A Theoretical Framework for the Mass Distribution of Gas Giant Planets forming through the Core Accretion Paradigm
AA Theoretical Framework for the Mass Distribution of Gas GiantPlanets forming through the Core Accretion Paradigm
Fred C. Adams, , Michael R. Meyer, and Arthur D. Adams Physics Department, University of Michigan, Ann Arbor, MI 48109 Astronomy Department, University of Michigan, Ann Arbor, MI 48109updated: January 2021
ABSTRACT
This paper constructs a theoretical framework for calculating the distributionof masses for gas giant planets forming via the core accretion paradigm. Startingwith known properties of circumstellar disks, we present models for the planetarymass distribution over the range 0 . M J ≤ M p < M J . If the circumstellar disklifetime is solely responsible for the end of planetary mass accretion, the observed(nearly) exponential distribution of disk lifetime would imprint an exponentialfall-off in the planetary mass function. This result is in apparent conflict withobservations, which suggest that the mass distribution has a (nearly) power-lawform dF/dM p ∼ M − p p , with index p ≈ .
3, over the relevant planetary mass range(and for stellar masses ∼ . − M (cid:12) ). The mass accretion rate onto the planetdepends on the fraction of the (circumstellar) disk accretion flow that enters theHill sphere, and on the efficiency with which the planet captures the incomingmaterial. Models for the planetary mass function that include distributions forthese efficiencies, with uninformed priors, can produce nearly power-law behavior,consistent with current observations. The disk lifetimes, accretion rates, andother input parameters depend on the mass of the host star. We show how thesevariations lead to different forms for the planetary mass function for differentstellar masses. Compared to stars with masses M ∗ = 0 . − M (cid:12) , stars withsmaller masses are predicted to have a steeper planetary mass function (fewerlarge planets). Subject headings: planetary systems — planets and satellites: formation — plan-ets and satellites: dynamical evolution and stability a r X i v : . [ a s t r o - ph . E P ] J a n
1. Introduction
The mass distribution of fundamental objects represents an important feature of anyastrophysical discipline, including studies of planets, stars, galaxies, dark matter halos, andgalaxy clusters. Moreover, a complete understanding of the issue requires both the obser-vational specification of the mass distribution and a predictive theoretical framework. Thestellar initial mass function (starting with Salpeter 1955), for example, affects a large numberof astronomical issues, including galactic chemical evolution, supernova rates, and feedbackduring star formation. With thousands of exoplanets now discovered (e.g., see Han et al.2014; Gaudi et al. 2020), we can start to address the planetary mass function (PMF), whichwill ultimately inform studies of planetary system architectures, planetary habitability, andmany other issues. In addition, the PMF will provide a consistency check on theories ofplanet formation, once they are sufficiently developed. The observational determination ofthis mass distribution is now coming into focus, although the database remains incomplete,and many uncertainties persist. On the other hand, our theoretical understanding of themass function remains in its infancy. The goal of this paper is to construct simple but phys-ically motivated models for the PMF for giant planets, i.e., companions in the mass interval M p = 0 . − M J (where M J is the mass of Jupiter).For the range of planetary masses of interest, most objects are thought to form via thecore accretion paradigm (Pollack et al. 1996), which is organized into three phases (see, e.g.,Benz et al. 2014 for a review). In the context of this mechanism, the accumulation of smallrocky bodies (phase I) leads to the production of planetary cores with masses of order 10 M ⊕ ,which takes place over a time scale t core . Note that core formation could take place eitherthrough two-body accumulation of icy planetesimals (Hansen & Murray 2013) or throughrapid accretion of small pebbles assisted by gas drag (Ormel & Klahr 2010; Lambrechts &Johansen 2012). In any case, the cores accrete gas from the surrounding disk after reachingtheir critical mass threshold. During the subsequent phase II, growth is relatively slow andis controlled by cooling processes (e.g., opacity effects). After the total mass reaches roughlytwice the core mass (mass scale M II ) at time t onset , gas accretion occurs rapidly and theplanets accumulate most of their mass. This phase III lasts until the planets reach theirfinal masses in the range M p ≈ . − M J , and then terminate their growth at time t end .The final mass of a given planet is thus given by the integral expression M p = M II + (cid:90) t end t onset ˙ M p ( t ) dt = M II + (cid:104) ˙ M p (cid:105) ( t end − t onset ) , (1)where ˙ M p ( t ) is the accretion rate onto the planet as a function of time, and where thefinal equality follows from the mean value theorem. Within the context of the core accretionparadigm, the mass accretion rate onto the planet is some fraction of the total mass accretion 3 –rate ˙ M d through the disk, so that (cid:104) ˙ M p (cid:105) = (cid:104) f ˙ M d (cid:105) ≡ (cid:15) (cid:104) ˙ M d (cid:105) , where the final equality definesa time-averaged efficiency factor (cid:15) . Since circumstellar disks have lifetimes ∼ −
10 Myr,it is natural to associate the end of planetary accretion ( t end ) with the disappearance of thedisk. One objective of this paper is to test the plausibility of this hypothesis.As written, equation (1) is exact , and depends only on the four variables ( M II , (cid:104) ˙ M p (cid:105) , t onset , t end ). In other words, for a particular planet-forming event, if we know the mass M II at the onset of gas accretion, the starting and ending times ( t onset , t end ), and the meanaccretion rate (cid:104) ˙ M p (cid:105) , the final mass of the planet is completely determined. If, in addition,we know the distributions of the variables over the collection of planet forming disks ofinterest, then the distribution of planetary masses is also determined. Unfortunately, themass accretion history of the planet, along with the time scales t onset and t end , depend on alarge number of physical processes that remain under study, including disk viscosity whichdrives accretion, the fraction of material moving through the disk captured by the growingplanet, disk and planetary magnetic fields, planetary migration, the role of circumplanetarydisks, and many others. In a complete theory, where all of the relevant subprocesses areknown and predictable, one could calculate t onset , t end , and ˙ M p ( t ), and their distributions,from first principles. In the absence of such a complete understanding, the more modest goalof this paper is to construct observationally motivated distributions of the relevant inputvariables, as encapsulated via equation (1), and use the results to predict the PMF. Wecan then determine the properties of these input distributions required to produce planetarymass functions that are consistent with observations. In addition, the framework developedherein can be readily generalized in future work.Although no rigorous, a priori theory of the planetary mass function has been estab-lished, a substantial amount of previous work has been carried out. One approach is to usepopulation synthesis models, wherein planet formation is simulated using a large numberof physical processes that are treated in an approximate manner (e.g., see Ida & Lin 2008;Benz et al. 2014). With the proper choice of input parameters, these models produce massdistributions for planets that are roughly consistent with the observed PMF (Thommes et al.2008; Mordasini et al. 2009), although they tend to underproduce sub-saturn mass planets(Suzuki et al. 2018). The approach of this paper is complementary to population synthesismodels. Instead of including as many physical processes as possible, our goal is to find theminimal level of complexity necessary to reproduce the observed PMF. This present approachis semi-empirical in that it relies on observational input to help specify the distributions ofinput parameters (see Adams & Fatuzzo 1996 for an analogous approach to the stellar initialmass function). In addition, this minimalistic approach allows for PMF expressions to beobtained semi-analytically. Other related studies include a statistical model based on orbitalspacing (Malhotra 2015) and the role of dynamical instabilities in shaping the planetary 4 –mass function (Carrera et al. 2018).The scope of this paper is limited to gaseous giant planets forming through the processof core accretion, in particular those objects for which the gas accretion phase determinesthe final planetary mass. As a result, this approach is limited to planets forming in themass range 0 . M J ≤ M p ≤ M J . The current observational sample, especially from the Kepler mission (Borucki et al. 2010; Batalha et al. 2011), indicates that the total planetpopulation includes a large number of smaller bodies, with a preference for superearths withmasses M p ∼ M ⊕ (e.g., Zhu et al. 2018). These objects are roughly analogous to thecores of the giant planets, but they are thought to contain relatively little gas, and theirmass distribution is not addressed here. These lower mass planets could represent a secondpopulation (Pascucci et al. 2018; see also Schlaufman 2015), which should be addressed infuture studies. In addition, this present analysis is confined to those star/disk systems wheregiant planets are able to form. The probability that a given system will produce giant planetsis a separate but important issue (Howard et al. 2010; Meyer et al. 2021).This paper is organized as follows. Section 2 discusses the observational constraintsutilized in our models, including a brief overview of the observed planetary mass function,distributions of disk lifetimes, constraints on core formation time scales, and the dependenceof these quantities on the mass of the host star. Section 3 presents a framework to constructthe PMF. A model using the observed distribution of disk lifetimes, the expected dependenceof the mass accretion rate on the Hill radius, and an additional random variable can reproducethe nearly power-law mass function that is observed. Since disk properties, and hence planetproperties, depend on the mass of the central star, Section 4 explores the effects of varying thestellar mass. The paper concludes in Section 5 with a summary of our results and discussionof their implications. For completeness, the paper includes an appendix that considers analternative model for the PMF.
2. Empirical Considerations
This section outlines the observational input used to inform and test our models of theplanetary mass function. We start with current constraints on the observed PMF. Unfor-tunately, the observed mass distribution function for gas giant planets is not fully specifiedwith current data. Studies carried out to date for FGK stars (starting with Cumming et al.2008; see also Mayor et al. 2011) suggest that the mass distribution has an approximatelypower-law form over the mass range of interest 0 . M J ≤ M p ≤ M J , i.e., dFdM p = AM p p where p ≈ . ± . , (2) 5 –where A is a normalization constant (and more data are needed at the ends of the massrange). Although significant uncertainties remain, this paper uses the power-law mass dis-tribution of equation (2) as its basis for comparison to observations. Note that this form forthe mass function is only applicable over the specified range of planetary masses. The dis-tribution of planet radii displays a break near R p = 4 R ⊕ (Schlaufman 2015), correspondingto planet mass M p = 10 − M ⊕ (Wolfgang et al. 2016), which falls just below the masses ofinterest here. Extending the range to smaller masses, down to M p ∼ M ⊕ , the planetary ra-dius distribution has a bimodal form (e.g., Fulton et al. 2017). Another complication is thatplanets of a given mass can display a distribution of radii (Otegi et al. 2020). Current resultsfrom microlensing surveys (Suzuki et al. 2016; Shvartzvald et al. 2016) also find power-lawdistributions for the ratios of companion masses q = M /M ∗ over the range 10 − < q < − ,with slopes p ≈ . − .
5, roughly consistent with equation (2). Keep in mind that thesesurveys primarily sample host stars of lower mass (which are the most common stars). Forcompleteness, we note that the observed mass distribution for wide-orbit planets with masses M p > M J follows a similar power-law (Schlaufman 2018; Wagner et al. 2019), although theGemini Planet Imager Exoplanet Survey hints at a steeper slope p ≈ . ± . dF/dt is often taken to have the general form dFdt = 1 τ exp( − t/τ ) , (3)which is consistent with observations of disk fractions in star forming regions of varyingages (Haisch et al. 2001; Hern´andez et al. 2007; Mamajek 2009; Fedele et al. 2010). Notethat different signatures are used to indicate the presence of disks, including signs of activegas accretion onto the stars, near-infrared excess emission at 1 − µ m (tracing the inner 6 –disk), infrared emission at 5 − µm (tracing disk scales 0.3 – 30 AU), and millimeter-wave observations of dust and gas (tracing the outer disks at 10 −
100 AU). As written, thedistribution of disk lifetimes is normalized such that (cid:90) ∞ dFdt dt = 1 . (4)Observations indicate that the time scale τ ≈ − τ represents the timerequired for the population of disks to decrease by a factor of e ≈ .
72. The corresponding‘half-life’, the time required for the population to decrease by a factor of 2, is only about τ / = τ ln 2 ∼ τ = τ m , (5)where τ ∼ m ≡ M ∗ M (cid:12) . (6)Mass accretion rates ˙ M d for the disks surrounding T Tauri stars depend on both time andthe mass of the central star. These dependences can be modeled with a function of the form˙ M d = ˙ M d (0) m (1 + t/t ) / , (7)where the time scale t ∼ α -disk mod-els (e.g., Hartmann 2009). The observed m dependence (in the numerator) follows fromobservational results (see the review of Hartmann et al. 2016). Although no simple expla-nation is currently accepted for this law, it follows if disk lifetimes decrease with stellarmass (equation [5]) and the starting disk mass M d ∝ M ∗ increases with stellar mass (seealso Hartmann 2006; Dullemond et al. 2006). This latter scaling law has been observed in anumber of surveys (e.g., Natta et al. 2007; Andrews et al. 2013; Ansdell et al. 2016), with theslope somewhat steeper than linear in some studies (e.g., Pascucci et al. 2016; Barenfeld etal. 2016). In general, the observations find a well-defined correlation between submillimeterdisk luminosity and stellar mass, whereas the conversion from the observed quantities to theinferred relationship M d ∝ M ∗ requires disk modeling. In addition, the observations corre-spond to dust emission, so that the dust-to-gas ratio must be assumed in order to obtain 7 –estimates for the total disk mass. For completeness, note that a weaker correlation is foundfor actively evaporating disks in the Orion Nebula Cluster (Eisner et al. 2018).Finally, the formation time t core for the cores of giant planets, and hence the time t onset before the start of rapid gas accretion, is also expected to depend on the mass of the centralstar. Here we adopt a scaling law of the form t onset = t c0 m / , (8)where we expect that t c0 ∼ L ∗ ∼ M µ ∗ , where µ ≈ / −
2, and a fixed sublimation temperature T ice , we can find the semi-major axis of theiceline from energy balance. Since T ∝ L ∗ /a ∝ M ∗ /a = constant , we find that a ice ∝ M ∗ .The larger radius of the formation site — for larger stars — leads to slower accumulation ofthe cores. The Safronov accumulation time (specifically, the doubling time in the absenceof gravitational focusing) depends on the surface density Σ and orbital frequency Ω (e.g.,Safronov 1977) according to t saf ∝ Σ − Ω − . Using a surface density Σ ∝ M ∗ /a / (e.g.,Andrews et al. 2009), along with a Keplerian rotation curve, we find the result given inequation (8). Note that steeper surface density profiles lead to more sensitive dependence ofthe core formation time on stellar mass.The particular scaling law (8) was derived for the case of classic planetesimal accretion.For the competing picture wherein pebble accretion accounts for the formation of giant planetcores (e.g., Ormel & Klahr 2010; Lambrechts & Johansen 2012; Lin et al. 2018; Rosenthalet al. 2018), the scaling exponent can be different. For example, Lambrechts & Johansen(2014) find that the pebble accretion rate depends on stellar mass according to ˙ M ∼ M − / ∗ ,which coresponds to the alternate scaling law t onset ∼ m / ∼ m / (compare with equation[8]). The goal of this present work is to explore the effects due to the core formation timeincreasing with stellar mass, but the exact scaling is not as important. For completeness,we note that pebble accretion can occur over a wider range of radial locations in the disk,compared with the classical picture. Nonetheless, studies suggest that pebble accretion is The index depends on the specific evolution time and mass range considered. This result follows fromconsiderations of Hayashi’s forbidden region (e.g., Hansen & Kawaler 1994), and can also be determinedfrom stellar evolution simulations (e.g., Paxton et al. 2011). We also note that the migration of solid material could affect this scaling law, if the migration ratesdepend on the stellar mass.
3. Planetary Mass Function for Exponential Distribution of Disk Lifetimes
This section constructs a working model for the planetary mass function based on theempirical considerations of the previous section. The models constructed here are indepen-dent of the mass of the host star. The effects of varying stellar mass on the resulting PMFsare addressed in Section 4.
Observations provide estimates for the distribution of disk lifetimes. For purposes ofdetermining planetary masses, however, we need the distribution of time remaining after theonset of rapid gas accretion. The starting time t onset is determined by the core formationtime scale and the slow cooling phase. It takes into account the fact that rapid gas accretiononto the planet occurs only after the body has built up a critical mass, which includes boththe rocky core and enough additional gas to become self-gravitating. For a given stellar M ∗ ,the time to build the core t onset depends on the surface density of solids and cooling timedepends on the gas opacity in the planetary atmosphere (e.g., see Pollack et al. 1996; Benzet al. 2014; Helled et al. 2014; Piso et al. 2015, and references therein).For the case of interest here, where disk lifetimes are exponentially distributed, theremaining accretion time also follows an exponential law. Consider a given value t onset of thestarting time and define a new effective time variable ˜ t according to˜ t = t − t onset , (9)which resets the zero point of time for gas accretion. The time t is the disk lifetime, whichrepresents the time when accretion ends, and which follows the exponential distributionof equation (3). For any value of t onset , the probability distribution of the corrected timevariable ˜ t is determined by dFd ˜ t = dFdt dtd ˜ t = 1 τ exp[ − (˜ t + t onset ) /τ ] = exp[ − t onset /τ ] τ exp[ − ˜ t/τ ] , (10)where dF/dt is the distribution for the original time variable t . The probability distributionsfor both t and ˜ t thus have the same exponential form. As written, however, this expressionfor the probability distribution (for ˜ t ) is not normalized. Its integral over all ˜ t > t disk < t onset will not produce gas giant planets. For purposes of calculating probabilitydistributions for the time remaining after the onset of rapid gas accretion, however, thenormalized form of the distribution must be used, i.e., dFd ˜ t = 1 τ exp[ − ˜ t/τ ] . (11)After disks with short lifetimes are removed from the sample, the remaining disk popula-tion will ‘continue to decay’ with an exponential law that has the same ‘half-life’ as before(analogous to the case of radioactive decay).If all of the disks produced planetary cores on the same time scale, then the factorexp[ − t onset /τ ] appearing in equation (10) would determine the fraction of star/disk systemsthat could produce Jovian planets. In the absence of additional mechanisms that suppressplanet formation, the expectation value (cid:104) exp[ − t onset /τ ] (cid:105) is proportional to the planetaryoccurrence fraction. In practice, however, the timescale t onset depends on stellar mass, so thecorresponding planetary occurrence fraction is also mass dependent. In addition, the massaccretion rate depends on stellar mass, so the resulting masses of the planets that form alsodepend on M ∗ (see Section 4). Note that the observed occurrence rate for Jovian planets( M p = 1 − M J ), while highly uncertain, is estimated to fall in the range 10 – 30 percentfor FGK stars (integrated over all separations; see Winn & Fabrycky 2015; Meyer et al.2021, and references therein). Consistency implies that the typical delay time for the onsetof rapid gas accretion is bounded by t onset < ∼ τ ∼ The mass accretion rate onto a forming planet represents a fundamental variable of theproblem. We can conceptually divide the process into three components. [a] On the largestscale of the circumstellar disk, an accretion flow moves mass inward at a rate ˙ M d . Somefraction of this mass flow will enter the sphere of influence of the planet, where this fractioncan vary from system to system. [b] Since the sphere of influence of the planet grows withplanetary mass, the mass accretion rate onto the planet is expected to increase as the planet 10 –becomes more massive. [c] Only some fraction of the material that enters the Hill spherewill actually be accreted onto the planet. This treatment includes these factors as decribedbelow:For the latter stages of gas accretion, the rate of accretion onto the planet depends onthe Hill radius R H raised to some power, where R H = a (cid:18) M p M ∗ (cid:19) / , (12)where a is the semimajor axis of the forming planet and M p is its instantaneous mass. Thefiducial cross section scales as R H (Zhu et al. 2011; D¨urmann & Kley 2017). However,numerical simulations indicate that the mass accretion rate onto forming planets scalesaccording to steeper power-law ˙ M ∼ R H . The additional factor of R H arises from the shockof the inward flowing gas at the boundary defined by the Hill radius (Tanigawa & Watanabe2002; Tanigawa & Tanaka 2016). The shock raises the density by a factor of ( v/v s ) , wherethe incoming speed v is given by v ∼ Ω R H and v s is the sound speed (see also Lee 2019).As a result, the mass accretion rate must scale with the mass of the forming planet˙ M ∼ R H ∼ M / p . (13)If additional processes are operational, then accretion onto the planet can be suppressedfurther. Possible mechanisms that come into play include suppression by gap opening (whichlimits the density of the incoming flow; Malik et al. 2015; Ginzburg & Chiang 2019), strongmagnetic fields on the planetary surface (Batygin 2018; Cridland 2018), suppression of directinfall due to the initial angular momentum of the incoming material (Machida et al. 2008),and processes taking place within circumplanetary disks (Szul´agyi et al. 2017; Fung et al.2019). The latter three processes are analogous to well-studied mechanisms involved inthe star formation process. Strong stellar magnetic fields truncate the disk and shut downdirect accretion, which occurs along field lines at an attenuated rate. The initial angularmomentum of the pre-collapse gas causes most of the incoming material to fall initially ontoa circumstellar disk, rather than directly onto the star. Although these processes are notwell-studied in the context of planet formation, they will act to suppress mass accretionand are likely to vary from source to source, thereby introducing a random variable into theproblem. Strictly speaking, this statement holds provided that the disk properties do not change rapidly enoughto compromise the mass supply to the planet. Since mass accretion rates through the disk can be episodic,for example, the mass accretion rate onto the planet can be non-monotonic.
11 –Putting all of the components together, we can write the mass accretion rate onto theplanet in the form ˙ M = ξ ˙ M (cid:18) M p M J (cid:19) / . (14)In this form, the benchmark value of the mass accretion rate is that expected when theplanet has a Jovian mass. In order of magnitude, for typical disk properties and accretionefficiencies, we expect ˙ M ∼ M J Myr − , but a range of values are possible (Helled et al.2014). The leading coefficient ξ is a random variable on [0,1] that takes into account boththe efficiency for material to enter the Hill sphere and the efficiency with which the planetaccretes the incoming material. In principle, one could keep these two efficiencies as separatevariables. In this treatment, however, we consider only a single random variable ξ . Moreover,given that the mass accretion process has a well-defined scale set by the mass accretion ratethrough the disk, the natural starting point is to consider the random variable ξ to have auniform distribution. The mass accretion rate onto the planet from equation (14) is a rapidly increasingfunction of planetary mass. Once the planet grows sufficiently large, the rate of accretiononto the planet can become comparable to the rate of mass accretion through the disk.At this point in time, ˙ M must saturate, and would no longer follow equation (14). Fortypical parameters ( ξ = 1 / M = 1 M J Myr − ; e.g., Helled et al. 2014), and a diskaccretion rate ˙ M d = 10 − M (cid:12) yr − (e.g., Gullbring et al. 1998; Hartmann et al. 2016), thiscrossover point corresponds to planetary mass M p ≈ M J . Larger disk accretion ratesrequire larger planetary masses to reach saturation (which would depend on stellar mass).Since these mass scales are outside the range of interest, and since including this effectrequires specification of additional parameters, we do not model the saturation of ˙ M in thispresent paper. Nonetheless, if the saturation of the mass accretion rate takes a specifiedform, it is straightforward to incorporate this complexity. Finally, we note that includingthis effect leads to lower mass accretion rates for sufficiently large planets, thereby leadingto somewhat lower masses. In contrast, for problems without a well-defined fundamental scale, the natural choice for the probabilitydensity function is log-uniform, or dP/dx ∝ /x (Jefferys 1939), which is often called the Jefferys prior .
12 –
Starting with the form for the mass accretion rate from equation (14), we can integrateover time to find an expression for the planet mass M p = M II (cid:34) − ξ M / II ˙ M ˜ t M / J (cid:35) − . (15)The scale M II is the mass of the planet at the end Phase II (start of Phase III), when gasaccretion takes place rapidly. For the sake of definiteness, we take M II = 20 M ⊕ (abouttwice the core mass). The expression (15) thus contains two random variables ( ξ, ˜ t ). Thecummulative probability that one will draw the two variables in order to obtain a planetmass less than M p is given by the integral P ( m < M p ) = (cid:90) t ∗ d ˜ tτ exp[ − ˜ t/τ ] + (cid:90) ∞ t ∗ d ˜ tτ exp[ − ˜ t/τ ] t ∗ ˜ t , (16)where we have defined t ∗ ≡ M / J M / II ˙ M (cid:34) − (cid:18) M II M p (cid:19) / (cid:35) . (17)The time scale t ∗ ( M p ) is the time required to build a planet of mass M p at the maximumrate (with ξ = 1). The differential probability (probability density function) is thus given by dPdM p = 1 τ ˙ M (cid:18) M J M p (cid:19) / (cid:90) ∞ t ∗ dtt exp[ − t/τ ] = 1 τ ˙ M (cid:18) M J M p (cid:19) / E ( t ∗ /τ ) , (18)where E ( z ) is the exponential integral (Abramowitz & Stegun 1972) and where t ∗ is givenby equation (17). Note that only the product ˙ M τ of the fiducial mass accretion rate andtime scale τ of the disk lifetime distribution appear in the final expression. To evaluate theabove expressions, it is thus useful to define a composite parameter – which is a mass scale— according to M ≡ (cid:18) M II M J (cid:19) / ˙ M τ . (19)For typical parameter values ( M II = 20 M ⊕ , ˙ M ∼ M J Myr − , and τ = 5 Myr), we find thescale M ≈ M J . Given that the parameters are uncertain, we can find the distributionsof planet masses with varying values of M . Note that the values of the starting mass M II and the time scale τ of the disk lifetime distribution are relatively well known. As a result,specification of the mass scale M is largely determined by the baseline value of the massaccretion rate ˙ M . 13 – Fig. 1.— Planetary Mass Function. The curves show the planetary mass function predictedfrom an exponential distribution of disk lifetimes and a uniform-random distribution ofmass accretion rates. The distributions are characterized by the mass scale M , which isdetermined by the the overall mass accretion rate ˙ M and the time scale τ of the disklifetime distribution. Results are shown for M = 1 M J (green) √ M J (red), and 10 M J (blue). For each case, the colored curves are determined by sampling from the distributions,whereas the underlying dashed black curves show the analytic result from equation (18). Forcomparison, the solid black curve shows a power-law mass distribution ( dP/dM ∼ M − . ). 14 –Given the above results, we can construct planetary mass functions in two ways: First,we can sample the variables in equation (15) using their specified distributions, calculatethe masses, and construct the histogram of results. Second, we can use equation (18). Bothresults are shown in Figure 1 for different choices of the fiducial mass scale in the range M = 1 – 10 M J . The sampling results are shown as the colored curves and the analyticresults are shown dashed black curves (note that the results are essentially the same). Forcomparison, the solid black curve shows the expected power-law mass function of the form dF/dM ∝ M − . . With low accretion rates (mass scale) M = 1 M J (green curve), themodel produces a PMF that is steeper than the target distribution, with a deficit of highmass planets. For larger values of M , the theoretical PMF approaches a power-law form overmost of the relevant mass range and is in reasonable agreement with the target distribution.Nonetheless, the theoretical distributions remain slightly steeper.As mentioned above, the mass accretion rate from equation (14) is an increasing functionof time. If the planet mass become sufficiently large, the predicted mass accretion ratebecomes larger than the rate supplied by the circumstellar disk, so that the planetary massaccretion rate must saturate and approach a constant (maximum) value. The PMF here isconstructed using the increasing form of the mass accretion rate. This assumption leads tosomewhat larger planets for the tail of the distribution. For completeness, we also considerthe opposite limit in Appendix A, where the mass accretion rate saturates and thus becomesconstant while most of the planetary mass is acquired. This Appendix also illustrates howthe PMF framework constructed in the paper can be modified or generalized.
4. Planetary Mass Functions for Varying Stellar Mass
The treatment thus far does not include the mass of the host star in the analysis. Thissection uses the observed properties of circumstellar disks, which depend on the mass of thestar, to inform the models of the planetary mass function constructed in the previous section.Within this framework, the PMF is specified up to the characteristic mass scale M definedin equation (19). Here we use the empirical scaling laws outlined in Section 2 to define howthe mass scale M varies as a function of the stellar mass, and use the result to specify thePMF for varying M ∗ .Consider the accretion of gas onto a forming giant planet where the rate is a constantfraction (cid:15) of the disk accretion rate. Including the time dependence of the disk mass accretionrate from equation (7), the total mass accreted — and hence the mass of the planet — must 15 –be given by the integral M p = (cid:90) t end t onset (cid:15) ˙ M d (0) m dt (1 + t/t ) / = 2 (cid:15) ˙ M d (0) t m (cid:20) t onset /t ) / − t end /t ) / (cid:21) , (20)where m is the dimensionless stellar mass from equation (6). In order to define the char-acteristic mass scale M as a function of m , we use the scaling laws from Section 2. Thecore formation time varies with stellar mass according to equation (8). The typical time foraccretion to end is given by the disk lifetime parameter τ , which varies with m according toequation (5). We can then define the mass scale M using the ansatz M = (cid:104) (cid:15) ˙ M d (0) t (cid:105) F ( m ) where F ( m ) ≡ m (cid:20) √ m ) / − /m ) / (cid:21) . (21)Notice that the function F ( m ) → m = ( τ /t c0 ) / ∼
3. Our interpretationof this finding is that relatively few planets should form in disks associated with larger stellarmasses (see the discussion below).The relative mass fraction F ( m ) from equation (21) is shown in Figure 2 for standardvalues of the input parameters (solid red curve) and corresponding cases where the parame-ters are allowed to vary (lighter cyan curves). All of the curves follow the same basic trend:At low stellar masses, the accretion rate through the disk ( ˙ M ∼ m ) becomes small, and thistrend leads to small relative mass scales and hence the partial suppression of giant planetformation. In the opposite limit of large stellar mass, m ∼
3, the relative mass becomes smalldue to the lack of time for accretion. The time required for core formation becomes larger,while the disk lifetimes become shorter, so that little time is left for the cores (planets) toaccrete gas. The observed scaling laws thus imply that larger stars ( m > ∼
3) have difficultyproducing giant planets via the core accretion paradigm. In the intermediate regime, for m ∼ . −
2, the relative mass fraction varies (almost) linearly with increasing mass. Thisresult suggests that distribution of companion mass ratios should be nearly the same forstars in this mass range. In constrast, the distribution should be steeper for stellar massesoutside this range ( m (cid:28) . m (cid:29) t . One can show that in the limit of large t , the quantities from equation (21) become M ≡ (cid:15) ˙ M d (0) τ and F ( m ) ≡ m (cid:20) − t c0 τ m / (cid:21) . (22)In this case, the function F ( m ) → m = ( τ /t c0 ) / , which is the same as 16 – Fig. 2.— Relative mass fraction F ( m ) as a function of stellar mass, where m = M ∗ /M (cid:12) .The thick red curve shows the relative mass for typical values of the core formation time anddisk lifetime, which are functions of stellar mass as outlined in the text. The collection ofcyan curves shows the corresponding function where the baseline parameters t c0 and τ areallowed to vary with a (uniform) random distribution and an amplitude of 10 percent. 17 –found above. We can now construct the planetary mass function for different values of the stellarmass. The stellar mass dependence is incorporated by assuming that the total availablemass for making a giant planet is proportional to the relative mass fraction F ( m ) definedabove. For given stellar mass, the characteristic mass scale M that determines the shape ofthe PMF is then taken to have the form M = F ( m ) × M J . (23)Note that we are assuming that the mass scale for forming giant planets is proportional tothe total mass available. However, since the disk can in principle form multiple planets, thecoefficient in equation (23) is chosen to be only a fraction of the disk mass. If the massscale M is used in the PMF constructed according to Section 3, we obtain the distributionsshown in Figure 3. The figure shows the mass functions for planets forming around starswith masses m = 0.25 – 2.5. The planetary mass distributions for host stars in the range m = 1 – 2.5 display a similar shape. In contrast, stars with lower masses show a significantdeficit of larger planets, as shown by the PMF for m = 0 . m = 0 .
25 (red curve). These steeper distributions imply that red dwarfs are less likely tohave large planets, but are still expected have some Jovian companions (see also the moredetailed planet formation models of Laughlin et al. 2004).For even larger host stars, the available mass supply decreases due to the longer coreformation times and shorter disk lifetimes (see Figure 2). Since F ( m ) → m > ∼
3, thispresent framework does not provide a mass distribution for planets around higher mass stars:such planets should be rare. This implication is roughly consistent with current observationalresults (Reffert et al. 2015), which find that the planet occurence rate drops significantly forstellar masses larger than M ∗ ∼ . − M (cid:12) .Emerging observational trends (Meyer et al. 2021) comparing low mass brown dwarfcompanions to gas giant planets, as a function of stellar mass, suggest that distributionsof companion mass ratios are more universal than the distributions of companion massesthemselves. In addition to the PMF, we thus consider the distribution of the mass ratios q = M p /M ∗ in Figure 4 (for the same host star masses as before; compare with Figure3). The resulting distributions of mass ratio are roughly similar across the range of stellarmasses for m ≥ .
5. Once again, however, the distribution for stars with the smallest mass, m = 0 .
25 (red curve), is steeper and shows a deficit of large- q planets compared to the others. This equivalence is expected. This point in parameter space corresponds to the core formation timebecoming larger than the disk lifetime, and the crossover point is the same in both approximations.
18 –
Fig. 3.— Planetary mass function for varying masses of the host star. The characteristicmass scale for the PMF is determined by the available mass supply, which depends on diskmasses, accretion rates, and time scales according to equations (21) and (23). Curves areshown for stellar masses of m = 0.25 (red), 0.5 (yellow), 1.0 (green), 1.5 (blue), 2.0 (cyan),and 2.5 (magenta). All of the mass distributions are normalized so that they are equal at M p = M J . 19 – Fig. 4.— Planetary mass function expressed in terms of companion mass ratio q = M p /M ∗ for varying masses of the host star. The mass scale M for the PMF is determined bythe available mass supply, which depends on disk masses, accretion rates, and time scalesaccording to equations (21) and 23). Curves are shown for stellar masses of m = 0.25 (red),0.5 (yellow), 1.0 (green), 1.5 (blue), 2.0 (cyan), and 2.5 (magenta). 20 –This deficit is due to the small fiducial mass scale, which results from the smaller availablemass supply in the disks surrounding low mass stars. Although the curves are shown downto ratios q = 3 × − for all stellar masses, for m = 0 .
25 this value corresponds to a planetwith only M p ∼ M ⊕ . Planets with such small masses can accrete little gas, so that theirproduction mechanism lies outside this framework for the formation of gas giants. Finally,we note that the similarity between the PMFs and the mass-ratio distributions (Figures 3and 4) arises because the distributions have a nearly power-law form. In the limit where thePMF is exactly a power-law, the two distributions would have the same shape.This finding has observational implications. Microlensing surveys measure companionmass ratios, and favor low mass stars (which are more common than larger stars). Onthe other hand, both RV and transit surveys have historically used our solar system as atemplate and favored solar-type stars. As a result, the observed distribution of companionmass ratios is predicted to be somewhat steeper for microlensing surveys than for the others.For the largest stellar mass considered here, m = 2 . m >
3) should have relativelyfewer large planets (because the relative mass fraction
F →
5. Conclusion
This paper constructs a series of models for the initial mass function for gas giant planetsforming through the core accretion paradigm, including its dependence on the mass of thehost star. This section provides a brief summary of our results (Section 5.1), followed by adiscussion of their implications (Section 5.2).
This paper develops a working framework for determining the planetary mass functionwithin the core accretion paradigm. In this regime, the final planet mass is determined byits mass accretion history (see equation [1]). The treatment is semi-empirical in that the dis-tributions of the input parameters are specified (in part) using observations of circumstellardisk properties. This approach can produce mass distributions that are roughly consistentwith current observations over the mass range M p = 0 . − M J (e.g., see Figure 1).Within current observational uncertainties, disk lifetimes are observed to be exponen- 21 –tially distributed, whereas the planetary mass function has a nearly power-law form. Theframework developed in this paper produces a nearly power-law PMF using the exponentialdistribution of disk lifetimes as input. In order to obtain results consistent with observations,other input parameters — in addition to the disk lifetime — must sample distributions ofvalues, and the mass accretion rate must increase as the planet grows. In the limiting caseof a constant mass accretion rate, the exponential distribution of disk lifetimes imprints anexponential fall-off in the PMF, which leads to a deficit of larger planets compared to theobserved distribution.This analysis reveals an interesting feature for the expected case where the total disklifetime has an exponential distribution: If we consider the time remaining after some initialtemporal offset, for example after the planet reaches the stage where runaway gas accretionoccurs, then the distribution of time remaining (after the offset) also has an exponentialform (see equations [9 – 11]). Moreover, the exponential distribution has the same decayconstant or half-life.In this formulation of the problem, flow of material onto the planet is separated into twoparts. The circumstellar disk provides a background reservoir of gas that ultimately suppliesthe forming planet. In general, disks support a (slow) radially inward mass accretion flow,and some fraction of this large-scale flux is intercepted by the planet, i.e., enters its sphereof influence as delineated by the Hill radius. In the vicinity of the planet, only a fraction ofthe incoming material is actually accreted by the planet. The overall efficiency of accretiononto the planet ξ , along with the disk lifetime t , vary from case to case and provide sufficientcomplexity to produce a nearly power-law PMF (Figure 1).Disk properties — including lifetimes, mass accretion rates, and expected/inferred coreformation times — are observed to vary with the mass M ∗ of the host star. These variationsimply that the total mass provided by the disk depends on stellar mass. In this approach,the fiducial mass scale M for the PMF is assumed to scale with the relative mass fraction F , thereby leading to planetary mass distributions that depend on M ∗ (Section 4).The PMFs constructed in this paper indicate that the probability of producing largeplanets around low mass stars is lower than for the case of solar-type stars, i.e., the PMFis steeper for small stars (Figure 3). This finding is consistent with observational surveys(Bonfils et al. 2013; Vigan et al. 2020) as well as theoretical expectations (Laughlin et al.2004)). On the other hand, for sufficiently massive stars, with M ∗ > ∼ M (cid:12) , the core formationtime scales become comparable to typical disk lifetimes, and the probability of producinggiant planets is reduced (Figure 2; see also Reffert et al. 2015). If we consider the distributionof mass ratios q = M p /M ∗ , instead of the PMF itself, the resulting distributions are roughlysimilar across the range of intermediate stellar masses M ∗ = 0 . − . M (cid:12) (Figure 4). However, 22 –the mass ratio distribution for low mass stars remains steeper than that for larger stars. This paper constructs models for the PMF using a combination of disk lifetimes anddisk accretion properties. The microphysics of gas accretion onto forming planets providesadditional degrees of freedom. Using the observed distribution of disk lifetimes and uniformdistributions for efficiency parameters that describe the accretion processes, this frameworkcan produce models that are roughly consistent with the observed PMF. In addition, ap-parent trends observed in the PMF for varying stellar mass can be reproduced. Given thecurrent state of both observations and theory regarding the distribution of planetary masses,however, this work represents only a first step toward a complete understanding of the PMF.Many issues should be addressed in greater detail.The current version of the theory is incomplete. For example, this paper accounts forthe accretion history of gas flow onto forming planets in a parametric manner, includingthe introduction of efficiency factors. As the process of disk accretion becomes better un-derstood (Hartmann 2009; Zhu et al. 2011; Szul´agyi et al. 2017), the fundamentals of diskphysics should be incorporated into the models, ultimately leading to specific probabilitydistributions for the input variables. Another complication is that several mechanisms canact to stop (or at least slow down) accretion onto forming planets. These processes includegap opening in the circumstellar disks, the effects of the rotating reference frame on accretiononto the planet surface, repulsion of the accretion flow by magnetic fields, and the physicsof accretion within the circumplanetary disk. This current treatment also ignores planetarymigration, which can induce variations in the accretion rate as the planets change their or-bital elements (particularly if accretion rates vary with radial location). In addition, disksoften produce multi-planet systems, and the effects of additional bodies can influence theformation of their neighbors. All of these effects should be integrated into future studies.The models considered here are relatively simple, in that they involve only the disklifetime t , an accretion efficiency factor ξ , and the manner in which accretion scales withplanetary mass. As the number of physical processes contributing to the determination ofplanet mass increases, thereby including more variables that sample distributions of values,the resulting PMF departs further from the functional forms of the individual distributions.In the limit where the number of independent contributing variables is large, the centrallimit theorem comes into play, and the resulting PMF approaches a log-normal form (e.g,Richtmyer 1978). Since the observed PMF is approximately a power-law, this feature arguesfor an intermediate number of relevant (and independent) variables. The process is not 23 –completely scale-free, however, and one should keep in mind that the power-law form of thePMF only manifests itself over a limited range of parameter space.The theoretical framework of this paper highlights two observational issues that areimportant for understanding the PMF — the distribution of disk lifetimes for ages t ∼ M p ∼ M J . The characteristic mass scale M is important for determining the shape of the PMF (see equation [19]), which dependson the time scale τ appearing in the distribution of disk lifetimes. Specification of the scale τ determines the fraction of ‘long-lived’ disks, which in turn determines the probability ofproducing large planets with M p ∼ M J . As a result, it will be useful for observations tomeasure the fraction of disks that live for ∼
10 Myr (Thanathibodee et al. 2018). If the valueof the mass scale M is small (e.g., due to a small time scale τ ), the formation of large planetsis suppressed. The degree of suppression, in turn, affects the apparent slope of the PMF forplanets more massive than Jupiter. The observational determination of the planetary massfunction for masses M p ∼ M J is thus crucial for testing the PMF produced by the coreaccretion paradigm (although such observations will be challenging).In addition to the core accretion paradigm, some planets could be produced throughgravitational instability in sufficiently massive disks (see, e.g., Boss 1997 to Wagner et al.2019). These planets tend to have larger masses M p ∼ M J (see, e.g., Adams & Benz 1992to Stamatellos & Inutsuka 2018) and could thus contribute to the planetary mass functionat the high-mass end. The results of this paper suggest that the exponential distribution ofdisk lifetimes can cause the core accretion paradigm to underproduce high mass planets, sothat additional planets produced via gravitational instability could fill this gap. Notice alsothe brown dwarf binaries can contribute to the ‘planetary’ mass distribution for M p ∼ M J (Meyer et al. 2021).As outlined above, gravitational instability tends to produce more massive planets, andthey tend to form with large semi-major axes (e.g., Rafikov 2005; see Vigan et al. 2017 forcurrent observational constraints). In contrast, the observed populations of Hot Jupiters andmore temperate gas giants tend to have smaller masses then their colder counterparts (e.g.,Howard et al. 2010). As a result, future studies (both observational and theoretical) shoulddetermine the PMF for different ranges of orbital separations.Finally, we note that theoretical descriptions of the planetary mass function suffer froma lack of uniqueness. An intrinsic aspect of the problem is that many physical processescontribute to the PMF, so that the distributions of many input variables combine to producea single function. As a result, although the models of this paper successfully reproduce thenearly power-law form of the PMF that is observed, alternate approaches could also work.Fortunately, the framework developed here is more robust than this initial application, so 24 –that future studies can improve the various steps of the calculation. Acknowledgment:
We would like to thank Veenu Seri for early work that helpedformulate the problem. We also thank Konstantin Batygin, Nuria Calvet, and Greg Laughlinfor many useful discussions, and thank the referees for their comments on the manuscript.This work was supported by the University of Michigan, the NASA JWST NIRCam project(contract number NAS5-02105), and the NASA Exoplanets Research Program (grant numberNNX16AB47G).
A. Planetary Mass Function for Constant Accretion Rate
In this Appendix, consider the limiting case where the mass accretion rate onto theplanet approaches a constant and calculate the corresponding PMF. In this context, theplanetary mass M p is given by M p = M II + ˙ M ξt = M II + M , (A1)where t is the disk life time, ξ is a uniform-random variable as before, and the mass accretionrate ˙ M is constant. The second equality defines the mass M accreted after the onset ofrapid accretion.The distribution of the mass M is determined by the cummulative probability P ( m < M ) = (cid:90) t ∗ dtτ exp[ − t/τ ] + (cid:90) t ∗ dtτ exp[ − t/τ ] t ∗ t , (A2)where the time scale t ∗ is defined by t ∗ = M/ ˙ M . (A3)The probability distribution function for the accreted mass M thus takes the form dPdM = 1˙ M τ E (cid:18) M ˙ M τ (cid:19) = 1 M E (cid:18) MM (cid:19) , (A4)where the second equality defines M = ˙ M τ . If the starting mass M II is constant, then thecorresponding probability distribution for the planetary mass M p has the form dPdM p = 1 M E (cid:18) M p − M II M (cid:19) . (A5) 25 – REFERENCES
Abramowitz, M., & Stegun, I. A. 1972, Handbook of Mathematical Functions (New York:Dover)Adams, F. C., & Benz, W. 1992, in ASP Conf. Proc. 32, Complementary Approaches toDouble and Multiple Star Research, ed. H. A. McAlister & W. I. Hartkopf (SanFrancisco: ASP), 185Adams, F. C., & Fatuzzo, M. 1996, ApJ, 464, 256Andrews, S. M., Wilner, D. J., Hughes, A. M., Qi, C., & Dullemond, C. P. 2009, ApJ, 700,1502Andrews, S. M., Rosenfeld, K. A., Kraus, A. L., & Wilner, D. J. 2013, ApJ, 771, 129Ansdell, M., Williams, J. P., van der Marel, N. et al. 2016, ApJ, 828, 46Batalha, N. M., Borucki, W. J., Bryson, S. T., et al. 2011, ApJ, 729, 27Barenfeld, S. A., Carpenter, J. M., Ricci, L., & Isella, A. 2016, ApJ, 827, 142Batygin, K. 2018, AJ, 155, 178Benz, W., Ida, S., Alibert, Y., Lin, D., & Mordasini, C. 2014, in Protostars and Planets VI,ed. H. Beuther et al. (Tucson, AZ: Univ. Arizona Press), 691Bitsch, B., Lambrechts, M., & Johansen, A. 2015, A&A, 582, 112Bonfils, X., Delfosse, X., Udry, S., Forveille, T., Mayor, M., Perrier, C., Bouchy, F., Gillon,M., Lovis, C., Pepe, F., Queloz, D., Santos, N. C., S´egransan, D., & Bertaux, J.-L.2013, A&A, 549, 109Borucki, W. J., Koch, D., Basri, G., et al. 2010, Sci, 327, 977Boss, A. P. 1997, Science, 276, 1836Carpenter, J. M., Mamajek, E. E., Hillenbrand, L. A., & Meyer, M. R. 2006, ApJ, 651, L49Carrera, D., Davies, M. B., & Johansen, A. 2018, MNRAS, 478, 961Chambers, J. E. 2016, ApJ, 825, 63Cridland, A. J. 2018, A&A, 619, 165 26 –Cumming, A., Butler, R. P., Marcy, G. W., Vogt, S. S., Wright, J. T., & Fischer, D. A. 2008,PASP, 120, 531Dullemond, C., Natta, A., & Testi, L. 2006, ApJ, 645, 69D¨urmann, C., & Kley, W. 2017, A&A, 598, A80Eisner, J. A., Arce, H. G., Ballering, N. P. et al. 2018, ApJ, 860, 77Fedele, D., van den Ancker, M. E., Henning, T., Jayawardhana, R., & Oliveira, J. M. 2010,A&A, 510, A72Fulton, B. J., Petigura, E. A., Howard, A. W., et al. 2017, AJ, 154, 109Fung, J., Zhu, Z., & Chiang E. 2019, ApJ, 887, 152Gaudi, S. B., Christiansen, J. L., & Meyer, M. R. 2020, in ExoFrontiers: Big ques-tions in exoplanetary science, Ed. N Madhusudhan (Bristol: IOP Publishing Ltd),arXiv:2011.04703Ginzburg, S., & Chiang E. 2019, MNRAS, 487. 681Gullbring, E., Hartmann, L, Brice˜no, C., & Calvet, N. 1998, ApJ, 492, 323Haisch, K.E., Lada, E.A., & Lada, C.J. 2001, ApJ, 553, L153Han, E., Wang, S., Wright, J., et al. 2014, PASP, 126, 827Hansen, C. J., & Kawaler, S. D. 1994, Stellar Structure and Evolution (Springer)Hansen, B.M.S., & Murray, N. 2013, ApJ, 775, 53Hartmann, L., D’Alessio, P., Calvet, N., & Muzerolle, J. 2006, ApJ, 648, 484Hartmann, L. W. 2009, Accretion Processes in Star Formation (Cambridge: CambridgeUniv. Press)Hartmann, L., Herczeg, G., & Calvet, N. 2016, ARA&A, 54, 135Helled, R., Bodenheimer, P., Podolak, M., Boley, A., Meru, F., Nayakshin, S., Fortney, J.J., Mayer, L., Alibert, Y., & Boss, A. P. 2014, in Protostars and Planets VI, ed. H.Beuther et al. (Tucson, AZ: Univ. Arizona Press), 643Hern´andez, J., Hartmann, L., Megeath, M. et al. 2007, ApJ, 662, 1067 27 –Hillenbrand, L. A., Strom, S. E., Calvet, N. et al. 1998, AJ, 116, 1816Howard, A. W., Marcy, G. W., Johnson, J. A. 2010, Science, 330, 653Ida, S., & Lin, D.N.C. 2008, ApJ, 685, 584Jeffreys, H. 1939, Theory of Probability (Clarendon Press)Lambrechts, M., & Johansen, A. 2012, A&A, 544, 32Lambrechts, M., & Johansen, A. 2014, A&A, 572, 107Laughlin, G., Bodenheimer, P., & Adams, F. C. 2004, ApJ, 612, L73Lee, E. J. 2019, ApJ, 878, 36Lin J. W., Lee E. J., & Chiang E. 2018, MNRAS, 480, 4338Machida, M. N., Kokubo, E., Inutsuka, S., & Matsumoto, T. 2008, ApJ, 685, 1220Malhotra, R. 2015, ApJ, 808, 71Malik, M., Meru, F., Mayer, L., & Meyer, M. 2015, ApJ, 802, 56Mamajek, E. E. 2009, in AIP Conference Series 1158, Exoplanets and Disks, eds. T. Usuda,M. Tamura, & M. Ishii (Melville, NY: AIP), 3Manzo-Mart´ınez, E., Calvet, N., Hern´andez, J., Lizano, S., Hern´andez, R. F., Miller, C. J.,Mauc´o, K., Brice˜no, C., & D’Alessio, P. 2020, ApJ, 893, 56Mayor, M., Marmier, M., Lovis, C., Udry, S., S´egransan, D., Pepe, F., Benz, W., Bertaux,J.-L., Bouchy, F., Dumusque, X., Lo Curto, G., Mordasini, C., Queloz, D., & Santos,N. C. 2011, arXiv:1109.2497Meyer, M. R., Backman, D. E., Weinberger, A. J., & Wyatt, M. C. 2007, in Protostars andPlanets V, ed. B. Reipurth, D. Jewitt, & K. Keil (Tucson, AZ: Univ. Arizona Press),573Meyer, M. R., Amara, A., Susemiehl, N., & Peterson, A. 2020, AAS Journals, in preparationMordasini, C., Alibert, Y., & Benz, W. 2009, A&A, 501, 1139Natta, A., Testi, L., Calvet, N., Henning, Th., Waters, R., & Wilner, D. 2007, in Protostarsand Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil (Tucson, AZ: Univ. ArizonaPress), 767 28 –Nielsen, E. L., De Rosa, R. J., Macintosh, B. et al. 2019, AJ, 158, 13Ormel, C. W. & Klahr, H. H. 2010, A&A, 520, 43Ormel, C. W. 2017, Astrophysics and Space Science Library, 445, 197Otegi, J. F., Bouchy, F., & Helled, R. 2020, A&A, 634, 43Pascucci, I., Testi, L., Herczeg, G. J., et al. 2016, ApJ, 831, 125Pascucci, I., Mulders, G. D., Gould, A., & Fernandes, R. 2018, ApJ, 856, L28Paxton, B., Bildsten, L., Dotter, A. et al. 2011, ApJ Suppl., 192, 3Piso A.-M. A., Youdin A. N., & Murray-Clay R. A., 2015, ApJ, 800, 82Pollack, J. B., Hubickyj, O., Bodenheimer, P., Lissauer, J. J., Podolak, M., & Greenzweig,Y. 1996, Icarus, 124, 62Rafikov, R. R. 2005, ApJ, 621, L69Reffert, S. Bergmann, C., Quirrenbach, A., Trifonov, T., & K¨unstler, A. 2015, A&A, 574,116Ribas, ´A., Bouy, H., & Mer´ın, B. 2015, A&A, 576, 52Richtmyer, R. D. 1978, Principles of Advanced Mathematical Physics (New York: Springer-Verlag)Rosenthal, M. M., Murray-Clay, R. A., Perets, H. B., & Wolansky, N. 2018, ApJ, 861, 74Safronov, V. S. 1977, NASSP, 370, 797Safronov, V. S., & Zvjagina, E. V. 1969, Icarus, 10, 109Salpeter, E. E. 1955, ApJ, 121, 161Schlaufman, K. C. 2015, ApJ, 799, L26Schlaufman, K. C. 2018, ApJ, 853, 37Shvartzvald, Y., Maoz, D., Udalski, A., et al. 2016, MNRAS, 457, 4089Stamatellos D., & Inutsuka S.-I., 2018, MNRAS, 477, 3110Suzuki, D., Bennett, D. P., Sumi, T., et al. 2016, ApJ, 833, 145 29 –Suzuki, D., Bennett, D. P., Ida, S. et al. 2018, ApJL, 869, L34Szul´agyi, J. 2017, ApJ, 842, 103Tanigawa, T., & Tanaka, H. 2016, ApJ, 823, 48Tanigawa, T., & Watanabe, S.-I. 2002, ApJ, 580, 506Thanathibodee, T., Calvet, N., & Herczeg, G. 2018, ApJ, 861, 73Thommes, E. W., Matsumura, S., & Rasio, F. A. 2008, Science, 321, 814Vigan, A., Bonavita, M., Biller, B. et al. 2017, A&A, 603, 3Vigan, A., Fontanive, C., Meyer, M. et al. 2020, A&A, submitted, arXiv:2007.06573Wagner, K., Apai, D., & Kratter, K. M. 2019, ApJ, 877, 46Williams, J. P., & Cieza, L. A. 2011, ARA&A, 49, 67Winn, J. N., & Fabrycky, D. C. 2015, ARA&A, 53, 409Wolfgang, A., Rogers, L. A., & Ford, E. B. 2016, ApJ, 825, 19Yao, Y., Meyer, M. R., Covey, K. R., Tan, J. C., & Da Rio, N. 2018, ApJ, 869, 72Yasui, C., Kobayashi, N., Tokunaga, A. T., & Saito, M. 2014, MNRAS, 442, 2543Zhu, W., Petrovich, C., Wu, Y., Dong., S., & Xie, J. 2018, ApJ, 860, 101Zhu, Z., Nelson, R. P., Hartmann, L., Espaillat, C., & Calvet, N., 2011, ApJ, 729, 47