adsabs/AstroCLIMB huggingface dataset:
Paper: Modeling Multiple X-Ray Reflection in Super-Eddington Winds
Authors: Zijian Zhang, 子健 张, Lars Lund Thomsen, Lixin Dai, Christopher S. Reynolds, Javier A. García, Erin Kara, Riley Connors, Megan Masterson, Yuhan Yao, Thomas Dauser
Figure caption: The illustration of the two parameters—namely, ΔE ≡ E 1 − E 2 and ﹩{F}_{21}\equiv {\nu }_{2}{F}_{{\nu }_{2}}/{\nu }_{1}{F}_{{\nu }_{1}}﹩—defined to quantify the significance of the double-peak line feature, based on a few characteristic modeled Fe line spectra. E 1 is the energy at which the total Fe line energy spectrum (black solid line) peaks and E 2 is the energy at which component 1 of the secondary reflection (red dashed line) peaks. ﹩{\nu }_{1}{F}_{{\nu }_{1}}﹩ and ﹩{\nu }_{2}{F}_{{\nu }_{2}}﹩ are the line energy fluxes at E 1 and E 2, respectively.
Paper: Modeling Multiple X-Ray Reflection in Super-Eddington Winds
Authors: Zijian Zhang, 子健 张, Lars Lund Thomsen, Lixin Dai, Christopher S. Reynolds, Javier A. García, Erin Kara, Riley Connors, Megan Masterson, Yuhan Yao, Thomas Dauser
Figure caption: The same spectra as in Figure 13, but fitted using an alternative model: tbabs∗(bbody+powerlaw+Gaussian+Gaussian)∗smedge. In the top panel, besides the observed data, we show the total modeled FPMA spectrum (blue solid curve) and its decomposed components bbody (blue dotted curve), powerlaw (blue dashed curve), and Gaussian (blue dashed–dotted curves). The bottom panel shows the residual between the observed and the modeled spectra.
Paper: Modeling Multiple X-Ray Reflection in Super-Eddington Winds
Authors: Zijian Zhang, 子健 张, Lars Lund Thomsen, Lixin Dai, Christopher S. Reynolds, Javier A. García, Erin Kara, Riley Connors, Megan Masterson, Yuhan Yao, Thomas Dauser
Figure caption: Best-fit model for the Fe Kα line that corresponds to the positive residual in the 5–8 keV region in the lower panel of Figure 13. The blue data points are the normalized NuSTAR residual in the line region. The orange curve is the best-fit super-Eddington Fe Kα model with multiple reflection. The fitted model parameters are listed in the legend.
Paper: Modeling Multiple X-Ray Reflection in Super-Eddington Winds
Authors: Zijian Zhang, 子健 张, Lars Lund Thomsen, Lixin Dai, Christopher S. Reynolds, Javier A. García, Erin Kara, Riley Connors, Megan Masterson, Yuhan Yao, Thomas Dauser
Figure caption: Best-fit model for the Fe Kα line that corresponds to the positive residual in the 5–8 keV region in the lower panel of Figure 15, after taking out the two Gaussian components. The line styles and colors are the same as in Figure 14.
Paper: The Alignment of High-resolution Solar Prominence Images Observed by the New Vacuum Solar Telescope
Authors: Yunfang Cai, Yongyuan Xiang, Kaifan Ji
Figure caption: The high-resolution prominence images at the Hα line center with the different postures observed by NVST. The lower panel images come from different groups of data with the same target on one day.
Paper: Modeling Multiple X-Ray Reflection in Super-Eddington Winds
Authors: Zijian Zhang, 子健 张, Lars Lund Thomsen, Lixin Dai, Christopher S. Reynolds, Javier A. García, Erin Kara, Riley Connors, Megan Masterson, Yuhan Yao, Thomas Dauser
Figure caption: The dependence of the absorption-edge profile on the wind terminal velocity v ∞. The black line shows the spectrum of all escaped photons that have only gone through scatterings in the wind, which is decomposed into several components: orange—photons that have undergone one scattering; blue—photons that have undergone two scatterings; and green—photons that have undergone three or more scatterings. Panel (b) is the result of the default set of model parameters (the same as in Figure 3), while panels (a) and (c) have different v ∞. As v ∞ increases, the absorption edge becomes less skewed, since the trough of the absorption edge is more blueshifted and the edge starts at lower energy.
Paper: The Alignment of High-resolution Solar Prominence Images Observed by the New Vacuum Solar Telescope
Authors: Yunfang Cai, Yongyuan Xiang, Kaifan Ji
Figure caption: The high-resolution imaging spectroscopic prominence images of three wavelengths (CENT corresponds to the Hα line center, R040 to the red wing 0.4 Å, and R070 to the red wing 0.7 Å).
Paper: Modeling Multiple X-Ray Reflection in Super-Eddington Winds
Authors: Zijian Zhang, 子健 张, Lars Lund Thomsen, Lixin Dai, Christopher S. Reynolds, Javier A. García, Erin Kara, Riley Connors, Megan Masterson, Yuhan Yao, Thomas Dauser
Figure caption: Line profiles at different ionization states. The x-axis is defined as g = E obs/(6.7 keV). The y-axis is the normalized flux. The gray dashed line is the total line flux as in Figure 3 (b), with 100% contribution from the Fe Kα line at 6.7 keV. The black line is the total line flux, which comes from a combination of 50% Fe Kα line at 6.7 keV (blue solid line) and 50% Fe Kα line at 6.97 keV (red solid line) in terms of photon number.
Paper: Modeling Multiple X-Ray Reflection in Super-Eddington Winds
Authors: Zijian Zhang, 子健 张, Lars Lund Thomsen, Lixin Dai, Christopher S. Reynolds, Javier A. García, Erin Kara, Riley Connors, Megan Masterson, Yuhan Yao, Thomas Dauser
Figure caption: The dependence of the absorption-edge profile on the half-open angle θ half. The color scheme and line styles are the same as in Figure B1. Panel (b) is the result of the default set of model parameters, while panels (a) and (c) have different θ half. As the funnel becomes narrower, the absorption edge becomes less skewed.
Paper: Modeling Multiple X-Ray Reflection in Super-Eddington Winds
Authors: Zijian Zhang, 子健 张, Lars Lund Thomsen, Lixin Dai, Christopher S. Reynolds, Javier A. García, Erin Kara, Riley Connors, Megan Masterson, Yuhan Yao, Thomas Dauser
Figure caption: The dependence of the absorption-edge profile on the wind acceleration radius R acc. The color scheme and line styles are the same as in Figure B1. Panel (b) is the result of the default set of model parameters, while panels (a) and (c) have different R acc. As R acc decreases, the absorption edge becomes slightly less skewed.
Paper: Modeling Multiple X-Ray Reflection in Super-Eddington Winds
Authors: Zijian Zhang, 子健 张, Lars Lund Thomsen, Lixin Dai, Christopher S. Reynolds, Javier A. García, Erin Kara, Riley Connors, Megan Masterson, Yuhan Yao, Thomas Dauser
Figure caption: The dependence of the absorption-edge profile on the height of corona H LP. The color scheme and line styles are the same as in Figure B1. Panel (b) is the result of the default set of model parameters, while panels (a) and (c) have different H LP. As H LP decreases, the absorption edge becomes less skewed.
Paper: Modeling Multiple X-Ray Reflection in Super-Eddington Winds
Authors: Zijian Zhang, 子健 张, Lars Lund Thomsen, Lixin Dai, Christopher S. Reynolds, Javier A. García, Erin Kara, Riley Connors, Megan Masterson, Yuhan Yao, Thomas Dauser
Figure caption: A comparison of the Fe line profiles calculated using elastic scattering (solid line) vs. Compton scattering (dotted line). The model parameters of (a), (b), and (c) are the same as Figure 3(b), Figure 4(c), and Figure 7(c), separately.
Paper: The Alignment of High-resolution Solar Prominence Images Observed by the New Vacuum Solar Telescope
Authors: Yunfang Cai, Yongyuan Xiang, Kaifan Ji
Figure caption: The rotation images of preprocessing. The left panel is a coarse rotation image and its angle, and the right panel is a fine rotation image and the total angle.
Paper: The Alignment of High-resolution Solar Prominence Images Observed by the New Vacuum Solar Telescope
Authors: Yunfang Cai, Yongyuan Xiang, Kaifan Ji
Figure caption: The local region of the reference prominence image and the contours of the solar edge of aligned images with two different methods. The black and white lines are the edge contours of the reference and the 100th image in the aligned sequence. The top and bottom panels are for the GFCT and our proposed methods, respectively.
Paper: The Alignment of High-resolution Solar Prominence Images Observed by the New Vacuum Solar Telescope
Authors: Yunfang Cai, Yongyuan Xiang, Kaifan Ji
Figure caption: The alignment accuracy of 100 time-sequence prominence images using two different methods. The red and blue solid lines represent the performance of the GFCT and our proposed methods, respectively. The residual errors of x- and y-directions are plotted in two panels.
Paper: The Alignment of High-resolution Solar Prominence Images Observed by the New Vacuum Solar Telescope
Authors: Yunfang Cai, Yongyuan Xiang, Kaifan Ji
Figure caption: Extracting the coordinations of solar edge from a high-resolution prominence image.
Paper: The Alignment of High-resolution Solar Prominence Images Observed by the New Vacuum Solar Telescope
Authors: Yunfang Cai, Yongyuan Xiang, Kaifan Ji
Figure caption: The optical flow field for the solar surface region of the image pair and their histogram statistics in X and Y directions, respectively.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: A schematic diagram showing how DCR affects light in the Earth's atmosphere; shorter wavelengths are refracted more by the atmosphere, which means that, to an observer on Earth, a blue star appears to be higher up in the sky than a red star located at the same position in space. R 450nm and R 650nm denote the amount of refraction by our atmosphere at an air mass (AM) of 1.4 for 450 nm and 650 nm light, respectively.
Paper: The Alignment of High-resolution Solar Prominence Images Observed by the New Vacuum Solar Telescope
Authors: Yunfang Cai, Yongyuan Xiang, Kaifan Ji
Figure caption: The aligned high-resolution imaging spectroscopic prominence images and their pseudo-color composite image. The three monochromatic images with different wavelengths correspond to the images in Figure 2.
Paper: The Alignment of High-resolution Solar Prominence Images Observed by the New Vacuum Solar Telescope
Authors: Yunfang Cai, Yongyuan Xiang, Kaifan Ji
Figure caption: The NVST high-resolution prominence image alignment flowchart.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: Binned residuals (bias), error on the mean (bold error bars), and the size of σ MAD (light error bars) as a function of z spec for Perfect and Realistic simulations for CDF50 (left) and PDFPeak (right). CDF50 and PDFPeak values are equivalent for Perfect simulations. For clarity, Realistic points (blue) are plotted at the midpoints of each redshift bin (0.05, 0.15, 0.25,...), while Perfect are displaced 0.02 to the right. The bias for the Realistic case noticeably deviates away from zero starting at z spec = 0.7, especially for CDF50, where z DCR ≈ 0.6 for the unconstrained candidates. PDFPeak, on the other hand, displays significantly larger σ MAD at higher z spec. This is due to z DCR being centered at 0.6 with little spread for CDF50, meaning that the redshift estimates are precise but not accurate.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption:  z DCR vs. z spec in the Perfect case for PDFPeak. As pointed out in Section 4, CDF50 results are equivalent to PDFPeak results for this case, and the 1σ uncertainties are zero. The right panels are the same as the left, but zoomed to z < 0.6. The outlier fraction η 0.10 is defined as the fraction where ∣Δz∣ > 0.10, with ﹩{\rm{\Delta }}z\equiv \tfrac{{z}_{\mathrm{estimate}}-{z}_{\mathrm{spec}}}{1+{z}_{\mathrm{spec}}}﹩. The imperfect z DCR estimates are due to degeneracies in the DCR shifts at several redshifts, which is most apparent at z spec > 0.8.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: Similar to Figure 7, but for the Realistic case, with z DCR vs. z spec and 1σ error bars for the Realistic case for CDF50 (top panels) and PDFPeak (bottom panels). We see noticeable degradation in the z DCR estimates, especially at z spec > 0.8. The z spec < 0.6 panels still show good agreement.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: Binned outlier fractions as a function of z spec for Perfect and Realistic simulations for CDF50 (left) and PDFPeak (right). The Perfect simulations show <5% outlier fractions at all redshifts. For the Realistic simulations, the outlier fractions remain below 40% for both CDF50 and PDFPeak until z spec = 0.7. At z spec ≥ 0.8, CDF50 estimates are mostly incorrect, as the PDFs are largely flat and unconstrained. Note that the outlier fractions for PDFPeak decreasing at the highest redshifts for the Realistic case is due to the possible estimated redshift range being limited from 0.0 to 1.2.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: Example SN Ia SED at peak epoch overplotted on top of the LSST g band at z = 0 (top) and z = 0.6 (bottom). As shown in Equation (1), the resulting DCR shifts at the two redshifts are different since the overlap between the filter and the SED changes significantly. Because the product of the filter function and the SED, F(λ)S(λ), is smaller for the z = 0.6 SN, the (absolute) DCR shift is also smaller, as shown in Figure 3.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: The resulting g-band DCR altitude shifts with respect to a K5V reference star for a z = 0 SN (blue) and z = 0.6 SN (orange) at different air masses (AMs), but both at peak epoch. As we expect, the shifts at the two redshifts are different and we see more shift at higher AM. Comparison with Figure 2 implies that the z = 0 SN is “bluer” (in the g band at least) than the reference star while the z = 0.6 SN is “redder.”
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: The relative posterior probabilities, true z, CDF50, and PDFPeak estimates for three cases highlighting the advantages and disadvantages for the two methods: (Left) CDF50 gives a better estimate (multimodal with similar peak amplitudes). (Middle) PDFPeak gives a better estimate (distinct peak but not large in amplitude). (Right) Both CDF50 and PDFPeak estimate the correct redshift.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: Redshift and air mass (AM) distributions for the LSST DDF simulations in SNANA. The peak redshift is around z = 0.9 for the No selection cut and the Epoch cut, but around 0.6 < z < 0.9 for the Epoch + S/N cut, while the AM peaks between 1.05 and 1.10 for all three selection cuts. Higher-AM observations lead to better astro-z estimates.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: LSST ugriz filters (from left to right) according to their relative transmission and wavelength range. We do not show the Y band, as it is not used for the DCR calculations. Multiple detections with different bands are crucial for accurate astrometric redshifts.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption:                          ﹩{z}_{\mathrm{SN}}﹩ vs. z spec (top row), ﹩{z}_{\mathrm{SN}+\mathrm{Host}}﹩ vs. z spec (second row), ﹩{z}_{\mathrm{SN}+\mathrm{DCR}}﹩ vs. z spec (third row), and ﹩{z}_{\mathrm{SN}+\mathrm{DCR}+\mathrm{Host}}﹩ vs. z spec (bottom row). Combining ﹩{z}_{\mathrm{SN}}﹩ with either z Host or z DCR significantly improves the estimates, more so at z spec > 0.6 for z Host and z spec < 0.6 for z DCR. Combining all three shows much better overall (throughout all redshifts) accuracy than any two combinations.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption:  z Host vs. z spec and the 1σ  error bars for CDF50 (top panels) and PDFPeak (bottom panels). For around 5% of the SNe that pass the Default selection cut, there are no z Host estimates, so we assume flat PDFs as discussed in Section 4. While we do not show these candidates as data points and error bars in this figure, for clarity, they are always included for the performance metrics. We see that the z Host are more accurate at higher redshifts, contrary to z DCR. We also note that CDF50 performs better here, although this could be the result of the PDFs being constructed from quantiles with 11 bins.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: Binned residuals (bias), error on the mean (bold error bars), and the size of σ MAD (light error bars) as a function of z spec for z DCR, z Host, and z DCR+Host for CDF50 (left) and PDFPeak (right). Again, z DCR is placed at the midpoints of each redshift bin, while z Host and z DCR+Host are displaced to the left and right, respectively. Contrary to z DCR, z Host display more bias and larger error bars at lower redshifts, but combining with z DCR alleviates this.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: Binned outlier fractions as a function of z spec for z DCR, z Host, and z DCR+Host for CDF50 (left) and PDFPeak (right). Combining with z DCR significantly reduces the outlier fractions of z Host at z spec < 0.6 for both CDF50 and PDFPeak, while the high-z outlier fractions are not affected by much.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: Binned η 0.10 as a function of z spec for ﹩{z}_{\mathrm{SN}}﹩, ﹩{z}_{\mathrm{SN}+\mathrm{Host}}﹩, ﹩{z}_{\mathrm{SN}+\mathrm{DCR}}﹩, and ﹩{z}_{\mathrm{SN}+\mathrm{DCR}+\mathrm{Host}}﹩. Combining all three lowers the outlier fractions throughout all redshifts, although ﹩{z}_{\mathrm{SN}+\mathrm{Host}}﹩ show slightly lower outlier fractions at z spec > 0.8.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: Binned residuals (bias), error on the mean (bold error bars) and the size of σ MAD (light error bars) as a function of z spec for ﹩{z}_{\mathrm{SN}}﹩, ﹩{z}_{\mathrm{SN}+\mathrm{Host}}﹩, ﹩{z}_{\mathrm{SN}+\mathrm{DCR}}﹩, and ﹩{z}_{\mathrm{SN}+\mathrm{DCR}+\mathrm{Host}}﹩. Combining all three reduces the bias throughout all redshifts.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: Histograms of the residuals Δz for the Realistic case for CDF50 (left) and PDFPeak (right). While the overall bias is significantly larger than in the Perfect case, we see little bias at z spec < 0.6.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption:  z DCR+Host vs. z spec and the 1σ error bars for CDF50 (top panels) and PDFPeak (bottom panels). We see noticeable improvement from Figure 12, especially at z spec < 0.6, where the z Host values are not very constraining.
Paper: Astrometric Redshifts of Supernovae
Authors: Jaemyoung (Jason) Lee, Masao Sako, Richard Kessler, Alex I. Malz
Figure caption: The celestial sphere for an observer in the Southern Hemisphere.
Paper: Large Dark Matter Content and Steep Metallicity Profile Predicted for Ultradiffuse Galaxies Formed in High-spin Halos
Authors: José A. Benavides, Laura V. Sales, Mario. G. Abadi, Mark Vogelsberger, Federico Marinacci, Lars Hernquist
Figure caption: Stellar mass as a function of halo virial mass for all simulated galaxies from TNG50. The color code indicates the redshift when the dark matter halo of each galaxy reached 50% (z 50) of its virial mass today (for satellites, the maximum virial mass as centrals is used instead). The population of galaxies has been separated into star forming (left panel) and quenched (right panel), with UDGs highlighted in open stars and circles, respectively. The median of all star-forming galaxies is indicated with a dashed black line in both panels, and the individual median for star-forming and quenched UDGs with solid blue and red lines, respectively. The median of all quenched galaxies is indicated with a thin dashed–dotted black line (only) in the right panel. Thin dotted lines correspond to multiplying the median of all star-forming dwarfs (long dashed line) by factors 0.5, 0.2, and 0.1 to show different levels of inefficiency at forming stars. Star-forming UDGs and massive quenched UDGs with M ⋆ ≥ 108.2 M ⊙ populate their halos with systematically less stellar mass than non-UDGs. The bias disappears for lower mass UDGs, whose median M ⋆–M 200 relation (thick solid red line) traces well the median of non-UDGs dwarfs. Note the abundance of UDGs forming two to five fewer stars than non-UDGs with similar halo mass. The green unfilled squares highlight three individual examples presented in Figure 4.
Paper: Large Dark Matter Content and Steep Metallicity Profile Predicted for Ultradiffuse Galaxies Formed in High-spin Halos
Authors: José A. Benavides, Laura V. Sales, Mario. G. Abadi, Mark Vogelsberger, Federico Marinacci, Lars Hernquist
Figure caption: Left: stellar mass vs. 3D stellar half-mass–radius relation. All simulated galaxies in TNG50 are presented in gray dots, as well as the catalog of simulated UDGs of J. A. Benavides et al. (2021, 2023), separated into star forming (blue stars) and quenched (red circles). For comparison with observations, we include a compilation of UDGs in clusters (P. G. van Dokkum et al. 2015a; S. Lim et al. 2020; A. Ferré-Mateu et al. 2023) and in isolation (L. Leisman et al. 2017; P. E. Mancera Piña et al. 2020; Y. Rong et al. 2020) with green symbols. Right: stellar mass assembly history of the population of simulated UDGs (solid lines) separated into star forming (blue) and quenched (red), as in the left panel. The profiles correspond to the medians of each sample, and the shaded regions to the 25th and 75th percentiles. The population of quenched UDGs assembled 50% (t 50) of their stellar mass ∼10 Gyr ago, while the star-forming galaxies just ∼4 Gyr ago. The simulated assembly histories seem in reasonably good agreement with quenched UDGs from low and high-density regions from A. Ferré-Mateu et al. (2023; orange/yellow dotted lines).
Paper: Large Dark Matter Content and Steep Metallicity Profile Predicted for Ultradiffuse Galaxies Formed in High-spin Halos
Authors: José A. Benavides, Laura V. Sales, Mario. G. Abadi, Mark Vogelsberger, Federico Marinacci, Lars Hernquist
Figure caption: Inefficiency of star formation Δ⋆ as a function of the redshift when each dark matter halo assembled 50% (z 50) of its final virial mass. Δ⋆ is the ratio between the expected stellar mass (M fit) and their true stellar mass, where M fit refers to the median of M ⋆ expected at a given M 200 from the non-UDG sample (thick black dashed curve in Figure 2). Star-forming galaxies are shown on the left panel, and quenched on the right. Blue stars and red circles highlight UDGs, and gray dots show non-UDGs galaxies. The median for each population is shown in dashed black (non-UDGs); blue or red solid line for UDGs star forming or quenched, respectively. Halos that assemble later show more stellar mass suppression (larger Δ⋆) for their halo mass.