surface values where the material is illuminated from basically one side.

An alternative way to illustrate this result is by substituting $M_{gas} = 2\sigma^2 f_g r_0/G$ in Equation (26), as is appropriate for an isothermal sphere, and $r_0 \approx (L_{EDD}/4\pi\sigma_{SB}T_{dust}^4)^{1/2}$, where $\sigma_{SB}$ is the Stefan-Boltzmann constant, yielding

$$L_{\text{EDD}} = \frac{16\pi c^2}{k_{\text{IR}}^2 \sigma_{\text{SB}} T_{\text{dust}}^4} \sigma^4 = 1.34 \times 10^{13} \times \left(\frac{100 \text{ K}}{T_{\text{dust}}}\right)^4 \\ \times \left(\frac{5 \text{ cm}^2/\text{g}}{k_{\text{IR}}}\right)^2 \times \left(\frac{\sigma}{200 \text{ km s}^{-1}}\right)^4 L_{\odot}. \quad (27)$$

Figure 32(c) shows the IR luminosity as a function of the velocity dispersion $\sigma$, which is here measured from Gaussian fits to the high-lying OH65 doublet to filter the nuclear regions, for all (U)LIRGs where OH65 is detected (GA15). The OH65 doublet is in part broadened owing to radial motions, but $\sigma$ is not corrected for inclination effects, and thus the intrinsic values are higher than in Figure 32(c); in addition, the $L_{IR}$ values are also upper limits because a fraction of the luminosity most likely arises from more extended regions. The lines indicate the $L_{EDD}$ values for $T_{dust} = 100$ and $150$ K as given by Equation (27). The results are roughly consistent with Figure 32(b), i.e., radiation pressure support in the vertical direction of the region sampled by OH65 is expected for $T_{dust} = 100-150$ K, coinciding with the highest values of $T_{dust}$ inferred from the high-lying OH and H$_2$O lines in some sources, but this is insufficient to drive an outflow.

Using now the cloud approach in Thompson et al. (2015, their Equation (38)), the ratio of Eddington luminosity to gas mass is

$$\frac{L_{\text{EDD}}}{M_{\text{gas}}} = \frac{2\pi c G \mu m_{\text{H}} N_{\text{H}}}{f_g} = 1.03 \times 10^4 \\ \times \left(\frac{0.15}{f_g}\right) \times \left(\frac{N_{\text{H}}}{10^{23} \text{ cm}^{-2}}\right) L_{\odot}/M_{\odot}, \quad (28)$$

where $N_{\text{H}}$ is the column density of the cloud, which is assumed to have $\tau_{IR} \gtrsim 1$. Writing again $M_{\text{gas}}$ in terms of $\sigma$ and $T_{dust}$, we find

$$L_{\text{EDD}} = \frac{4\pi c^2 \mu^2 m_{\text{H}}^2 N_{\text{H}}^2}{\sigma_{\text{SB}} T_{\text{dust}}^4} \sigma^4 \\ = 4.60 \times 10^{12} \times \left(\frac{100 \text{ K}}{T_{\text{dust}}}\right)^4 \\ \times \left(\frac{N_{\text{H}}}{10^{23} \text{ cm}^{-2}}\right)^2 \times \left(\frac{\sigma}{200 \text{ km s}^{-1}}\right)^4 L_{\odot}. \quad (29)$$

We have assumed here that the impinging radiation field on the cloud is UV dominated (Thompson et al. 2015), but we note that the outflows are observed in absorption lines against a strong far-IR field, and thus the radiation has been already reprocessed and reemitted at far-IR wavelengths. Hence, Equations (28) and (29) probably underestimate $L_{EDD}/M_{gas}$ and $L_{EDD}$ by a factor of $\sim 2$.

## Appendix D: Velocity Field

In GA14, we favored a decelerating velocity field in Mrk 231 based on three arguments: (i) OH84 and OH65 showed absorption at extreme velocities ($\lesssim -1000 \text{ km s}^{-1}$), indicating the presence of high-velocity gas close to the nuclear source of far-IR radiation, while the extended outflowing component probed by OH119 and OH79 indicated expanding velocities significantly lower, up to $\approx 900 \text{ km s}^{-1}$. (ii) CO (1–0) and (2–1) emission in the line wings, tracing an extended outflow, was found up to $\approx 800 \text{ km s}^{-1}$ (Cicone et al. 2012), significantly lower than OH. (iii) In our nonclumpy spherical models, the line profiles are better fit with decelerating flows.

We examine here whether the decelerating flows are also applicable to the other ULIRGs of our sample. Figure 10 indicates that, in all sample sources except Mrk 273 and IRAS 08572+3915, the ground-state OH119 and OH79 doublets peak at more blueshifted velocities than the excited OH84 and OH65. In addition, Figure 7 shows that, in all sources except IRAS 03158+4227, Mrk 231, IRAS 08572+3915, and IRAS 20100–4156 (and IRAS 09022–3615, where OH65 is not detected), OH119 shows absorption well in excess of $500 \text{ km s}^{-1}$, but the OH65 absorption is restricted to expanding velocities $v < 500 \text{ km s}^{-1}$. Since the very excited OH65 is formed very close to the nuclear source of strong far-IR emission and OH119 is generally generated farther from it, does this observational evidence indicate that the gas is accelerated as it moves away from the nuclear region?

Not necessarily. Besides high $T_{dust}$, significant absorption of OH65 requires high columns of gas. If the gas column per unit of velocity decreases with increasing outflowing velocity, it is expected that the high-velocity gas will not be detected in OH65 owing to insufficient columns, even if a significant fraction of it is located close to the nuclear region. In addition, absorption and reemission in the ground-state doublets over large volumes can bring the intensity close to zero at central velocities, thus shifting the velocity of the peak absorption to high velocities, while this effect cannot happen in the excited lines (Section 4.5).

For instance, based on our models of IRAS 14378–3651 (see Table 2), the high-velocity component has a moderate column of $N_{OH} = 3 \times 10^{16} \text{ cm}^{-2}$ and is not expected to generate any OH65 absorption even though it is located only 200–300 pc from the center. On the other hand, in IRAS 20551–4250 and IRAS 10565+2448, both the high- and low-velocity components have similar spatial distributions. Although our model components also include accelerated flows, the best fits are mostly found for decelerating or constant-velocity fields (except in two components), though this may be an artificial effect of our nonclumpy approach.

Our favored interpretation for the apparent higher spatial extent of the high-velocity relative to the low-velocity gas in our sample galaxies is not necessarily that the gas is smoothly accelerated from the central region, but that the different doublets trace components with different columns that have been accelerated to different velocities. The escape velocity is given by $v_{esc} \approx 200 \sqrt{M_9/R_{200}} \text{ km s}^{-1}$, where $M_9 = M/10^9 M_{\odot}$ and $R_{200} = R/200$ pc, so that a significant fraction of the low-velocity gas may not escape the central potential well of the host galaxy, while the high-velocity gas will escape and be observed at larger distances.
References

Aalto, S., García-Burillo, S., Muller, S., et al. 2012, A&A, 537, A44

Aalto, S., García-Burillo, S., Muller, S., et al. 2015, A&A, 574, A85

Adams, F. C., Graff, D. S., & Richstone, D. O. 2001, ApJL, 551, L31

Alatalo, K., Blitz, L., Young, L. M., et al. 2011, ApJ, 735, 88

Alatalo, K., Lacy, M., Lanz, L., et al. 2015, ApJ, 798, 31

Arav, N., Borguet, B., Chamberlain, C., Edmonds, D., & Danforth, C. 2013, MNRAS, 436, 3286

Baldry, I. K., Glazebrook, K., Brinkmann, J., et al. 2004, ApJ, 600, 681

Beifiorsi, A., Courteau, S., Corsini, E. M., & Zhu, Y. 2012, MNRAS, 419, 2497

Bolatto, A. D., Downes, S. R., Leroy, A. K., et al. 2013, *Natur*, 499, 450

Borguet, B. C. J., Arav, N., Edmonds, D., Chamberlain, C., & Benn, C. 2013, ApJ, 762, 49

Burkert, A., & Silk, J. 2001, ApJL, 554, L151

Bushouse, H. A., Borne, K. D., Colina, L., et al. 2002, ApJS, 138, 1

Cicone, C., Feruglio, C., Maiolino, R., et al. 2012, A&A, 543, A99

Cicone, C., Maiolino, R., Sturm, E., et al. 2014, A&A, 562, A21

Combes, F., García-Burillo, S., Braine, J., et al. 2013, A&A, 550, A41

Costagliola, F., Aalto, S., Sakamoto, K., et al. 2013, A&A, 556, A66

Dasyra, K. M., Tacconi, L. J., Davies, R. I., et al. 2006, ApJ, 651, 835

Davies, R. I., Tacconi, L. J., & Genzel, R. 2004, ApJ, 613, 781

DeBuhr, J., Quataert, E., & Ma, C.-P. 2012, MNRAS, 420, 2221

Dekel, A., & Birnboim, Y. 2006, MNRAS, 368, 2

di Matteo, T., Springel, V., & Hernquist, L. 2005, *Natur*, 433, 604

Díaz-Santos, T., Assef, R. J., Blain, A. W., et al. 2016, ApJL, 816, L6

Downes, D., & Solomon, P. M. 1998, ApJ, 507, 615

Downes, D., Solomon, P. M., & Radford, S. J. E. 1993, ApJL, 414, L13

Duc, P.-A., Mirabel, I. F., & Maza, J. 1997, A&AS, 124, 533

Efstathiou, A., Pearson, C., Farrah, D., et al. 2014, MNRAS, 437, L16

Evans, A. S., Mazzarella, J. M., Surace, J. A., & Sanders, D. B. 2002, ApJ, 580, 749

Evans, A. S., Surace, J. A., & Mazzarella, J. M. 2000, ApJL, 529, L85

Fabian, A. C. 1999, MNRAS, 308, L39

Fabian, A. C. 2012, ARA&A, 50, 455

Fabian, A. C., & Iwasawa, K. 1999, MNRAS, 303, L34

Falstad, N., González-Alfonso, E., Aalto, S., et al. 2015, A&A, 580, A52

Falstad, N., González-Alfonso, E., Aalto, S., & Fischer, J. 2017, A&A, 597, A105

Farrah, D., Bernard-Salas, J., Spoon, H. W. W., et al. 2007, ApJ, 667, 149

Farrah, D., Lebouteiller, V., Spoon, H. W. W., et al. 2013, ApJ, 776, 38

Farrah, D., Urrutia, T., Lacy, M., et al. 2012, ApJ, 745, 178

Faucher-Giguère, C.-A., & Quataert, E. 2012, MNRAS, 425, 605

Ferrarese, L., & Ford, H. 2005, SSRv, 116, 523

Ferrarese, L., & Merritt, D. 2000, ApJL, 539, L9

Feruglio, C., Fiore, F., Carniani, S., et al. 2015, A&A, 583, A99

Feruglio, C., Maiolino, R., Piconcelli, E., et al. 2010, A&A, 518, L155

Fischer, J., Sturm, E., González-Alfonso, et al. 2010, A&A, 518, L41

Franceschini, A., Braito, V., Persic, M., et al. 2003, MNRAS, 343, 1181

García-Burillo, S., Combes, F., Usero, A., et al. 2015, A&A, 580, A35

García-Marín, M., Colina, L., Arribas, S., Alonso-Herrero, A., & Mediavilla, E. 2006, ApJ, 650, 850

Gebhardt, K., Bender, R., Bower, G., et al. 2000, ApJL, 539, L13

Genzel, R., Lutz, D., Sturm, E., et al. 1998, ApJ, 498, 579

Genzel, R., Tacconi, L. J., Rigopoulou, D., Lutz, D., & Tecza, M. 2001, ApJ, 563, 527

Goicoechea, J. R., & Cernicharo, J. 2002, ApJL, 576, L77

Goicoechea, J. R., Cernicharo, J., Lerate, M. R., et al. 2006, ApJL, 641, L49

Goicoechea, J. R., Joblin, C., Contursi, A., et al. 2011, A&A, 530, L16

González-Alfonso, E., & Cernicharo, J. 1997, A&A, 322, 938

González-Alfonso, E., & Cernicharo, J. 1999, ApJ, 525, 845

González-Alfonso, E., Fischer, J., Aalto, S., & Falstad, N. 2014a, A&A, 567, A91

González-Alfonso, E., Fischer, J., Bruderer, S., et al. 2013, A&A, 550, A25

González-Alfonso, E., Fischer, J., Sturm, E., et al. 2015, ApJ, 800, 69

González-Alfonso, E., Fischer, J., Graciá-Carpio, J., et al. 2012, A&A, 541, A4

González-Alfonso, E., Fischer, J., Graciá-Carpio, J., et al. 2014b, A&A, 561, A27 (GA14)

Graciá-Carpio, J., Sturm, E., Hailey-Dunsheath, S., et al. 2011, ApJL, 728, L7

Graham, A. W., & Driver, S. P. 2007, ApJ, 655, 77

Graham, A. W., Erwin, P., Caon, N., & Trujillo, I. 2001, ApJL, 563, L11

Guillard, P., Boulanger, F., Lehnert, M. D., et al. 2015, A&A, 574, A32

Haan, S., Surace, J. A., Armus, L., et al. 2011, AJ, 141, 100

Harrison, C. M., Alexander, D. M., Mullaney, J. R., & Swinbank, A. M. 2014, MNRAS, 441, 3306

Harvey-Smith, L., Allison, J. R., Green, J. A., et al. 2016, MNRAS, 460, 2180

Heckman, T. M., Alexandroff, R. M., Borthakur, S., Overzier, R., & Leitherer, C. 2015, ApJ, 809, 147

Heckman, T. M., Armus, L., & Miley, G. K. 1990, ApJS, 74, 833

Hopkins, P. F., Cox, T. J., Younger, J. D., & Hernquist, L. 2009, ApJ, 691, 1168

Hopkins, P. F., Hernquist, L., Cox, T. J., et al. 2006a, ApJS, 163, 1

Hopkins, P. F., Hernquist, L., Cox, T. J., Robertson, B., & Springel, V. 2006b, ApJS, 163, 50

Imanishi, M., Nakagawa, T., Ohyama, Y., et al. 2008, PASJ, 60, 489

Imanishi, M., Nakanishi, K., & Izumi, T. 2016, AJ, 152, 218

Ishibashi, W., & Fabian, A. C. 2015, MNRAS, 451, 93

Iwasawa, K., Sanders, D. B., Teng, S. H., et al. 2011, A&A, 529, A106

Jahnke, K., & Macciò, A. V. 2011, ApJ, 734, 92

Janssen, A. W., Christopher, N., Sturm, E., et al. 2016, ApJ, 822, 43

Khochfar, S., & Ostriker, J. 2008, ApJ, 680, 54

Kim, D.-C., Veilleux, S., & Sanders, D. B. 1998, ApJ, 508, 627

King, A. 2003, ApJL, 596, L27

King, A. 2005, ApJL, 635, L121

King, A., & Pounds, K. 2015, ARA&A, 53, 115

Lang, P., Wuyts, S., Somerville, R. S., et al. 2014, ApJ, 788, 11

Lee, J. C., Hwang, H. S., Lee, M. G., Kim, M., & Kim, S. C. 2011, MNRAS, 414, 702

Lee, J. C., Hwang, H. S., Lee, M. G., Kim, M., & Lee, J. H. 2012, ApJ, 756, 95

Leitherer, C., Kraeher, D., Goldader, J. D., et al. 1999, ApJS, 123, 3

Lindberg, J. E., Aalto, S., Muller, S., et al. 2016, A&A, 587, A15

Lípari, S., Sánchez, S. F., Bergmann, M., et al. 2009, MNRAS, 392, 1295

Lípari, S., Terlevich, R., Zheng, W., et al. 2005, MNRAS, 360, 416

Magorrian, J., Tremaine, S., Richstone, D., et al. 1998, AJ, 115, 2285

Maiolino, R., Gallerani, S., Neri, R., et al. 2012, MNRAS, 425, 66

Marconi, A., & Hunt, L. K. 2003, ApJL, 589, L21

Martin, C. L., & Soto, K. 2016, ApJ, 819, 49

Martín, S., Aalto, S., Sakamoto, K., et al. 2016, A&A, 590, A25

Martizzi, D., Fielding, D., Faucher-Giguère, C.-A., & Quataert, E. 2016, MNRAS, 459, 2311

Meijerink, R., Spaans, M., Loenen, A. F., & van der Werf, P. P. 2011, A&A, 525, A119

Meusinger, H., Stecklum, B., Theis, C., & Brunzendorf, J. 2001, A&A, 379, 845

Murphy, T. W., Jr., Armus, L., Matthews, K., et al. 1996, AJ, 111, 1025

Murray, N., Quataert, E., & Thompson, T. A. 2005, ApJ, 618, 569

Nardini, E., Risaliti, G., Watabe, Y., Salvati, M., & Sani, E. 2010, MNRAS, 405, 2505

Peng, C. Y. 2007, ApJ, 671, 1098

Peng, Y., Maiolino, R., & Cochrane, R. 2015, *Natur*, 521, 192

Pereira-Santaella, M., Colina, L., García-Burillo, S., et al. 2016, A&A, 594, A81

Pilbratt, G. L., Riedinger, J. R., Passvogel, T., et al. 2010, A&A, 518, L1

Poglitsch, A., Waelkens, C., Geis, N., et al. 2010, A&A, 518, L2

Privon, G. C., Aalto, S., Falstad, N., et al. 2016, ApJ, in press (arXiv:1612.04401)

Risaliti, G., Gilli, R., Maiolino, R., & Salvati, M. 2000, A&A, 357, 13

Rodríguez Zaurín, J., Tadhunter, C. N., Rose, M., & Holt, J. 2013, MNRAS, 432, 138

Roth, N., Kasen, D., Hopkins, P. F., & Quataert, E. 2012, ApJ, 759, 36

Rupke, D. S., & Veilleux, S. 2013a, ApJ, 768, 75

Rupke, D. S., & Veilleux, S. 2013b, ApJL, 775, L15

Rupke, D. S., Veilleux, S., & Sanders, D. B. 2002, ApJ, 570, 588

Rupke, D. S., Veilleux, S., & Sanders, D. B. 2005a, ApJ, 632, 751

Rupke, D. S., Veilleux, S., & Sanders, D. B. 2005b, ApJS, 160, 87

Rupke, D. S., Veilleux, S., & Sanders, D. B. 2005c, ApJS, 160, 115

Sakamoto, K., Aalto, S., Costagliola, F., et al. 2013, ApJ, 764, 42

Sakamoto, K., Aalto, S., Wilner, D. J., et al. 2009, ApJL, 700, L104

Sanders, D. B., Mazzarella, J. M., Kim, D.-C., Surace, J. A., & Soifer, B. T. 2003, AJ, 126, 1607

Sanders, D. B., Soifer, B. T., Elias, J. H., et al. 1988, ApJ, 325, 74

Sanders, R. H. 1989, IAUS, 136, 77

Sani, E., & Nardini, E. 2012, AdAst, 2012, 783451

Schawinski, K., Urry, C. M., Simmons, B. D., et al. 2014, MNRAS, 440, 889

Scoville, N. Z., Evans, A. S., Thompson, R., et al. 2000, AJ, 119, 991

Severgnini, P., & Risaliti, G. 2000, MemSAI, 72, 63

Shankar, F., Bernardi, M., Sheth, R. K., et al. 2016, MNRAS, 460, 3119

Shih, H.-Y., & Rupke, D. S. N. 2010, ApJ, 724, 1430

Shirahata, M., Nakagawa, T., Usuda, T., et al. 2013, PASJ, 65, 5

Silk, J., & Rees, M. J. 1998, A&A, 331, L1

Solomon, P. M., Downes, D., Radford, S. J. E., & Barrett, J. W. 1997, ApJ, 478, 144
Spoon, H. W. W., Farrah, D., Lebouteiller, V., et al. 2013, ApJ, 775, 127

Spoon, H. W. W., & Holt, J. 2009, ApJL, 702, L42

Springel, V., Di Matteo, T., & Hernquist, L. 2005, ApJL, 620, L79

Stern, J., Faucher-Giguère, C.-A., Zakamska, N. L., & Hennawi, J. F. 2016, ApJ, 819, 130

Sternberg, A., & Dalgarno, A. 1995, ApJS, 99, 565

Stierwalt, S., Armus, L., Surace, J. A., et al. 2013, ApJS, 206, 1

Stone, M., Veilleux, S., Meléndez, M., et al. 2016, ApJ, 826, 111

Strateva, I., Ivezić, Z., Knapp, G., et al. 2001, AJ, 122, 1861

Sturm, E., González-Alfonso, E., Veilleux, S., et al. 2011, ApJL, 733, L16

Surace, J. A., Sanders, D. B., & Mazzarella, J. M. 2004, AJ, 127, 3235

Tacchella, S., Carollo, C. M., Renzini, A., et al. 2015, Sci, 348, 314

Tacconi, L. J., Genzel, R., Lutz, D., et al. 2002, ApJ, 580, 73

Teng, S. H., Rigby, J. R., Stern, D., et al. 2015, ApJ, 814, 56

Thompson, T. A., Fabian, A. C., Quataert, E., & Murray, N. 2015, MNRAS, 449, 147

Thompson, T. A., Quataert, E., & Murray, N. 2005, ApJ, 630, 167

Tombesi, F., Meléndez, M., Veilleux, S., et al. 2015, Natur, 519, 436

Tremaine, S., Gebhardt, K., Bender, R., et al. 2002, ApJ, 574, 740

Tunnard, R., Greve, T. R., García-Burillo, S., et al. 2015, ApJ, 800, 25

Veilleux, S., Cecil, G., & Bland-Hawthorn, J. 2005, ARA&A, 43, 769

Veilleux, S., Kim, D.-C., & Sanders, D. B. 1999, ApJ, 522, 113

Veilleux, S., Kim, D.-C., Sanders, D. B., Mazzarella, J. M., & Soifer, B. T. 1995, ApJS, 98, 171

Veilleux, S., Meléndez, M., Sturm, E., et al. 2013, ApJ, 776, 27

Veilleux, S., Rupke, D. S. N., Kim, D.-C., et al. 2009, ApJS, 182, 628

Veilleux, S., Sanders, D. B., & Kim, D.-C. 1997, ApJ, 484, 92

Véron-Cetty, M.-P., & Véron, P. 2006, A&A, 455, 773

Weaver, R., McCray, R., Castor, J., Shapiro, P., & Moore, R. 1977, ApJ, 218, 377

Zubovas, K., & King, A. 2012, ApJL, 745, L34

Zubovas, K., & Nayakshin, S. 2014, MNRAS, 440, 2625