Stars are hot objects, but they form within cold dense molecular clouds. A cold phase before star formation is necessary because at a given density gravity can take over only when the temperature is low enough. Molecular clouds are highly irregular objects. As gravitational collapse proceeds, a small amount of residual angular momentum can prevent the material from falling right into the forming star, and a flattened distribution of gas and dust surrounding the star is thus formed; that is a circumstellar disk. Matter in the disk may still be fed to the star, which is an important way for the star to gain mass. Solids collide and grow within these disks. As they become bigger and bigger, they form objects that are called planetesimals, protoplanets, and planets, some of which may be massive enough to collect a large amount of gas, and become a gas giant planet. We may say that planets are by-products of star formation.
With the advent of powerful observational facilities, including but not limited to the Kepler space mission  and the Atacama Large Millimeter/submillimeter Array (ALMA) , it is established that planets exist beyond our solar system — on average nearly every star hosts a planet , and signpost of early stages of planet formation are speculated based on rings and gaps alleged to be ripples created by protoplanets in dusty disks [4-6]. More than 30 different molecules have been detected in these disks . It is the task of theoreticians to explain the chemical richness (and sometimes scarcity) of these objects with a consistent picture.
Ultimately all gas-rich circumstellar disks will disappear, and solar systems like our own will emerge. The disk matter can be removed by accretion onto the star or onto forming planets, and by photoevaporation due to energetic photons from nearby massive stars. The exact timelines of these events are not clear yet.Ⅱ. BASICS OF DISK THERMOCHEMICAL MODELS
Astronomy is an observational science, but it has never been short of theoretical models ever since ancient times. Models are instrumental for extracting knowledge about reality from observational data. Without theoretical modeling a field of study would just be data-collecting.
At the moment, thermochemical modeling of protoplanetary disks are usually done assuming an axisymmetric geometry, and the disk is in hydrostatic equilibrium. Namely, with a cylindrical coordinate system, all physical quantities are only a function of r and z, and do not depend on the azimuthal angle. The word "thermochemical" means the chemical composition and temperature of each disk parcel may evolve with time, but the density and location of each parcel is taken to be static. This way of modeling is of course an approximation to reality1. As will be described in this review, there are signs that models of this type are approaching their limit, and need to be improved. There is another type of models that focus on the dynamical evolution instead of the chemical composition of the disk, in which the magnetohydrodynamic (MHD) equations are numerically solved. To evolve the disk dynamical structure in tandem with thermochemical structure (and maybe the radiation field also) is computationally challenging at present, though there are groups of modelers working on this.
1A quote attributed to the statistician George Box: "All models are wrong, but some are useful."
When modeling protoplanetary disks, there are some parameters that are essential for the calculation. Since the disk mass is usually not greater than a few percent of the central star mass, the disk gravitation can be neglected, and the central star provides gravitation for the whole system and holds the surrounding gas and dust together. So the stellar mass
For chemical modeling, the most essential input is the reaction network, which is usually a text file listing reactants, products, and rate parameters of each reaction. The networks for modeling protoplanetary disks are usually the same as those used for modeling the chemistry of interstellar molecular clouds. Common sources of networks are the UMIST network , the KIDA network , and the OSU network . Besides gas phase reactions among the neutrals, ions, and anions, adsorption and desorption from dust grains as well as chemical reactions on the dust grain surfaces have to be included.
Depending on the purpose of modeling and the intended degree of sophistication, there are many other parameters that could be included in the calculation. For example, the detailed dust size distribution, the cosmic ray intensity, and the background UV intensity, may be important in some situations.
These parameters are constrained by comparison between observational data and model results. For example, the stellar spectrum can be directly obtained with telescopes (usually by combining results from a few different telescopes, each working at different wavelengths). Comparing observed spectrum with model evolution tracks of stars of different mass, the mass (and age) of the star under study can be inferred. The procedure is similar for disks. The observed spectral energy distribution (SED) is compared with the modeled SED with some assumed dust and gas distribution. Through an iterative process the parameters describing the distribution can be obtained. These are the general ideas. In practice things are not as straightforward as it may seem, as will be detailed below.Ⅲ. MEASURING THE DISK MASS
As mentioned in the previous section, the dust and gas mass in a disk is important for modeling its chemistry. The dust mass distribution can be determined through fitting to the SED and/or image (possibly in Fourier space) . For the determination of gas mass, it is usually more challenging.
The paper by Bergin and Williams  is a very informative review on the mass determination of protoplanetary disks. When talking about disk masses, what is meant is usually the total mass, namely,
One choice is dust. At millimeter wavelengths of the electromagnetic spectrum, in most part of disks the dust emission is optically thin (however, an optically thick disk may appear to be optically thin due to dust scattering; see Zhu et al. ), which means that the emission flux from dust is proportional to the total dust mass . Multiplying the dust mass with a factor of 100 gives the gas mass. One caveat here is that the conversion from dust flux to dust mass, the opacity of the dust must be known, which is subject to uncertainty . Another point of caution is that measurement like this is insensitive to grains with size larger than
Carbon monoxide (CO) rotational transition lines are also commonly used. In the ISM the abundance of CO relative to hydrogen is
The HD molecule is arguably a better tracer of molecular hydrogen . It does not undergo complex chemical changes in well-shielded regions as we currently understand, and does not get adsorbed onto dust grain surface easily. The D/H abundance ratio is relatively well-measured to be
Molecular line studies of the protoplanetary disks are important in at least three aspects: (ⅰ) They provide chemical repository of the disks, which are raw materials for forming planets, and presumably, also for the origin of life. (ⅱ) Combined with thermochemical models, molecular spectra and images constrain the disk structure and evolutionary state. (ⅲ) Detailed inspection of the line profiles provides information about the dynamics of the disk, e.g., to infer the strength of turbulence  or the existence of planets in the disk . What will be described here touches upon the first and second points.
What has been found is that, for many sources, the observed intensity of a few molecular or atomic species can only be reproduced by models with low disk gas mass [18, 23, 24], or equivalently speaking, the abundances of these species must be significantly lower than model prediction if disk gas masses based on dust measurement are adopted in the models. Spectral lines involved in such types of studies include different transitions of O I , C I , C II , CO [18, 28, 29], and H
One subtlety needs to be emphasized. When talking about depletion in the context of molecular cloud chemistry, that usually means the species under study freezes out onto the surfaces of dust grains to form ice. The situation here is not exactly the same. In disk thermochemical models, the freezing out of gas phase species are consistently included, together with thermal desorption and non-desorption mechanisms involving cosmic rays and UV photons. So the problem is not that those species are not allowed to efficiently freeze out. On the surface layers of the disk, the scattered UV photons are always able to bring enough of them from the dust grain ice mantle into gas phase to become overabundant than the observed. So freezing-out alone is not enough to explain the disparity between observations and models.
The degree of depletion for different species may not be the same. In parameterized studies of Du et al.  and Bergin et al. , depletion is modeled through depletion of elemental carbon and oxygen in the calculation of chemical abundances. It is found that to fit the data, oxygen has to be depleted more than carbon. This is somewhat understandable, since the main bearer of oxygen— water— is more likely to freeze out than the bearers of carbon (CO, CO
The parameterized studies of Du et al.  and Bergin et al.  did not model the mechanism by which carbon and oxygen are reduced (and possibly by a different degree). Kama et al.  gave a semi-quantitative picture for the underlying mechanism. Dust grains in the disk tend to settle down towards the midplane of the disk. During this process, they grow in size by coagulation, and form ice mantle by adsorption. In this way carbon and oxygen are removed from the upper atmosphere of the disk. Some of the dust grains may be brought back to the upper layers by turbulence, but the absolute flux of upward flow of dust mass is lower than the downward one, because at the midplane the dust grains are bigger, and big grains cannot be easily brought into motion by gas flows.
Schwarz et al. [36, 37] studied the potential using chemical evolution alone to resolve the low-CO-abundance problem, and arrived at the conclusion that chemistry alone is not capable of reproducing the observed low CO emission, and other mechanisms are needed. The effect of dust evolution on gas phase chemistry is a candidate mechanism.
At present there are still technical challenges to create a physically and chemically consistent model in which the dust dynamical evolution and the evolution of gas chemistry are coupled. There are a few studies toward this direction. Krijt et al.  modeled the coupled evolution of dust grains and gas phase and solid phase water. For the dust grain dynamics, the super-particle and representative particle approach. Besides other findings, the C/O ratio in the disk upper layers is indeed found to increase to become greater than 1. This is qualitatively consistent with the phenomenological result of Du et al. . Radial drift of dust grains is not included in Krijt et al.'s work , though it may also play a role since there is indication that the C/O ratio has a radial gradient. This effect has been confirmed by the recent work of Krijt et al. , in which the CO abundance is found to be increased by a factor of a few in the inner disk. Cridland et al.  studied the evolution of ice lines by coupling dust evolution with chemistry. However, they did not include the effect that dust grains may act as vehicles to transport icy molecules over the disk, and the issue of volatile abundances is not within their theme of discussion.
Besides invoking dust evolution to change the distribution of different species in the disk, another possible solution is to incorporate a three-phase chemical model of gas-grain chemistry. Here "three-phase" means to include chemical processes in gas phase, on dust grain surfaces, and in dust grain mantles [41-43]. Apparently, since molecules are "hidden" in the dust grain mantles and cannot be directly evaporated into the gas phase, their gas phase abundances will be reduced. In fact, the effect of reduced evaporation rate due to mantle formation is taken into account in disk thermochemical models, e.g., in Du and Bergin  and subsequent papers of these authors. To be clear, these studies adopted a fairly complete surface reaction network at the level of Hasegawa et al.  and Hasegawa and Herbst  updated with recent experimental results. Nonthermal and photo-induced processes are also included. It appears that a simplistic three-phase model is not able to quantitatively reproduce observations, because molecules in the bulk of ice mantle can be turned into surface species once the surface layer species are removed. However, if mantle species are able to react with each other to form less volatile species, the gas phase abundances of many molecules of interest can indeed be significantly reduced . A detailed comparison between such a model and observational data is yet to be published to the best of knowledge of the author.Ⅵ. CONCLUSION
There are mismatches between the results of thermochemical models of protoplanetary disks — configured with disk gas mass constrained by observations — and observational data of different species. Nontrivial explanations are required. The most likely missing ingredient in the models is that the dynamical evolution of dust grains has to be coupled with the chemistry of gas phase species. While a lot of work still needs to be done in this direction, we may conclude that, by observing spectral lines of different species in the disk, it is possible to extract important information about the evolution status of the disk. Namely, in many disks the dust grains may have grown to a large size in the midplane, and they are likely to aggregate into planets.Ⅶ. ACKNOWLEDGMENTS
This work was supported by the "Hundred Talents Program" of Chinese Academy of Sciences, and the National Natural Science Foundation of China (No.11873094).
W. J. Borucki, D. Koch, G. Basri, N. Batalha, T. Brown, D. Caldwell, J. Caldwell, J. Christensen-Dalsgaard, W. D. Cochran, E. DeVore, E. W. Dunham, A. K. Dupree, T. N. Gautier, J. C. Geary, R. Gilliland, A. Gould, S. B. Howell, J. M. Jenkins, Y. Kondo, D. W. Latham, G. W. Marcy, S. Meibom, H. Kjeldsen, J. J. Lissauer, D. G. Monet, D. Morrison, D. Sasselov, J. Tarter, A. Boss, D. Brownlee, T. Owen, D. Buzasi, D. Charbonneau, L. Doyle, J. Fortney, E. B. Ford, M. J. Holman, S. Seager, J. H. Steffen, W. F. Welsh, J. Rowe, H. Anderson, L. Buchhave, D. Ciardi, L. Walkowicz, W. Sherry, E. Horch, H. Isaacson, M. E. Everett, D. Fischer, G. Torres, J. A. Johnson, M. Endl, P. MacQueen, S. T. Bryson, J. Dotson, M. Haas, Jeffrey Kolodziejczak, J. Van Cleve, H. Chandrasekaran, J. D. Twicken, E. V. Quintana, B. D. Clarke, C. Allen, J. Li, H. Wu, P. Tenenbaum, E. Verner, F. Bruhweiler, J. Barnes, and A. Prsa, Science 327, 977 (2010).
C. L. Brogan, L. M. Pérez, T. R. Hunter, W. R. F. Dent, A. S. Hales, R. E. Hills, S. Corder, E. B. Fomalont, C. Vlahakis, Y. Asaki, D. Barkats, A. Hirota, J. A. Hodge, C. M. V. Impellizzeri, R. Kneissl, E. Liuzzo, R. Lucas, N. Marcelino, S. Matsushita, K. Nakanishi, N. Phillips, A. M. S. Richards, I. Toledo, R. Aladro, D. Broguiere, J. R. Cortes, P. C. Cortes, D. Espada, F. Galarza, D. Garcia-Appadoo, L. Guzman-Ramirez, E. M. Humphreys, T. Jung, S. Kameno, R. A. Laing, S. Leon, G. Marconi, A. Mignano, B. Nikolic, L. A. Nyman, M. Radiszcz, A. Remijan, J. A. Rodón, T. Sawada, S. Takahashi, R. P. J. Tilanus, B. Vila Vilaro, L. C. Watson, T. Wiklind, E. Akiyama, E. Chapillon, I. de Gregorio-Monsalvo, J. Di Francesco, F. Gueth, A. Kawamura, C. F. Lee, Q. Nguyen Luong, J. Mangum, V. Pietu, P. Sanhueza, K. Saigo, S. Takakuwa, C. Ubach, T. van Kempen, A. Wootten, A. Castro-Carrizo, H. Francke, J. Gallardo, J. Garcia, S. Gonzalez, T. Hill, T. Kaminski, Y. Kurono, H. Y. Liu, C. Lopez, F. Morales, K. Plarre, G. Schieven, L. Testi, L. Videla, E. Villard, P. Andreani, J. E. Hibbard, and K. Tatematsu, Astrophys. J. Lett. 808, L3 (2015).
A. Cassan, D. Kubas, J. P. Beaulieu, M. Dominik, K. Horne, J. Greenhill, J. Wambsganss, J. Menzies, A. Williams, U. G. Jørgensen, A. Udalski, D. P. Bennett, M. D. Albrow, V. Batista, S. Brillant, J. A. R. Caldwell, A. Cole, Coutures Ch., K. H. Cook, S. Dieters, D. Dominis Prester, J. Donatowicz, P. Fouqué, K. Hill, N. Kains, S. Kane, J. B. Marquette, R. Martin, K. R. Pollard, K. C. Sahu, C. Vinter, D. Warren, B. Watson, M. Zub, T. Sumi, M. K. Szymański, M. Kubiak, R. Poleski, I. Soszynski, K. Ulaczyk, and G. Pietrzyński, and Ł. Wyrzykowski, Nature 481, 167(2012).
S. M. Andrews, J. Huang, L. M. Pérez, A. Isella, C. P. Dullemond, Nicolás T. Kurtovic, V. V. Guzmán, J. M. Carpenter, D. J. Wilner, S. Zhang, Z. Zhu, T. Birnstiel, X. N. Bai, M. Benisty, A. Meredith Hughes, K. I. Öberg, and L. Ricci, Astrophys. J. Lett. 869, L41 (2018).
S. Zhang, Z. Zhu, J. Huang, V. V. Guzmán, S. M. Andrews, T. Birnstiel, C. P. Dullemond, J. M. Carpenter, A. Isella, L. M. Pérez, M. Benisty, D. J. Wilner, C. Baruteau, X. N. Bai, and L. Ricci, Astrophys. J. Lett. 869, L47 (2018).
Y. Liu, G. Dipierro, E. Ragusa, G. Lodato, G. J. Herczeg, F. Long, D. Harsono, Y. Boehler, F. Menard, D. Johnstone, I. Pascucci, P. Pinilla, C. Salyk, G. van der Plas, S. Cabrit, W. J. Fischer, N. Hendler, C. F. Manara, B. Nisini, E. Rigliaco, H. Avenhaus, A. Banzatti, and M. Gully-Santiago, A & A 622, A75 (2019).
B. A. McGuire, Astrophys. J. Suppl. Ser. 239, 17(2018). DOI:10.3847/1538-4365/aae5d2
D. McElroy, C. Walsh, A. J. Markwick, M. A. Cordiner, K. Smith, and T. J. Millar, A & A 550, A36 (2013).
V. Wakelam, J. C. Loison, E. Herbst, B. Pavone, A. Bergeat, K. Béroff, M. Chabot, A. Faure, D. Galli, W. D. Geppert, D. Gerlich, P. Gratier, N. Harada, K. M. Hickson, P. Honvault, S. J. Klippenstein, S. D. Le Picard, G. Nyman, M. Ruaud, S. Schlemmer, I. R. Sims, D. Talbi, J. Tennyson, and R. Wester, Astrophys. J. Suppl. Ser. 217, 20(2015). DOI:10.1088/0067-0049/217/2/20
R. T. Garrod, S. L. W. Weaver, and E. Herbst, ApJ 682, 283(2008). DOI:10.1086/588035
S. M. Andrews, D. J. Wilner, A. M. Hughes, C. Qi, and C. P. Dullemond, ApJ 700, 1502(2009). DOI:10.1088/0004-637X/700/2/1502
E. A. Bergin and J. P. Williams, The Determination of Protoplanetary Disk Masses, M. Pessah and O. Gressel Eds. Formation, Evolution, and Dynamics of Young Solar Systems, volume 445, Springer, (2017).
Z. Zhu, S. Zhang, Y. F. Jiang, A. Kataoka, T. Birnstiel, C. P. Dullemond, S. M. Andrews, J. Huang, L. M. Pérez, J. M. Carpenter, X. N. Bai, D. J. Wilner, and L. Ricci, Astrophys. J. Lett. 877, L18 (2019).
B. T. Draine, ApJ 636, 1114 (2006).
S. V. W. Beckwith, A. I. Sargent, R. S. Chini, and R. Guesten, Astron. J. 99, 924(1990). DOI:10.1086/115385
I. Kamp, W. F. Thi, G. Meeus, P. Woitke, C. Pinte, R. Meijerink, M. Spaans, I. Pascucci, G. Aresu, and W. R. F. Dent, A & A 559, A24 (2013).
J. P. Williams, and W. M. J. Best, Astrophys. J. 788, 59(2014). DOI:10.1088/0004-637X/788/1/59
C. Favre, L. I. Cleeves, E. A. Bergin, C. Qi, and G. A. Blake, Astrophys. J. Lett. 776, L38 (2013).
E. A. Bergin, L. I. Cleeves, U. Gorti, K. Zhang, G. A. Blake, J. D. Green, S. M. Andrews, N. J. Evans Ⅱ, T. Henning, K. Öberg, K. Pontoppidan, C. Qi, C. Salyk, and E. F. van Dishoeck, Nature 493, 644
J. L. Linsky, B. T. Draine, H. W. Moos, E. B. Jenkins, B. E. Wood, C. Oliveira, W. P. Blair, S. D. Friedman, C. Gry, D. Knauth, J. W. Kruk, S. Lacour, N. Lehner, S. Redfield, J. M. Shull, G. Sonneborn, and G. M. Williger, Astrophys. J. 647, 1106(2006). DOI:10.1086/505556
A. M. Hughes, D. J. Wilner, S. M. Andrews, C. Qi, and M. R. Hogerheijde, Astrophys. J. 727, 85(2011). DOI:10.1088/0004-637X/727/2/85
R. Teague, J. Bae, and E. A. Bergin, Nature 574, 378(2019). DOI:10.1038/s41586-019-1642-0
F. Du, E. A. Bergin, M. Hogerheijde, E. F. van Dishoeck, G. Blake, S. Bruderer, I. Cleeves, C. Dominik, D. Fedele, D. C. Lis, G. Melnick, D. Neufeld, J. Pearson, and U. Yıldız, Astrophys. J. 842, 98(2017). DOI:10.3847/1538-4357/aa70ee
M. K. McClure, arXiv e-prints, art.arXiv: 1910.06029, (2019).
G. Aresu, I. Kamp, R. Meijerink, M. Spaans, S. Vicente, L. Podio, P. Woitke, F. Menard, W. F. Thi, M. Güdel, and A. Liebhart, A & A 566, A14 (2014).
T. Tsukagoshi, M. Momose, M. Saito, Y. Kitamura, Y. Shimajiri, and R. Kawabe, Astrophys. J. Lett. 802, L7 (2015).
W. F. Thi, G. Mathews, F. Ménard, P. Woitke, G. Meeus, P. Riviere-Marichalar, C. Pinte, C. D. Howard, A. Roberge, G. Sandell, I. Pascucci, B. Riaz, C. A. Grady, W. R. F. Dent, I. Kamp, G. Duchêne, J. C. Augereau, E. Pantin, B. Vandenbussche, I. Tilling, J. P. Williams, C. Eiroa, D. Barrado, J. M. Alacid, S. Andrews, D. R. Ardila, G. Aresu, S. Brittain, D. R. Ciardi, W. Danchi, D. Fedele, I. de Gregorio-Monsalvo, A. Heras, N. Huelamo, A. Krivov, J. Lebreton, R. Liseau, C. Martin-Zaidi, I. Mendigutía, B. Montesinos, A. Mora, M. Morales-Calderon, H. Nomura, N. Phillips, L. Podio, D. R. Poelman, S. Ramsay, K. Rice, E. Solano, H. Walker, G. J. White, and G. Wright, A & A 518, L125 (2010).
A. Dutrey, S. Guilloteau, and M. Guelin, A & A 317, L55 (1997).
K. R. Schwarz, E. A. Bergin, L. I. Cleeves, G. A. Blake, K. Zhang, K. I. Öberg, E. F. van Dishoeck, and C. Qi, Astrophys. J. 823, 91(2016). DOI:10.3847/0004-637X/823/2/91
K. Zhang, K. M. Pontoppidan, and C. Salyk, and G. A. Blake. Astrophys. J. 766, 82(2013).
M. R. Hogerheijde, E. A. Bergin, C. Brinch, L. I. Cleeves, J. K. J. Fogel, G. A. Blake, C. Dominik, D. C. Lis, G. Melnick, D. Neufeld, O. Panić, J. C. Pearson, L. Kristensen, U. A. Yıldız, and E. F. van Dishoeck, Science 334, 338 (2011).
M. K. McClure, E. A. Bergin, L. I. Cleeves, E. F. van Dishoeck, G. A. Blake, N. J. Evans Ⅱ, J. D. Green, T. Henning, K. I. Öberg, K. M. Pontoppidan, and C. Salyk, Astrophys. J. 831, 167(2016). DOI:10.3847/0004-637X/831/2/167
F. Du, E. A. Bergin, and M. R. Hogerheijde, Astrophys. J. Lett. 807, L32 (2015).
E. A. Bergin, F. Du, L. I. Cleeves, G. A. Blake, K. Schwarz, R. Visser, and K. Zhang, Astrophys. J. 831, 101(2016). DOI:10.3847/0004-637X/831/1/101
M. Kama, S. Bruderer, E. F. van Dishoeck, M. Hogerheijde, C. P. Folsom, A. Miotello, D. Fedele, A. Belloche, R. Güsten, and F. Wyrowski, A & A 592, A83 (2016).
K. R. Schwarz, E. A. Bergin, L. I. Cleeves, K. Zhang, K. I. Öberg, G. A. Blake, and D. Anderson, Astrophys. J. 856, 85(2018). DOI:10.3847/1538-4357/aaae08
K. R. Schwarz, E. A. Bergin, L. I. Cleeves, K. Zhang, K. I. Öberg, G. A. Blake, and D. E. Anderson, Astrophys. J. 877, 131(2019). DOI:10.3847/1538-4357/ab1c5e
S. Krijt, F. J. Ciesla, and E. A. Bergin, Astrophys. J. 833, 285(2016). DOI:10.3847/1538-4357/833/2/285
S. Krijt, K. R. Schwarz, E. A. Bergin, and F. J. Ciesla, Astrophys. J. 864, 78(2018). DOI:10.3847/1538-4357/aad69b
A. J. Cridland, R. E. Pudritz, and T. Birnstiel, MNRAS 465, 3865(2017). DOI:10.1093/mnras/stw2946
Q. Chang, and E. Herbst, Astrophys. J. 787, 135(2014). DOI:10.1088/0004-637X/787/2/135
M. Ruaud, and U. Gorti, Astrophys. J. 885, 146(2019). DOI:10.3847/1538-4357/ab4996
V. Wakelam, E. Chapillon, A. Dutrey, S. Guilloteau, W. Iqbal, A. Coutens, and L. Majumdar, MNRAS 484, 1563(2019). DOI:10.1093/mnras/stz068
F. Du, and E. A. Bergin, Astrophys. J. 792, 2(2014). DOI:10.1088/0004-637X/792/1/2
T. I. Hasegawa, E. Herbst, and C. M. Leung, Astrophys. J. Suppl. Ser. 82, 167(1992). DOI:10.1086/191713
T. I. Hasegawa, and E. Herbst, MNRAS 261, 83(1993). DOI:10.1093/mnras/261.1.83