pulsar emission mechanism (Torres 2018).

PSR J1846-0258 is the only pulsar with a published gamma-ray detection (pulsed $4.2\sigma$, via unweighted $Z_{m=1}^2$, see Equation (2) in Section 3) that is not detected using the methods of this catalog (Kuiper et al. 2018). It has no 4FGL-DR3 counterpart source, and thus no $L_\gamma$ measurement. Following Kuiper et al.’s (2018) prescription, we confirmed this detection. It increases to $4.3\sigma$ using Pass 8 data. As discussed in Section 3, timing solutions may produce significance versus time curves that peak outside the ephemeris validity range. This is indeed the case with PSR J1846-0258, with a maximum H-Test ($Z_{m=1}^2$) of $4.4\sigma$ ($5.0\sigma$) occurring ~484 days after the end of the published 3295 day timing solution (including a 153 day loss of coherence). We also used NICER to construct an independent ~770 days timing solution outside the time range covered by Kuiper et al. (2018), which we applied to similarly filtered LAT data. This did not yield an LAT detection, which, given the 4× shorter span of the NICER data, is consistent with expectations from the previous detection (Limyansky 2022).

## 7.3. Flux Upper Limits and Sensitivity

Discovery of most of the pulsars in this list started with an initial phase-integrated source detection of the type applied to

<footnote n="123">¹²³ http://zmtt.bao.ac.cn/GPPS/GPPSnewPSR.html</footnote>

<footnote n="124">¹²⁴ http://groups.bao.ac.cn/ism/CRAFTS/</footnote>
<figure><img src="image_25.png" /><figcaption>**Figure 25.** Hammer–Aitoff projection of the estimated LAT 12 yr sky-survey energy flux pulsar detection sensitivity.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**Figure 26.** Integral energy flux from 0.1–100 GeV, $G_{100}$, vs. Galactic latitude $b$ (scaled as $b^{0.65}$ for clarity). The shaded area shows the 5th–95th percentile range of the 12 yr pulsar detection sensitivity in the sky directions at the corresponding latitude. The points correspond to the gamma-ray pulsars in this catalog.</figcaption></figure>

all Fermi-LAT catalog entries. We summarize the 4FGL procedure: for an all-sky model defined by *pointlike* (Kerr 2010), which includes all current sources, a new trial source is introduced at the position of each of HEALPix nside = 256 (~1/4 deg) pixels. Its flux is optimized using each of five fixed trial spectral shapes. Three are power laws, but two are peaked, at 1 and 2 GeV. The one with the highest TS, if greater than 10, is used as a seed to add to the model. The model is re-optimized with now free spectral parameters, including the new source or sources. All sources are also relocalized. All that survive with TS > 10 are used for additional iterations. Finally, the list is used as seeds for the full catalog *gtlike* analysis. After this stage, only sources with TS > 25 are included in the final catalog. Similarly, 2PC started from the 3FGL catalog, which however used only a single power-law trial spectrum.

Construction of an estimated sensitivity threshold map, shown in Figure 25, followed a similar procedure. For each HEALPix pixel location, a trial source was introduced, now with a pulsar-like spectrum, and the flux likelihood function determined. Instead of optimizing it as before, the value corresponding to its 95% cumulative probability was recorded

for that point. A more precise procedure would have instead inserted simulated trial data with varying intensity, applying the full subsequent analysis procedure, to determine the threshold curve.

We assume that our procedure yields the correct position dependence. Converting the sensitivity estimates to a 50% completeness threshold requires an ad hoc scale adjustment, which we obtain by comparing the distribution of sensitivity values to the detected pulsar fluxes, shown in Figure 26 as a function of latitude. This detection threshold analysis considers only phase-integrated source detections.

As discussed in 2PC and by Hou et al. (2014), and further demonstrated by the 15 pulsars in this catalog that were detected via their pulsations but have TS below the 4FGL threshold, narrow gamma-ray pulses can facilitate source discovery. Such pulsars appear in Figure 26 near the shaded area.

Finally, a requirement for inclusion in the catalog is that no new candidate source be within 0.5° of an existing one. Therefore, in the map in Figure 25, we set such pixels around each 4FGL source to *nan*. The all-sky sensitivity threshold map is provided in FITS format.
<figure><img src="image_26.png" /><figcaption>**Figure 25.** Hammer–Aitoff projection of the estimated LAT 12 yr sky-survey energy flux pulsar detection sensitivity.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>**Figure 26.** Integral energy flux from 0.1–100 GeV, $G_{100}$, vs. Galactic latitude $b$ (scaled as $b^{0.65}$ for clarity). The shaded area shows the 5th–95th percentile range of the 12 yr pulsar detection sensitivity in the sky directions at the corresponding latitude. The points correspond to the gamma-ray pulsars in this catalog.</figcaption></figure>

all Fermi-LAT catalog entries. We summarize the 4FGL procedure: for an all-sky model defined by *pointlike* (Kerr 2010), which includes all current sources, a new trial source is introduced at the position of each of HEALPix nside = 256 (~1/4 deg) pixels. Its flux is optimized using each of five fixed trial spectral shapes. Three are power laws, but two are peaked, at 1 and 2 GeV. The one with the highest TS, if greater than 10, is used as a seed to add to the model. The model is re-optimized with now free spectral parameters, including the new source or sources. All sources are also relocalized. All that survive with TS > 10 are used for additional iterations. Finally, the list is used as seeds for the full catalog *gtlike* analysis. After this stage, only sources with TS > 25 are included in the final catalog. Similarly, 2PC started from the 3FGL catalog, which however used only a single power-law trial spectrum.

Construction of an estimated sensitivity threshold map, shown in Figure 25, followed a similar procedure. For each HEALPix pixel location, a trial source was introduced, now with a pulsar-like spectrum, and the flux likelihood function determined. Instead of optimizing it as before, the value corresponding to its 95% cumulative probability was recorded

for that point. A more precise procedure would have instead inserted simulated trial data with varying intensity, applying the full subsequent analysis procedure, to determine the threshold curve.

We assume that our procedure yields the correct position dependence. Converting the sensitivity estimates to a 50% completeness threshold requires an ad hoc scale adjustment, which we obtain by comparing the distribution of sensitivity values to the detected pulsar fluxes, shown in Figure 26 as a function of latitude. This detection threshold analysis considers only phase-integrated source detections.

As discussed in 2PC and by Hou et al. (2014), and further demonstrated by the 15 pulsars in this catalog that were detected via their pulsations but have TS below the 4FGL threshold, narrow gamma-ray pulses can facilitate source discovery. Such pulsars appear in Figure 26 near the shaded area.

Finally, a requirement for inclusion in the catalog is that no new candidate source be within 0.5° of an existing one. Therefore, in the map in Figure 25, we set such pixels around each 4FGL source to *nan*. The all-sky sensitivity threshold map is provided in FITS format.
<figure><img src="image_27.png" /><figcaption>**Figure 27.** One realization of the cutoff energy, $\epsilon_{37,hi}$, takes on a nearly constant value, shown here in the estimates for the *b*-free sample.</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>**Figure 28.** The spectral energy density, $S(E)$, for 116 pulsars with an estimate of *b*. All pulsars have a valid $E_p$, marked with a circle ($b > 0$) or a cross (six pulsars with $b < 0$). The spectra are normalized such that $S(E_p) = \dot{E}$. Curve colors in $\dot{E}$ decades guide the eye. Curves are dashed below $\epsilon_{37,lo}$ and end at $\epsilon_{37,hi}$.</figcaption></figure>

# 8. Discussion

## 8.1. Phase-averaged Spectral Trends

In Section 6, we established that the energy where the SED culminates, $E_p$, and the SED's curvature at that energy, $d_p$, are robust quantities that can be reliably estimated when using the general *b*-free SED model or the restricted $b = 2/3$ case. $E_p$, $d_p$, and *b* all evolve with $\dot{E}$, but the scatter of *b* and $E_p$ is much larger compared to $d_p$. For 90% of the pulsar sample, $0.5 \text{ GeV} < E_p < 2.5 \text{ GeV}$, with young, high-$\dot{E}$ pulsars at the bottom of the range and older, unrecycled pulsars and MSPs at the top.

A consequence of the opposed $\dot{E}$ evolution of $E_p$ and $d_p$ is a nearly universal cutoff energy, $\epsilon_{37,hi}$, at which the SED has fallen $1/e \approx 0.37$ from the peak (Figure 27). Another way of looking at this universality is to consider individual spectral shapes. Figure 28 shows the best-fit SED shapes for the *b*-free pulsars, encoded according to $\dot{E}$. The region between $\epsilon_{37,hi}$ and $\epsilon_{37,lo}$ (the mirror $1/e$ point below $E_p$) is highlighted, allowing for a visual estimate of the effective maximum energy and width. The upper peak bounds cluster in the 3–10 GeV range, independently of $\dot{E}$. Thus, we have the simple descriptive result: almost all $\gamma$-ray pulsars emit their power within a two-decade (0.1–10 GeV) envelope, with the Crab-like, flat-
<figure><img src="image_28.png" /><figcaption>**Figure 27.** One realization of the cutoff energy, $\epsilon_{37,hi}$, takes on a nearly constant value, shown here in the estimates for the *b*-free sample.</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>**Figure 28.** The spectral energy density, $S(E)$, for 116 pulsars with an estimate of *b*. All pulsars have a valid $E_p$, marked with a circle ($b > 0$) or a cross (six pulsars with $b < 0$). The spectra are normalized such that $S(E_p) = \dot{E}$. Curve colors in $\dot{E}$ decades guide the eye. Curves are dashed below $\epsilon_{37,lo}$ and end at $\epsilon_{37,hi}$.</figcaption></figure>

# 8. Discussion

## 8.1. Phase-averaged Spectral Trends

In Section 6, we established that the energy where the SED culminates, $E_p$, and the SED's curvature at that energy, $d_p$, are robust quantities that can be reliably estimated when using the general *b*-free SED model or the restricted $b = 2/3$ case. $E_p$, $d_p$, and *b* all evolve with $\dot{E}$, but the scatter of *b* and $E_p$ is much larger compared to $d_p$. For 90% of the pulsar sample, $0.5 \text{ GeV} < E_p < 2.5 \text{ GeV}$, with young, high-$\dot{E}$ pulsars at the bottom of the range and older, unrecycled pulsars and MSPs at the top.

A consequence of the opposed $\dot{E}$ evolution of $E_p$ and $d_p$ is a nearly universal cutoff energy, $\epsilon_{37,hi}$, at which the SED has fallen $1/e \approx 0.37$ from the peak (Figure 27). Another way of looking at this universality is to consider individual spectral shapes. Figure 28 shows the best-fit SED shapes for the *b*-free pulsars, encoded according to $\dot{E}$. The region between $\epsilon_{37,hi}$ and $\epsilon_{37,lo}$ (the mirror $1/e$ point below $E_p$) is highlighted, allowing for a visual estimate of the effective maximum energy and width. The upper peak bounds cluster in the 3–10 GeV range, independently of $\dot{E}$. Thus, we have the simple descriptive result: almost all $\gamma$-ray pulsars emit their power within a two-decade (0.1–10 GeV) envelope, with the Crab-like, flat-
spectrum, high-$\dot{E}$ pulsars nearly filling the envelope, while sharply peaked MSPs concentrate their power into only the higher decade. In contrast with this absolute energy scale, the spectral curvature as estimated by $d_p$ varies strongly with $\dot{E}$. Selection bias, if any, contributes weakly to this result, as we discuss below.

In Section 6, we also showed that the low-energy spectral index $\Gamma_{100}$ could not be measured reliably for $b = 2/3$ models, and of the population of 116 pulsars for which it can be measured, the values tend to cluster around 1.2 with a scatter of a few tenths and only a very modest evolution with $\dot{E}$. This result is clearly visible in Figure 28, in which it can further be seen that many of these SEDs have not transitioned to a power law at 100 MeV, making these measurements effectively upper limits on the asymptotic spectral index. The low-energy spectral shape is thus another frequent property of gamma-ray pulsars, taking on values that are well in excess of the monoenergetic synchrotron radiation spectral index ($\Gamma_0 = \Gamma_{sr} = 2/3$) for all but few of the lowest-$\dot{E}$ MSPs.

As stated, the curvature evolves strongly with $\dot{E}$, and depends on the spectral steepness beyond $E_p$ and $\epsilon_{37,hi}$. Indeed, Lyutikov et al. (2012) and Richards & Lyutikov (2018) argued that the shape of the high-energy spectral tail is particularly rich in information about the highest energies reached by the particle acceleration mechanisms, and thus, whether and when the radiation-reaction limit is reached. The tail’s shape can reveal, for example, the presence of an inverse Compton “rebound,” and also informs about the best candidates for >10 GeV emission to be observed by Cherenkov telescopes. Barnard et al. (2022) calculated the very high-energy Vela spectrum radiated by primary particles in the pulsar magnetosphere and current sheet. Measuring the high-energy spectral shape with Fermi alone is, however, challenging: the photon rates decrease rapidly, and possible contributions from a surrounding PWN and/or SNR must be handled carefully. We envisage future studies dedicated specifically to these measurements.

The low- and high-energy spectral shapes, and thus the curvature, depend on the underlying acceleration of leptons; on variations in the local conditions that govern the balance between acceleration and radiation, observational convolutions of which depend on pulse phase; and on the extent and reacceleration of pair cascades. Disentangling these effects may not be possible with only GeV data, but the robust evolution of spectral shape demonstrated here is an important input in constraining models of pulsar emission. Future successful theories should also reproduce the approximately constant spectral indices at 100 MeV, the weak decline of $E_p$ with pulsar $\dot{E}$, curvatures which track $\dot{E}$, and the overall shape that evolves from flat, Crab-like spectra to MSP-like spectra with well-defined power-law indices and sharp peaks. These measured trends also provide new ingredients for population syntheses: by adopting a particular spectral shape for a given synthesized $\dot{E}$, more reliable estimates of the LAT sensitivity and selection effects can be obtained.

The physical implication of these results may be somewhat diluted since the spectra are phase-averaged, with multiple distinct acceleration regions (whose observational sampling depends on the pulse phase) contributing to and broadening the measured spectral features. On the other hand, we can point to the relatively tight clustering of $d_p$ about the $\dot{E}$ scaling relation, which suggests that these accelerators must have broadly

similar properties. Indeed, while we established a general trend of harder trailing peaks in Section 5.3, these differences do not seem to be large enough to add appreciable scatter to the observed $d_p$ values. Furthermore, at least some MSPs approach the shape of monoenergetic curvature radiation, further limiting the possible variations in spectral shape with phase. On the other hand, the preference of $b < 1$ for high-$\dot{E}$ pulsars may indicate a richer dependence on phase.

Kalapotharakos et al. (2019) examined the mechanisms and configurations at play in the current sheet just outside the light cylinder, and conclude that curvature radiation dominates over synchrotron radiation yielding a “fundamental plane” (FP) such that $L_\gamma \propto \epsilon_{cut}^{4/3} B_s^{1/6} \dot{E}^{5/12}$. The spectral cutoff energy $\epsilon_{cut}$ tracks the pulsar’s highest photon energies, which reflect the highest attained particle energies. Kalapotharakos et al. (2022) developed a proxy for $\epsilon_{cut}$ called $\epsilon_{c1}$, obtained by imposing a PLEC shape with $b=1$ on the 4FGL spectra and then finding the energy $\epsilon_{c1} > E_p$ where the SED falls to 10% of its maximum. This is similar to our $\epsilon_{37,hi}$ shown in Figure 27, and we include PLEC\_E10\_b23 and PLEC\_E10\_bfr in the 3PC catalog FITS file, with the important difference that we use the $b=2/3$ and $b$-free PLEC4 fits that give the best LAT photon data likelihoods. Kalapotharakos et al. (2022, 2023) thus improved the FP data fits as compared to their earlier work, finding good agreement with their FP exponent predictions. Further improvement could include selecting pulsars with the most reliable distances, and reviewing the choice of the $\epsilon_{cut}$ estimator in light of what we established here regarding measurement bias, and improving the uncertainties used in the FP fits. Applying $f_\Omega$ corrections to $L_\gamma$ for pulsars with reliable estimates of the inclination angles $\alpha$, $\zeta$, including Shklovskii corrections for the MSPs, and perhaps correcting $\dot{E}$ for pulsars with neutron star mass measurements may also improve FP tests.

## 8.2. Selection Bias

Are our results representative of all gamma-ray pulsars? Here we address whether observation bias might skew our $E_p$ distribution, and prospects for discovering pulsars with lower $E_p$. In 4FGL-DR3, over 2000 sources have $E_p < 500$ MeV. They were fit with the LogParabola function rather than PLEC4, but that affects the $E_p$ values only marginally. The LAT detection and analysis chain is clearly sensitive to SEDs with maxima below what we see in our sample.

Half of our gamma-ray pulsars were discovered using radio rotation ephemerides of pulsars discovered independently of the LAT, code “R,”<fnref n="125" /> generally in radio surveys covering large sky areas, with selection biases a priori unrelated to gamma-ray properties. Conceivably, neutron star orientations such that both radio and gamma-ray beams sweep the Earth could correlate with gamma-ray spectral shape. The “X” pulsars in our sample mitigate such a putative radio-selection bias somewhat.

On the other hand, the average $E_p$ of “U” pulsars is higher than for “R” pulsars, with “P” MSPs having higher $E_p$ than the “G” pulsars. MSPs generally have lower $\dot{E}$ than young radio-quiet pulsars, so this reflects the $E_p$ versus $\dot{E}$ trend we report. Our target selection procedures may have tended to favor larger

<footnote n="125">¹²⁵ The discovery codes U (LAT unidentified target), P (Pulsar Search Consortium), G (gamma-ray blind search), and R, X (radio or X-ray ephemeris) are defined in the caption for Table 2.</footnote>
spectrum, high-$\dot{E}$ pulsars nearly filling the envelope, while sharply peaked MSPs concentrate their power into only the higher decade. In contrast with this absolute energy scale, the spectral curvature as estimated by $d_p$ varies strongly with $\dot{E}$. Selection bias, if any, contributes weakly to this result, as we discuss below.

In Section 6, we also showed that the low-energy spectral index $\Gamma_{100}$ could not be measured reliably for $b = 2/3$ models, and of the population of 116 pulsars for which it can be measured, the values tend to cluster around 1.2 with a scatter of a few tenths and only a very modest evolution with $\dot{E}$. This result is clearly visible in Figure 28, in which it can further be seen that many of these SEDs have not transitioned to a power law at 100 MeV, making these measurements effectively upper limits on the asymptotic spectral index. The low-energy spectral shape is thus another frequent property of gamma-ray pulsars, taking on values that are well in excess of the monoenergetic synchrotron radiation spectral index ($\Gamma_0 = \Gamma_{sr} = 2/3$) for all but few of the lowest-$\dot{E}$ MSPs.

As stated, the curvature evolves strongly with $\dot{E}$, and depends on the spectral steepness beyond $E_p$ and $\epsilon_{37,hi}$. Indeed, Lyutikov et al. (2012) and Richards & Lyutikov (2018) argued that the shape of the high-energy spectral tail is particularly rich in information about the highest energies reached by the particle acceleration mechanisms, and thus, whether and when the radiation-reaction limit is reached. The tail’s shape can reveal, for example, the presence of an inverse Compton “rebound,” and also informs about the best candidates for >10 GeV emission to be observed by Cherenkov telescopes. Barnard et al. (2022) calculated the very high-energy Vela spectrum radiated by primary particles in the pulsar magnetosphere and current sheet. Measuring the high-energy spectral shape with Fermi alone is, however, challenging: the photon rates decrease rapidly, and possible contributions from a surrounding PWN and/or SNR must be handled carefully. We envisage future studies dedicated specifically to these measurements.

The low- and high-energy spectral shapes, and thus the curvature, depend on the underlying acceleration of leptons; on variations in the local conditions that govern the balance between acceleration and radiation, observational convolutions of which depend on pulse phase; and on the extent and reacceleration of pair cascades. Disentangling these effects may not be possible with only GeV data, but the robust evolution of spectral shape demonstrated here is an important input in constraining models of pulsar emission. Future successful theories should also reproduce the approximately constant spectral indices at 100 MeV, the weak decline of $E_p$ with pulsar $\dot{E}$, curvatures which track $\dot{E}$, and the overall shape that evolves from flat, Crab-like spectra to MSP-like spectra with well-defined power-law indices and sharp peaks. These measured trends also provide new ingredients for population syntheses: by adopting a particular spectral shape for a given synthesized $\dot{E}$, more reliable estimates of the LAT sensitivity and selection effects can be obtained.

The physical implication of these results may be somewhat diluted since the spectra are phase-averaged, with multiple distinct acceleration regions (whose observational sampling depends on the pulse phase) contributing to and broadening the measured spectral features. On the other hand, we can point to the relatively tight clustering of $d_p$ about the $\dot{E}$ scaling relation, which suggests that these accelerators must have broadly

similar properties. Indeed, while we established a general trend of harder trailing peaks in Section 5.3, these differences do not seem to be large enough to add appreciable scatter to the observed $d_p$ values. Furthermore, at least some MSPs approach the shape of monoenergetic curvature radiation, further limiting the possible variations in spectral shape with phase. On the other hand, the preference of $b < 1$ for high-$\dot{E}$ pulsars may indicate a richer dependence on phase.

Kalapotharakos et al. (2019) examined the mechanisms and configurations at play in the current sheet just outside the light cylinder, and conclude that curvature radiation dominates over synchrotron radiation yielding a “fundamental plane” (FP) such that $L_\gamma \propto \epsilon_{cut}^{4/3} B_s^{1/6} \dot{E}^{5/12}$. The spectral cutoff energy $\epsilon_{cut}$ tracks the pulsar’s highest photon energies, which reflect the highest attained particle energies. Kalapotharakos et al. (2022) developed a proxy for $\epsilon_{cut}$ called $\epsilon_{c1}$, obtained by imposing a PLEC shape with $b=1$ on the 4FGL spectra and then finding the energy $\epsilon_{c1} > E_p$ where the SED falls to 10% of its maximum. This is similar to our $\epsilon_{37,hi}$ shown in Figure 27, and we include PLEC_E10_b23 and PLEC_E10_bfr in the 3PC catalog FITS file, with the important difference that we use the $b=2/3$ and $b$-free PLEC4 fits that give the best LAT photon data likelihoods. Kalapotharakos et al. (2022, 2023) thus improved the FP data fits as compared to their earlier work, finding good agreement with their FP exponent predictions. Further improvement could include selecting pulsars with the most reliable distances, and reviewing the choice of the $\epsilon_{cut}$ estimator in light of what we established here regarding measurement bias, and improving the uncertainties used in the FP fits. Applying $f_\Omega$ corrections to $L_\gamma$ for pulsars with reliable estimates of the inclination angles $\alpha$, $\zeta$, including Shklovskii corrections for the MSPs, and perhaps correcting $\dot{E}$ for pulsars with neutron star mass measurements may also improve FP tests.

## 8.2. Selection Bias

Are our results representative of all gamma-ray pulsars? Here we address whether observation bias might skew our $E_p$ distribution, and prospects for discovering pulsars with lower $E_p$. In 4FGL-DR3, over 2000 sources have $E_p < 500$ MeV. They were fit with the LogParabola function rather than PLEC4, but that affects the $E_p$ values only marginally. The LAT detection and analysis chain is clearly sensitive to SEDs with maxima below what we see in our sample.

Half of our gamma-ray pulsars were discovered using radio rotation ephemerides of pulsars discovered independently of the LAT, code “R,”<fnref n="125" /> generally in radio surveys covering large sky areas, with selection biases a priori unrelated to gamma-ray properties. Conceivably, neutron star orientations such that both radio and gamma-ray beams sweep the Earth could correlate with gamma-ray spectral shape. The “X” pulsars in our sample mitigate such a putative radio-selection bias somewhat.

On the other hand, the average $E_p$ of “U” pulsars is higher than for “R” pulsars, with “P” MSPs having higher $E_p$ than the “G” pulsars. MSPs generally have lower $\dot{E}$ than young radio-quiet pulsars, so this reflects the $E_p$ versus $\dot{E}$ trend we report. Our target selection procedures may have tended to favor larger

<footnote n="125">¹²⁵ The discovery codes U (LAT unidentified target), P (Pulsar Search Consortium), G (gamma-ray blind search), and R, X (radio or X-ray ephemeris) are defined in the caption for Table 2.</footnote>
$E_p$, to be addressed in future work. Because the $E_p-\dot{E}$ anticorrelation is fairly modest with substantial scatter, its slope does not change substantially when excluding LAT-induced discoveries.

We thus confirm that our highly diverse pulsar sample is striking in its spectral homogeneity: almost all gamma-ray pulsars are “GeV pulsars,” with $0.5 < E_p < 2.5$ GeV and following the same $\dot{E}$ trends.

Exceptions exist. Upon its EGRET discovery, Ramana-murthy et al. (1996) highlighted the low gamma-ray efficiency of PSR J0659+1414. Romani et al. (2011) cast it as the archetype of a hypothesized category of pulsars subluminous in gamma-rays. It stands out here as having very low $E_p$ and $d_p$ (Figure 20) and the lowest $\epsilon_{37,hi}$ (the isolated dot at lower left in Figure 27).

Another clear exception is the Crab: although its measured $E_p = 90 \pm 80$ MeV and spectral shape might be construed as a high-$\dot{E}$ continuation of the trend in Figure 28, its true SED peak occurs in the soft gamma-ray band (Kuiper & Hermsen 2015). A further two pulsars, PSRs B1509-58 and J1846-0258, have spectral shapes similar to GeV pulsars but with values of $E_p$ well outside the continuum of GeV pulsars. Both pulsars have $\dot{E} \approx 10^{37}$ erg s$^{-1}$ and little or no LAT signal above 100 MeV. Kuiper & Hermsen (2015) highlighted them as archetypes of a pulsar population with SED peaks in the MeV range, and show 18 high-$\dot{E}$ pulsars with SEDs rising in the keV range. Half of the 18 are in this catalog, as are two of five similar pulsars they discuss. PSR B1509-58 has the best-defined MeV peak. They argue that some of the others may not be detected by the LAT because their SEDs also peak below the LAT's range, as for PSR J1846-0258, at the very limit of our instrument's sensitivity (see Section 7.2). However, the SEDs shown by Kuiper & Hermsen (2015) leave open an alternate explanation: PSRs J0205+6449 and J2229+6114 indeed have keV slopes similar to PSR J1846-0258, yet are bright up to a GeV. The as-yet undetected pulsars may simply be too faint, located as they are in very high background regions. Phase-connected rotation ephemerides for a few of the radio-quiet, noisy pulsars are unavailable. Others may be GeV pulsars, but with gamma-ray beams that simply miss the Earth. Harding & Kalapotharakos (2017) further discussed the MeV pulsars.

To see whether the paucity of $E_p < 500$ MeV pulsars reflects magnetospheric emission processes or stems from observational limits, we start with a tally of LAT candidate sources. There are 238 unidentified nonvariable LAT sources with $E_p < 500$ MeV, undetected beyond 10 GeV, and significantly curved (well fit with a LogParabola function). Of these, 122 have localization ellipse semimajor axes with 95% C.L. <10', suitable for radio searches (see Table 4). Restricting the search to the 87 sources with $|b| < 10^\circ$ would favor young, energetic pulsars. Blind searches for radio-quiet pulsars require a large gamma-ray count rate, even more so if timing noise, glitch rates, or background are high. Unfortunately, only four (25) of the 238 targets have TS > 200 (100). The four sources are 4FGL J0340.4+5302, J0426.5+5434, J1749.8-0303, and J1805.7+3401. All were searched several times for radio pulsations but none have been reported to date.

Many high-$\dot{E}$ pulsars have already been found in radio and X-ray searches, effective at finding such sources because high-$\dot{E}$ pulsars tend to have broad radio beams, and because searches can target the supernova remnants near to which such young sources likely remain. But high-$\dot{E}$ pulsars are young, and

therefore rare, limited by the Galactic supernova rate. What is rare is far, and thus faint. The subset of high-$\dot{E}$ pulsars for which only the spectral tail reaches the LAT's sensitivity range is thus even rarer. Kuiper et al.'s (2018) detection of PSR J1846-0258 importantly demonstrates the LAT's ability to see such objects. Learning how many more $E_p < 500$ MeV pulsars exist will likely have to wait for a sensitive, large field-of-view MeV instrument such as AMEGO-X (Caputo et al. 2022) and ASTROMeV (De Angelis et al. 2021). It would allow the missing SED segments for the known and candidate low-$E_p$ pulsars to be filled in, and to search for pulsations from new ones. But again, the ultimate low $E_p$ population is likely to be small compared to that for GeV pulsars; thus, we propose that the trends shown here do capture the spectral character of most gamma-ray pulsars.

### 8.3. Light-curve Trends

The 237 energy-integrated profile fits in this catalog generally confirm the broad categories of light-curve shapes known since EGRET, consolidated in 1PC and 2PC: a majority of gamma-ray pulsars have two principal peaks separated by $\Delta \approx 0.4 \pm 0.15$ rotations (see Figure 10). For radio-detected pulsars, the first peak generally trails the radio pulse by $\delta \approx 0.2 \pm 0.2$ in phase. Gamma-ray emission from near or beyond the light cylinder can produce the observed anticorrelation between $\Delta$ and $\delta$, as discussed in 2PC and explicitly reproduced by the models of Kalapotharakos et al. (2014, 2023). About 40 pulsars have a single broad gamma-ray peak. Very few pulsars deviate from these trends: only two young pulsars in Table 9 are flagged as “complex,” appearing to have a third peak half-way between the “standard” two peaks (one is EGRET pulsar B1055-52). This number increases to seven for the MSPs in Table 10. Overall, the MSP and young pulsar profiles are more similar than they differ.

On the other hand, 167 pulsars have enough accumulated photons to allow profile fits in multiple energy sub-bands, fits not done for 2PC. For the first time, we quantify how profiles evolve with energy for a large sample. Figures 13 and 14 highlight the 28 brightest profiles, fit in at least six energy sub-bands. Reminiscent of the spectral results, profile morphology and evolution are remarkably homogeneous given the large diversity of pulsar types and environments. We describe two examples below.

In the second panel from the top,<fnref n="126" /> the energy-resolved fit summaries show the phase evolution for P1 and P2, and P3 for the few cases where it is a strong component. Changes in the peak separation $\Delta$ are nearly always small, $\lesssim 0.1$ at most. For <5 pulsars, such as PSR J2229+6114 shown in Figure Set 12, $\Delta$ appears to change with energy. However, inspection of the profiles in Figure Set 12 shows that individual profile components vary to accommodate slight pulse shape changes, but $\Delta$ in fact varies little.

The summary plots also illustrate the other example. Whereas P3 for the Vela pulsar increases by 0.1 in phase over the LAT energy range, visibly detaching from P1 and moving across the bridge region, this turns out to be the only known pulsar where a gamma-ray peak phase varies substantially with $E_\gamma$. Three pulsars, PSRs J0633+1736 (Geminga), J1747-2958, and J1836+5925 (“the next Geminga”), show P3 evolution in the summary plots. Again, inspection of the profiles shows it

<footnote n="126">¹²⁶ Summary plot examples are in Figure Set 12.</footnote>
$E_p$, to be addressed in future work. Because the $E_p-\dot{E}$ anticorrelation is fairly modest with substantial scatter, its slope does not change substantially when excluding LAT-induced discoveries.

We thus confirm that our highly diverse pulsar sample is striking in its spectral homogeneity: almost all gamma-ray pulsars are “GeV pulsars,” with $0.5 < E_p < 2.5$ GeV and following the same $\dot{E}$ trends.

Exceptions exist. Upon its EGRET discovery, Ramana-murthy et al. (1996) highlighted the low gamma-ray efficiency of PSR J0659+1414. Romani et al. (2011) cast it as the archetype of a hypothesized category of pulsars subluminous in gamma-rays. It stands out here as having very low $E_p$ and $d_p$ (Figure 20) and the lowest $\epsilon_{37,hi}$ (the isolated dot at lower left in Figure 27).

Another clear exception is the Crab: although its measured $E_p = 90 \pm 80$ MeV and spectral shape might be construed as a high-$\dot{E}$ continuation of the trend in Figure 28, its true SED peak occurs in the soft gamma-ray band (Kuiper & Hermsen 2015). A further two pulsars, PSRs B1509-58 and J1846-0258, have spectral shapes similar to GeV pulsars but with values of $E_p$ well outside the continuum of GeV pulsars. Both pulsars have $\dot{E} \approx 10^{37}$ erg s$^{-1}$ and little or no LAT signal above 100 MeV. Kuiper & Hermsen (2015) highlighted them as archetypes of a pulsar population with SED peaks in the MeV range, and show 18 high-$\dot{E}$ pulsars with SEDs rising in the keV range. Half of the 18 are in this catalog, as are two of five similar pulsars they discuss. PSR B1509-58 has the best-defined MeV peak. They argue that some of the others may not be detected by the LAT because their SEDs also peak below the LAT's range, as for PSR J1846-0258, at the very limit of our instrument's sensitivity (see Section 7.2). However, the SEDs shown by Kuiper & Hermsen (2015) leave open an alternate explanation: PSRs J0205+6449 and J2229+6114 indeed have keV slopes similar to PSR J1846-0258, yet are bright up to a GeV. The as-yet undetected pulsars may simply be too faint, located as they are in very high background regions. Phase-connected rotation ephemerides for a few of the radio-quiet, noisy pulsars are unavailable. Others may be GeV pulsars, but with gamma-ray beams that simply miss the Earth. Harding & Kalapotharakos (2017) further discussed the MeV pulsars.

To see whether the paucity of $E_p < 500$ MeV pulsars reflects magnetospheric emission processes or stems from observational limits, we start with a tally of LAT candidate sources. There are 238 unidentified nonvariable LAT sources with $E_p < 500$ MeV, undetected beyond 10 GeV, and significantly curved (well fit with a LogParabola function). Of these, 122 have localization ellipse semimajor axes with 95% C.L. <10', suitable for radio searches (see Table 4). Restricting the search to the 87 sources with $|b| < 10^\circ$ would favor young, energetic pulsars. Blind searches for radio-quiet pulsars require a large gamma-ray count rate, even more so if timing noise, glitch rates, or background are high. Unfortunately, only four (25) of the 238 targets have TS > 200 (100). The four sources are 4FGL J0340.4+5302, J0426.5+5434, J1749.8-0303, and J1805.7+3401. All were searched several times for radio pulsations but none have been reported to date.

Many high-$\dot{E}$ pulsars have already been found in radio and X-ray searches, effective at finding such sources because high-$\dot{E}$ pulsars tend to have broad radio beams, and because searches can target the supernova remnants near to which such young sources likely remain. But high-$\dot{E}$ pulsars are young, and

therefore rare, limited by the Galactic supernova rate. What is rare is far, and thus faint. The subset of high-$\dot{E}$ pulsars for which only the spectral tail reaches the LAT's sensitivity range is thus even rarer. Kuiper et al.'s (2018) detection of PSR J1846-0258 importantly demonstrates the LAT's ability to see such objects. Learning how many more $E_p < 500$ MeV pulsars exist will likely have to wait for a sensitive, large field-of-view MeV instrument such as AMEGO-X (Caputo et al. 2022) and ASTROMeV (De Angelis et al. 2021). It would allow the missing SED segments for the known and candidate low-$E_p$ pulsars to be filled in, and to search for pulsations from new ones. But again, the ultimate low $E_p$ population is likely to be small compared to that for GeV pulsars; thus, we propose that the trends shown here do capture the spectral character of most gamma-ray pulsars.

### 8.3. Light-curve Trends

The 237 energy-integrated profile fits in this catalog generally confirm the broad categories of light-curve shapes known since EGRET, consolidated in 1PC and 2PC: a majority of gamma-ray pulsars have two principal peaks separated by $\Delta \approx 0.4 \pm 0.15$ rotations (see Figure 10). For radio-detected pulsars, the first peak generally trails the radio pulse by $\delta \approx 0.2 \pm 0.2$ in phase. Gamma-ray emission from near or beyond the light cylinder can produce the observed anticorrelation between $\Delta$ and $\delta$, as discussed in 2PC and explicitly reproduced by the models of Kalapotharakos et al. (2014, 2023). About 40 pulsars have a single broad gamma-ray peak. Very few pulsars deviate from these trends: only two young pulsars in Table 9 are flagged as “complex,” appearing to have a third peak half-way between the “standard” two peaks (one is EGRET pulsar B1055-52). This number increases to seven for the MSPs in Table 10. Overall, the MSP and young pulsar profiles are more similar than they differ.

On the other hand, 167 pulsars have enough accumulated photons to allow profile fits in multiple energy sub-bands, fits not done for 2PC. For the first time, we quantify how profiles evolve with energy for a large sample. Figures 13 and 14 highlight the 28 brightest profiles, fit in at least six energy sub-bands. Reminiscent of the spectral results, profile morphology and evolution are remarkably homogeneous given the large diversity of pulsar types and environments. We describe two examples below.

In the second panel from the top,<fnref n="126" /> the energy-resolved fit summaries show the phase evolution for P1 and P2, and P3 for the few cases where it is a strong component. Changes in the peak separation $\Delta$ are nearly always small, $\lesssim 0.1$ at most. For <5 pulsars, such as PSR J2229+6114 shown in Figure Set 12, $\Delta$ appears to change with energy. However, inspection of the profiles in Figure Set 12 shows that individual profile components vary to accommodate slight pulse shape changes, but $\Delta$ in fact varies little.

The summary plots also illustrate the other example. Whereas P3 for the Vela pulsar increases by 0.1 in phase over the LAT energy range, visibly detaching from P1 and moving across the bridge region, this turns out to be the only known pulsar where a gamma-ray peak phase varies substantially with $E_\gamma$. Three pulsars, PSRs J0633+1736 (Geminga), J1747-2958, and J1836+5925 (“the next Geminga”), show P3 evolution in the summary plots. Again, inspection of the profiles shows it

<footnote n="126">¹²⁶ Summary plot examples are in Figure Set 12.</footnote>
<figure><img src="image_29.png" /><figcaption>**Figure 29.** Duty cycle *D* of the gamma-ray pulse profiles (fraction of a rotation that the fitted profile exceeds 25% of the maximum amplitude, measured from the background level). More young radio-quiet pulsars have large duty cycles than do the young radio-loud pulsars.</figcaption></figure>

comes from weak broadening of P2 rather than a peak drift as seen for Vela.

Such weak dependence of pulse phase on energy does not necessarily imply that the region illuminating an observer is uniform. Relativistic aberrations and time delays create caustics such that photons emitted at different distances from the neutron star remain in narrow phase ranges. Thus, whether different electron energies occur at different magnetospheric depths or not, we could see little light-curve evolution with photon energy. Nevertheless, since the SEDs suggest populations of gamma-ray emitting electrons with a narrow range of maximum energies for the lowest *E*̇ pulsars, broadening gradually for systems with more available power, the light curves also encourage vision of emission zones along caustic viewing lines that change little with electron energy. Figure 22 attempts to relate spectral and profile properties: it suggests that when the line of sight cuts across the emission region such that the viewer sees two closely spaced peaks (small Δ), then the line of sight sometimes samples a region with the highest electron energies (large *E*<sub>p</sub>).

Indeed, how the line of sight intersects the emission region is key. Beginning with Watters et al. (2009), and continuing as mastery of magnetospheric processes evolves (e.g., Venter et al. 2009; Bai & Spitkovsky 2010; Contopoulos & Kalapotharakos 2010; Arka & Dubus 2013; Kalapotharakos et al. 2014, 2018, 2023; Cerutti et al. 2016, 2020; Harding 2016; Philippov & Spitkovsky 2018; Pétri 2019), modelers have provided “atlases” allowing for comparison between observed and calculated pulse profiles. Sky maps show beam intensity versus neutron star latitude and longitude, that is, as a function of the inclination ζ between the rotation axis and the line of sight, and the rotational phase φ, with other parameters fixed. A cut across a sky map yields the profile sampled by a given line of sight. The dominant “other” parameter is the magnetic inclination α, but, e.g., period or magnetic field strength have more or less importance for different models. Some theories favor emission from plasmons in current sheets

beyond the light cylinder, where magnetic reconnection accelerates electrons to high energies to then radiate synchrotron gamma-rays (Pétri & Lyubarsky 2007; Cerutti et al. 2020). Other models favor electric field acceleration near the current sheet and curvature radiation. In either case, the picture is quite different from the various “gaps” that previously dominated the field (Philippov & Kramer 2022), and the resulting sky maps differ in consequence. The particle-in-cell calculations of the relativistic magnetohydrodynamic processes consume substantial computer resources. Realistic atlases are currently out of reach, with progress in kinetic plasma simulations expected in the coming years.

Figure 29 shows histograms of the gamma-ray duty cycle *D*, the total phase for which the fitted profile is >25% of the range from its baseline to the maximum. (The supplementary catalog also includes *D* with 10% and 50% thresholds, and *psrcat* includes radio pulse width.) Hou et al. (2014) highlighted some pulsars with large duty cycle, and discussed how detection sensitivity depends on *D*. With two exceptions, the young radio-loud pulsars have *D* < 0.4. In contrast, three times more young radio-quiet pulsars have 0.3 < *D* < 0.5 as loud ones.

Radio detection occurs when $|\zeta - \alpha| \lesssim \rho$, that is, along a broad diagonal swath of the plane of neutron star inclinations (α, ζ) (see Figure 3 of Johnston et al. 2020). The opening angle of the radio cone is $\rho = 3\sqrt{\pi h/2Pc}$, for emission height h. Gamma-ray detection favors the α > 30° and ζ > 30° part of the plane. Finally, long duty cycles arise when the line of sight slices a skymap in a ζ range with emission extended in φ, but these “flat” beam shapes favor small α (see, e.g., Figure 15 of Kalapotharakos et al. 2018). The tallies in Figure 29 are in good qualitative agreement with this simple geometric beam model. Thus, study of the high duty cycle pulsars may help constrain (α, ζ) and/or discriminate between models. The small light cylinders characteristic of MSPs make for broader beams, the likely reason for the MSP *D* tail in Figure 29.

If *D* could constrain the “beaming fraction” $f_\Omega$ used to calculate luminosity $L_\gamma$ (Equation (24)), the dispersion in the $L_\gamma$
<figure><img src="image_30.png" /><figcaption>**Figure 29.** Duty cycle *D* of the gamma-ray pulse profiles (fraction of a rotation that the fitted profile exceeds 25% of the maximum amplitude, measured from the background level). More young radio-quiet pulsars have large duty cycles than do the young radio-loud pulsars.</figcaption></figure>

comes from weak broadening of P2 rather than a peak drift as seen for Vela.

Such weak dependence of pulse phase on energy does not necessarily imply that the region illuminating an observer is uniform. Relativistic aberrations and time delays create caustics such that photons emitted at different distances from the neutron star remain in narrow phase ranges. Thus, whether different electron energies occur at different magnetospheric depths or not, we could see little light-curve evolution with photon energy. Nevertheless, since the SEDs suggest populations of gamma-ray emitting electrons with a narrow range of maximum energies for the lowest *E*̇ pulsars, broadening gradually for systems with more available power, the light curves also encourage vision of emission zones along caustic viewing lines that change little with electron energy. Figure 22 attempts to relate spectral and profile properties: it suggests that when the line of sight cuts across the emission region such that the viewer sees two closely spaced peaks (small Δ), then the line of sight sometimes samples a region with the highest electron energies (large *E*<sub>p</sub>).

Indeed, how the line of sight intersects the emission region is key. Beginning with Watters et al. (2009), and continuing as mastery of magnetospheric processes evolves (e.g., Venter et al. 2009; Bai & Spitkovsky 2010; Contopoulos & Kalapotharakos 2010; Arka & Dubus 2013; Kalapotharakos et al. 2014, 2018, 2023; Cerutti et al. 2016, 2020; Harding 2016; Philippov & Spitkovsky 2018; Pétri 2019), modelers have provided “atlases” allowing for comparison between observed and calculated pulse profiles. Sky maps show beam intensity versus neutron star latitude and longitude, that is, as a function of the inclination ζ between the rotation axis and the line of sight, and the rotational phase φ, with other parameters fixed. A cut across a sky map yields the profile sampled by a given line of sight. The dominant “other” parameter is the magnetic inclination α, but, e.g., period or magnetic field strength have more or less importance for different models. Some theories favor emission from plasmons in current sheets

beyond the light cylinder, where magnetic reconnection accelerates electrons to high energies to then radiate synchrotron gamma-rays (Pétri & Lyubarsky 2007; Cerutti et al. 2020). Other models favor electric field acceleration near the current sheet and curvature radiation. In either case, the picture is quite different from the various “gaps” that previously dominated the field (Philippov & Kramer 2022), and the resulting sky maps differ in consequence. The particle-in-cell calculations of the relativistic magnetohydrodynamic processes consume substantial computer resources. Realistic atlases are currently out of reach, with progress in kinetic plasma simulations expected in the coming years.

Figure 29 shows histograms of the gamma-ray duty cycle *D*, the total phase for which the fitted profile is >25% of the range from its baseline to the maximum. (The supplementary catalog also includes *D* with 10% and 50% thresholds, and *psrcat* includes radio pulse width.) Hou et al. (2014) highlighted some pulsars with large duty cycle, and discussed how detection sensitivity depends on *D*. With two exceptions, the young radio-loud pulsars have *D* < 0.4. In contrast, three times more young radio-quiet pulsars have 0.3 < *D* < 0.5 as loud ones.

Radio detection occurs when |ζ - α| ≲ ρ, that is, along a broad diagonal swath of the plane of neutron star inclinations (α, ζ) (see Figure 3 of Johnston et al. 2020). The opening angle of the radio cone is ρ = 3√πh/2Pc, for emission height h. Gamma-ray detection favors the α > 30° and ζ > 30° part of the plane. Finally, long duty cycles arise when the line of sight slices a skymap in a ζ range with emission extended in φ, but these “flat” beam shapes favor small α (see, e.g., Figure 15 of Kalapotharakos et al. 2018). The tallies in Figure 29 are in good qualitative agreement with this simple geometric beam model. Thus, study of the high duty cycle pulsars may help constrain (α, ζ) and/or discriminate between models. The small light cylinders characteristic of MSPs make for broader beams, the likely reason for the MSP *D* tail in Figure 29.

If *D* could constrain the “beaming fraction” *f*<sub>Ω</sub> used to calculate luminosity *L*<sub>γ</sub> (Equation (24)), the dispersion in the *L*<sub>γ</sub>
<figure><img src="image_30.png" /><figcaption>**Figure 30.** The cumulative number of pulsars detected with >100 MeV flux over $G_{100}$, "log N-log S." The lines show slopes of -1 and -3/2.</figcaption></figure>

versus $\dot{E}$ could be reduced. Large *D* pulsars, with their distinctive skymaps, could be particularly useful. But they are rare: the $\alpha$ range producing the right beam shape is limited, and the line of sight must fall into a narrow $\zeta$ range. PSR J1857+0143, the young radio-loud outlier in Figure 29, may be an example. Serylak et al. (2021) showed its polarization position angle versus phase at 1.4 GHz, but unfortunately scattering excludes an RVM estimate of ($\alpha, \zeta$) at this frequency while its narrow radio pulse width poorly constrains the RVM at higher frequencies. Table 6 points out unidentified LAT sources colocated with known pulsars for which we have not seen gamma-ray pulsations: some may be *D* ≈ 1 pulsars and could benefit from a ($\alpha, \zeta$) study.

The differences in spectral hardness (Figure 13) between the peaks nearer (P1) and farther (P2) from the magnetic pole, from the observer's point of view, are another powerful discriminant, mentioned above. Harding et al. (2021) and Barnard et al. (2022) addressed this specifically. Early LAT studies measured spectra in as many phase bins as the statistics would allow (Abdo et al. 2010c, 2010g, 2010h; DeCesar 2013). The observed phase evolution of the spectral parameters is smooth, and the range of parameter values is moderate (the cutoff energy can vary by a factor of 2). A simple analysis, measuring on-peak spectral shape in the peak phase intervals we provide, is easier to perform than a full phase-resolved spectral analysis that samples light curves at many points. Furthermore, a larger sample of pulsars is bright enough for on-pulse measurements than for full phase-resolved analysis; yet, a simple analysis is well suited to the current model accuracy. Such future work is encouraged.

## 8.4. Luminosity and Pulsar Populations

Our large gamma-ray pulsar sample, along with the known radio pulsars, is a starting point for a census of the total Milky Way population. Characterizing the population sheds light on many open questions. These include the supernova rate and neutron star evolution. The large number of "spider" binary MSPs caused a renaissance in the study of binary evolution and

pulsar recycling (Strader et al. 2019; van der Merwe et al. 2020; Swihart et al. 2022b). The number and luminosity of high-energy pulsars helps determine the relative contributions of pulsars and putative dark matter annihilation to the observed cosmic positron spectrum (Venter et al. 2015; Cholis & Krommydas 2022) and, via PWNs, to the leptonic cosmic-ray spectrum generally. Applying emission models to a synthetic population and then simply tallying the observed numbers of, for example, radio-loud versus radio-quiet pulsars tests predictions of beam geometry (Johnston et al. 2020).

The strong dependence of $L_\gamma$ on $\dot{E}$ (Figure 23) and the LAT's sensitivity (Figure 25) are key inputs. Key tests of a synthesis's fidelity are the pulsar latitude distribution (Figure 3) and the cumulative log N-log S distribution of our pulsar sample, shown in Figure 30. The slopes in the log N-log S plot are related to the spatial distributions of the sources: a flux-limited sample of a population of sources having a narrow luminosity distribution has slope -3/2 if the population is isotropic, and -1 if the distribution is disk-like. Indeed, in Figure 30 the MSPs have a steeper slope, being more isotropic, while the young pulsars have a slope close to -1. The turnover at low fluxes is the detection threshold, lower for objects away from the intense background in the plane. Distance uncertainties do not affect the log N-log S plot. However, an extrapolation from the observed to the unresolved population needs to take selection biases into account. One example is that most of our young radio-loud pulsars were discovered in radio surveys near the plane—results sensitive to details of the high-latitude population may suffer from bias. Another is that target sources for gamma-ray blind searches for radio-quiet pulsars have a higher threshold than shown in Figure 25, explained in Section 3.4.

The scatter around the $L_\gamma$ versus $\dot{E}$ correlation (or, equivalently, of efficiency versus $\dot{E}$ shown in Figure 24) is a combination of physics and uncertainties. The “physics” bias includes beaming, since $f_\Omega = 1$ in our $L_\gamma$ computation. We use a canonical moment of inertia $I = 10^{45}$ g cm$^2$ for all pulsars, but the true average is probably twice that, since neutron star radii
<figure><img src="image_31.png" /><figcaption>**Figure 30.** The cumulative number of pulsars detected with >100 MeV flux over $G_{100}$, "log N-log S." The lines show slopes of -1 and -3/2.</figcaption></figure>

versus $\dot{E}$ could be reduced. Large *D* pulsars, with their distinctive skymaps, could be particularly useful. But they are rare: the $\alpha$ range producing the right beam shape is limited, and the line of sight must fall into a narrow $\zeta$ range. PSR J1857+0143, the young radio-loud outlier in Figure 29, may be an example. Serylak et al. (2021) showed its polarization position angle versus phase at 1.4 GHz, but unfortunately scattering excludes an RVM estimate of ($\alpha, \zeta$) at this frequency while its narrow radio pulse width poorly constrains the RVM at higher frequencies. Table 6 points out unidentified LAT sources colocated with known pulsars for which we have not seen gamma-ray pulsations: some may be *D* ≈ 1 pulsars and could benefit from a ($\alpha, \zeta$) study.

The differences in spectral hardness (Figure 13) between the peaks nearer (P1) and farther (P2) from the magnetic pole, from the observer's point of view, are another powerful discriminant, mentioned above. Harding et al. (2021) and Barnard et al. (2022) addressed this specifically. Early LAT studies measured spectra in as many phase bins as the statistics would allow (Abdo et al. 2010c, 2010g, 2010h; DeCesar 2013). The observed phase evolution of the spectral parameters is smooth, and the range of parameter values is moderate (the cutoff energy can vary by a factor of 2). A simple analysis, measuring on-peak spectral shape in the peak phase intervals we provide, is easier to perform than a full phase-resolved spectral analysis that samples light curves at many points. Furthermore, a larger sample of pulsars is bright enough for on-pulse measurements than for full phase-resolved analysis; yet, a simple analysis is well suited to the current model accuracy. Such future work is encouraged.

## 8.4. Luminosity and Pulsar Populations

Our large gamma-ray pulsar sample, along with the known radio pulsars, is a starting point for a census of the total Milky Way population. Characterizing the population sheds light on many open questions. These include the supernova rate and neutron star evolution. The large number of "spider" binary MSPs caused a renaissance in the study of binary evolution and

pulsar recycling (Strader et al. 2019; van der Merwe et al. 2020; Swihart et al. 2022b). The number and luminosity of high-energy pulsars helps determine the relative contributions of pulsars and putative dark matter annihilation to the observed cosmic positron spectrum (Venter et al. 2015; Cholis & Krommydas 2022) and, via PWNs, to the leptonic cosmic-ray spectrum generally. Applying emission models to a synthetic population and then simply tallying the observed numbers of, for example, radio-loud versus radio-quiet pulsars tests predictions of beam geometry (Johnston et al. 2020).

The strong dependence of $L_\gamma$ on $\dot{E}$ (Figure 23) and the LAT's sensitivity (Figure 25) are key inputs. Key tests of a synthesis's fidelity are the pulsar latitude distribution (Figure 3) and the cumulative log N-log S distribution of our pulsar sample, shown in Figure 30. The slopes in the log N-log S plot are related to the spatial distributions of the sources: a flux-limited sample of a population of sources having a narrow luminosity distribution has slope -3/2 if the population is isotropic, and -1 if the distribution is disk-like. Indeed, in Figure 30 the MSPs have a steeper slope, being more isotropic, while the young pulsars have a slope close to -1. The turnover at low fluxes is the detection threshold, lower for objects away from the intense background in the plane. Distance uncertainties do not affect the log N-log S plot. However, an extrapolation from the observed to the unresolved population needs to take selection biases into account. One example is that most of our young radio-loud pulsars were discovered in radio surveys near the plane—results sensitive to details of the high-latitude population may suffer from bias. Another is that target sources for gamma-ray blind searches for radio-quiet pulsars have a higher threshold than shown in Figure 25, explained in Section 3.4.

The scatter around the $L_\gamma$ versus $\dot{E}$ correlation (or, equivalently, of efficiency versus $\dot{E}$ shown in Figure 24) is a combination of physics and uncertainties. The “physics” bias includes beaming, since $f_\Omega = 1$ in our $L_\gamma$ computation. We use a canonical moment of inertia $I = 10^{45}$ g cm$^2$ for all pulsars, but the true average is probably twice that, since neutron star radii
are closer to 14 km than to 10 km, shifting the X-axis of all $\dot{E}$ plots to the right. Furthermore, MSPs especially have a range of masses, of the order of only 60%, neglected in our single I value. Luminosity may also depend on parameters other than $\dot{E}$, such as the inclination angles, which would contribute to the scatter. The correlation is stronger at higher spindown powers, and is less pronounced for the MSPs. It remains to be understood whether this is due to high-energy processes “shrinking” as $\dot{E}$ tends toward the “deathline” near $10^{33}$ erg s$^{-1}$, or is because, e.g., of the compactness of the MSP magnetosphere.

Highly polarized radio beams generally occur only for pulsars with $\dot{E} > 10^{34}$ erg s$^{-1}$ (Weltevrede & Johnston 2008; Johnston & Kerr 2018). Smith et al. (2019) suggested it is an unlikely coincidence that both radio polarization and gamma-ray emission require the same minimum spin parameters for independent reasons—that these two phenomena share the same deathline may mean that the same electron populations produce both signals.

## 8.5. Toward 4PC and 400 Gamma-Ray Pulsars

A small number (~20) of our gamma-ray pulsars are below the 4FGL catalog detection threshold. Thus, we expect that the bulk of future pulsed discoveries will correspond to the 1000 or more currently unidentified LAT sources. Requiring nonvariability and a pulsar-like spectrum reduces this to a few hundred. It thus appears unlikely that the LAT will ever more than double the size of the current sample.

However, another 100 gamma-ray pulsars is possible, and dozens are likely. Machine-learning methods applied to the 4FGL sources typically find ~70 pulsar-like unassociated objects with $|b| > 10°$ (Mirabal & Bonaca 2021). Future high-sensitivity radio surveys with SKA are predicted to find 27,000 young pulsars and 3000 MSPs (Keane et al. 2015). Radio timing solutions for those colocated with LAT source would allow for gamma-ray folding, at least for the years following the radio discovery. A radio position, period, and orbital period for the binaries would improve the sensitivity and decrease the computing cost for gamma-ray blind searches. Without waiting for SKA, TRAPUM at MeerKAT and FAST are already finding new pulsars. Sustained timing of the new discoveries will require thoughtful management of available resources. The insights we have achieved in this work into the shapes of pulsar spectra will help quantify how many of the unidentified LAT sources are likely to be pulsars without pulsed detections.

We intend to maintain the catalog (see the footnote at the end of Section 1). We will add new gamma-ray pulsars as they accumulate. If new analyses are performed, e.g., on-peak spectral analysis, we will add those results.

Pulsar observations enable investigations over a spectacular range of topics, from fundamental physics to variations of the interstellar medium, including several aspects of stellar evolution, and more. Our gamma-ray pulsar sample is so large that even subsamples selected for specific studies have more members than did the entire collection acquired by Compton Gamma-ray Observatory, over 20 yr ago. Discoveries enabled by exploiting the sample will expand for many years to come, at the same time that Fermi LAT keeps finding more, and better characterizes those already known.

## Acknowledgments

The Fermi-LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l'Energie Atomique and the Centre National de la Recherche Scientifique/Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden.

Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'Études Spatiales in France. This work performed in part under DOE contract DE-AC02-76SF00515.

The Parkes Radio Telescope is part of the Australia Telescope, which is funded by the Commonwealth Government for operation as a National Facility managed by CSIRO. The Green Bank Telescope is operated by the National Radio Astronomy Observatory, a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The Arecibo Observatory is part of the National Astronomy and Ionosphere Center (NAIC), a national research center operated by Cornell University under a cooperative agreement with the National Science Foundation. The Nançay Radio Observatory is operated by the Paris Observatory, associated with the French Centre National de la Recherche Scientifique (CNRS). The Lovell Telescope is owned and operated by the University of Manchester as part of the Jodrell Bank Centre for Astrophysics with support from the Science and Technology Facilities Council of the United Kingdom. The International LOFAR Telescope and the Westerbork Synthesis Radio Telescope are operated by ASTRON, the Netherlands Institute for Radio Astronomy. This work made extensive use of the ATNF pulsar catalog (Manchester et al. 2005). I.G., A.K.H., and D.A.S. thank the ISSI (Bern, Switzerland) for financial support of Team 459 meetings that helped to improve the present work.

*Facility:* Fermi(LAT).

*Software:* Tempo2 (Hobbs et al. 2006), PINT (Luo et al. 2021), astropy (Astropy Collaboration et al. 2018), numpy (Harris et al. 2020).

## Appendix A

### Radio Timing for This Work

PSR J0653+4706: PSR J0653+4706 was discovered with LOFAR (van Haarlem et al. 2013) at 135 MHz in a targeted pulsar survey toward unidentified Fermi LAT gamma-ray sources (Bassa et al. 2017b; Pleunis et al. 2017). The survey used semicoherent dedispersion to minimize the effects of dispersive smearing at these low frequencies while retaining sensitivity to MSPs (Bassa et al. 2017a). The pulsar has a 4.76 ms spin period and a dispersion measure (DM) = 25.74 ± 0.14 pc cm$^{-3}$, placing it at a distance of ~900 pc
are closer to 14 km than to 10 km, shifting the X-axis of all $\dot{E}$ plots to the right. Furthermore, MSPs especially have a range of masses, of the order of only 60%, neglected in our single I value. Luminosity may also depend on parameters other than $\dot{E}$, such as the inclination angles, which would contribute to the scatter. The correlation is stronger at higher spindown powers, and is less pronounced for the MSPs. It remains to be understood whether this is due to high-energy processes “shrinking” as $\dot{E}$ tends toward the “deathline” near $10^{33}$ erg s$^{-1}$, or is because, e.g., of the compactness of the MSP magnetosphere.

Highly polarized radio beams generally occur only for pulsars with $\dot{E} > 10^{34}$ erg s$^{-1}$ (Weltevrede & Johnston 2008; Johnston & Kerr 2018). Smith et al. (2019) suggested it is an unlikely coincidence that both radio polarization and gamma-ray emission require the same minimum spin parameters for independent reasons—that these two phenomena share the same deathline may mean that the same electron populations produce both signals.

## 8.5. Toward 4PC and 400 Gamma-Ray Pulsars

A small number (~20) of our gamma-ray pulsars are below the 4FGL catalog detection threshold. Thus, we expect that the bulk of future pulsed discoveries will correspond to the 1000 or more currently unidentified LAT sources. Requiring nonvariability and a pulsar-like spectrum reduces this to a few hundred. It thus appears unlikely that the LAT will ever more than double the size of the current sample.

However, another 100 gamma-ray pulsars is possible, and dozens are likely. Machine-learning methods applied to the 4FGL sources typically find ~70 pulsar-like unassociated objects with $|b| > 10°$ (Mirabal & Bonaca 2021). Future high-sensitivity radio surveys with SKA are predicted to find 27,000 young pulsars and 3000 MSPs (Keane et al. 2015). Radio timing solutions for those colocated with LAT source would allow for gamma-ray folding, at least for the years following the radio discovery. A radio position, period, and orbital period for the binaries would improve the sensitivity and decrease the computing cost for gamma-ray blind searches. Without waiting for SKA, TRAPUM at MeerKAT and FAST are already finding new pulsars. Sustained timing of the new discoveries will require thoughtful management of available resources. The insights we have achieved in this work into the shapes of pulsar spectra will help quantify how many of the unidentified LAT sources are likely to be pulsars without pulsed detections.

We intend to maintain the catalog (see the footnote at the end of Section 1). We will add new gamma-ray pulsars as they accumulate. If new analyses are performed, e.g., on-peak spectral analysis, we will add those results.

Pulsar observations enable investigations over a spectacular range of topics, from fundamental physics to variations of the interstellar medium, including several aspects of stellar evolution, and more. Our gamma-ray pulsar sample is so large that even subsamples selected for specific studies have more members than did the entire collection acquired by Compton Gamma-ray Observatory, over 20 yr ago. Discoveries enabled by exploiting the sample will expand for many years to come, at the same time that Fermi LAT keeps finding more, and better characterizes those already known.

## Acknowledgments

The Fermi-LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l'Energie Atomique and the Centre National de la Recherche Scientifique/Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden.

Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'Études Spatiales in France. This work performed in part under DOE contract DE-AC02-76SF00515.

The Parkes Radio Telescope is part of the Australia Telescope, which is funded by the Commonwealth Government for operation as a National Facility managed by CSIRO. The Green Bank Telescope is operated by the National Radio Astronomy Observatory, a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. The Arecibo Observatory is part of the National Astronomy and Ionosphere Center (NAIC), a national research center operated by Cornell University under a cooperative agreement with the National Science Foundation. The Nançay Radio Observatory is operated by the Paris Observatory, associated with the French Centre National de la Recherche Scientifique (CNRS). The Lovell Telescope is owned and operated by the University of Manchester as part of the Jodrell Bank Centre for Astrophysics with support from the Science and Technology Facilities Council of the United Kingdom. The International LOFAR Telescope and the Westerbork Synthesis Radio Telescope are operated by ASTRON, the Netherlands Institute for Radio Astronomy. This work made extensive use of the ATNF pulsar catalog (Manchester et al. 2005). I.G., A.K.H., and D.A.S. thank the ISSI (Bern, Switzerland) for financial support of Team 459 meetings that helped to improve the present work.

*Facility:* Fermi(LAT).

*Software:* Tempo2 (Hobbs et al. 2006), PINT (Luo et al. 2021), astropy (Astropy Collaboration et al. 2018), numpy (Harris et al. 2020).

## Appendix A

### Radio Timing for This Work

PSR J0653+4706: PSR J0653+4706 was discovered with LOFAR (van Haarlem et al. 2013) at 135 MHz in a targeted pulsar survey toward unidentified Fermi LAT gamma-ray sources (Bassa et al. 2017b; Pleunis et al. 2017). The survey used semicoherent dedispersion to minimize the effects of dispersive smearing at these low frequencies while retaining sensitivity to MSPs (Bassa et al. 2017a). The pulsar has a 4.76 ms spin period and a dispersion measure (DM) = 25.74 ± 0.14 pc cm$^{-3}$, placing it at a distance of ~900 pc
with both the NE2001 and the YMW16 models. It is in a 5.84 day binary system with an $M_c \gtrsim 0.21 M_\odot$ binary companion (Bassa et al. 2018). The timing solution used in the catalog is obtained from LOFAR timing observations covering a 2.8 yr timespan. Subsequently, a 5.2 yr solution using Jodrell Bank Observatory L-band observations confirmed the model.

The LOFAR data indicate a flux density of $5 \pm 2$ mJy at 150 MHz. An 1800 s Green Bank Telescope observation at 820 MHz, with 200 MHz bandwidth, yielded a $37.2\sigma$ detection. For two polarizations, gain $G = 2$ K Jy$^{-1}$, system temperature $T_{\text{sys}} = 29$ K, and a 12% duty cycle seen in both the 150 and 820 MHz profiles, we obtain a flux density of $260$ µJy, calculated using the PRESTO sum_profiles.py routine.

Analysis of 101 positive detections at 1400 MHz with the Lovell telescope at Jodrell Bank Observatory yields flux densities of $89 \pm 11$ and $10 \pm 3$ µJy for the main peak and interpulse, respectively. The uncertainties are statistical, and include variations due to interstellar scintillation. The total is $S_{1400} = 99 \pm 23$ µJy, having added the 20% systematic calibration uncertainty in quadrature. The spectral index obtained from these three measurements agrees well with the average value of 1.7 of the pulsar population.

The JBO ROACH timing data show no significant pulsar proper motion. We found $\text{PMDEC} = -1.5(2)$ and $\text{PMRA} = 0.5$ (1.8). The Doppler correction is hence $0.98 < (1 - \xi) < 1$, with $(1 - \xi) = \dot{E}^{\text{int}}/\dot{E}$, as in Section 4.3.

PSR J1023+0038: PSR J1023+0038 was the first transitional MSP to be discovered (Archibald et al. 2009), and it underwent a transition from a rotationally powered MSP state to an accretion-powered state in 2013 (Stappers et al. 2014). A faint source coincident with the pulsar was detected by the LAT in the pre-transition data (Nolan et al. 2012), and the flux from this source increased five-fold when accretion began (Stappers et al. 2014).

Using the radio timing ephemeris of Archibald et al. (2013), we folded the pre-transition data (from the beginning of the LAT data to 2013 June), discovering significant pulsations, with $H = 62$. This ephemeris contains a single orbital period derivative to account for long-term trends in the orbital period, but does not account for shorter-timescale orbital phase variations seen in radio timing observations. We refined this solution using an unbinned timing analysis on the LAT data, also using only one orbital period derivative, resulting in an increased $H = 92$.

We also folded the post-transition data using a timing model built from X-ray timing by Jaodand et al. (2016) and X-ray and optical timing from Illiano et al. (2023). We fit a polynomial function to the orbital phase deviations measured by Illiano et al. (2023), and folded the LAT data using this orbital model and the spindown model of Jaodand et al. (2016). No pulsations were detected. We also searched for pulsations with slightly different spin periods and spin-down rates, and with slight shifts in orbital phase, using a sliding window approach, but again did not detect any significant pulsations.

This nondetection implies that the flux increase due to accretion is not an enhancement of the pulsed flux (seen in other nonaccreting redbacks; e.g., An et al. 2020; Clark et al. 2021), but is instead an additional unpulsed component. If the five-fold increase in gamma-ray flux is indeed due to unpulsed emission from the accretion process, and the gamma-ray pulsations remain unaffected by the accretion, then the pulsed

fraction would be reduced to 20% of the pre-transition level, and this would leave the pulsations undetectable to our analysis. Like PSR J1227-4853 (Johnson et al. 2015), it therefore remains unclear whether or not this system emits gamma-ray pulsations in the accretion-powered state.

PSR J1827-0849: Applying the methods of Clark et al. (2017) and Section 3.4 to LAT data around the position of a steep-spectrum radio source coincident with unidentified LAT source 3FGL J1827.6-0846, found by Frail et al. (2016), we discovered gamma-ray pulsations with spin period 2.24 ms in 2016. The radio positional uncertainties of $\sim 2''$ have greatly reduced the computing cost and allowed us to increase our sliding coherence window length to $2^{23}$ s (or $\sim 97$ days) and still perform the search on the ATLAS cluster$^{127}$ in about 1 day per source.

The associated radio source has flux density $229.9 \pm 14.8$ mJy in the GMRT 150 MHz All-Sky Survey, and $777 \pm 126$ mJy in VLSS$_r$ at 74 MHz. NVSS has no detection, giving $S_{1400} < 0.8$ mJy per beam. Table 8 lists an insignificant proper-motion estimate from LAT timing as per Section 3.5. C.J. Clark and coworkers are studying other LAT unidentified sources colocated with steep-spectrum radio sources and will detail their methods and results in future work.

PSR J1833-3840: This 1.87 ms pulsar was discovered in 2015 by F. Camilo, in observations of unidentified LAT sources using the Parkes radio telescope, using the methods reported by Camilo et al. (2015). Later, Frail et al. (2018) independently highlighted it in a search for steep radio spectrum pulsar candidates in LAT unidentified sources using the GMRT 150 MHz all-sky survey (TGSS ADR1). Fewer than 10% of known pulsars have such high spindown power, $\dot{E} = 1.1 \times 10^{35}$ erg s$^{-1}$, in the Milky Way or in globular clusters. The orbital period is 0.9 day. In 2020, then-undergraduate Camryn Phillips phase-connected GBT timing observations during PINT code development she was conducting. We phase-folded the gamma-ray data using her ephemeris and obtained the pulsed detection reported here.

PSR J1852-1310: The $P = 4.31$ ms isolated pulsar PSR J1852-1310 was discovered in a 1900s pointing on MJD 57793 toward an LAT unidentified source with a pulsar-like spectrum, at 800 MHz using the GUPPI backend on the Green Bank Radio Telescope. It was initially named PSR J1852-13. Routine timing observations at 1400 MHz with the Lovell telescope at Jodrell Bank Observatory began on MJD 57818. As of MJD 59927, 124 arrival times have been acquired using ROACH, allowing a phase-connected rotation ephemeris. The spindown power is $\dot{E} = 4.95 \times 10^{33}$ erg s$^{-1}$. For $DM = 44.95 \pm 0.01$ pc cm$^{-3}$ in the direction of this pulsar, the YMW16 model gives a distance of $1.27 \pm 0.5$ kpc. The integral energy flux of colocated 4FGL J1852.2-1309 thus gives $L_{\gamma} = (5.8 \pm 1.0) \times 10^{32}$ erg s$^{-1}$, for a gamma-ray efficiency of 11%.

GBT timing observations at 1400 MHz between MJD 58698 and 59050 yielded 49 times of arrival. Combining these with the JBO data slightly improves the MSP's proper-motion measurement, and we obtain $\text{PMDEC} = -6.6 \pm 1.4$ and $\text{PMRA} = -1.84 \pm 0.25$. The Doppler correction is hence $(1 - \xi) = 0.91$, as defined in Section 4.3, such that $\dot{E}^{\text{int}} = (4.6 \pm 0.2) \times 10^{33}$ erg s$^{-1}$.

<footnote n="127">127 https://www.aei.mpg.de/atlas</footnote>