# Minimum Orbital Periods of H-Rich Bodies

S. Rappaport Department of Physics, and Kavli Institute for Astrophysics and
Space Research, M.I.T., Cambridge, MA 02139, USA S. Rappaport<EMAIL_ADDRESS>A.
Vanderburg Department of Astronomy, University of Wisconsin, Madison, WI
53715, USA  J. Schwab Department of Astronomy and Astrophysics, University of
California, Santa Cruz, CA 95064, USA  L. Nelson Department of Physics and
Astronomy, Bishop’s University, 2600 College St., Sherbrooke, QC J1M 1Z7,
Canada

(Received March 9, 2021; Revised —; Accepted April 12, 2021)

###### Abstract

In this work we derive the minimum allowed orbital periods of H-rich bodies
ranging in mass from Saturn’s mass to 1 $M_{\odot}$, emphasizing gas giants
and brown dwarfs over the range $0.0003-0.074\,M_{\odot}$. Analytic fitting
formulae for $P_{\rm min}$ as a function of the mass of the body and as a
function of the mean density are presented. We assume that the density of the
host star is sufficiently high so as not to limit the minimum period. In many
instances this implies that the host star is a white dwarf. This work is
aimed, in part, toward distinguishing brown dwarfs from planets that are found
transiting the host white dwarf without recourse to near infrared or radial
velocity measurements. In particular, orbital periods of $\lesssim 100$
minutes are very likely to be brown dwarfs. The overall minimum period over
this entire mass range is $\simeq 37$ minutes.

stars : binaries – interacting binaries – white dwarfs, subdwarfs, brown
dwarfs, planets

††journal: ApJ

##  1 Introduction

Orbital periods in systems containing white dwarfs (‘WDs’) can be extremely
short, especially if both of the stars are H-exhausted objects. The best
examples of this are WD+WD binaries with periods of 7 and 9 minutes (Burdge et
al., 2019, 2020a). Such systems almost certainly involve one or more phases of
mass transfer. In this work, we raise the question of the minimum allowed
orbital periods when at least one of the stars is still H-rich. In this latter
category are the brown dwarf (BD) plus WD binaries. The minimum orbital
periods of those systems were discussed extensively in Nelson et al. (2018).
Table 1 in that paper provides a list of 25 systems with WD primaries and
either a BD or lower-main-sequence (MS) companion. These systems are thought
to have had no mass transfer episodes from the BD or MS star to the WD. The
orbital periods listed there range from 250 minutes down to 68 minutes. The
minimum allowed orbital period for these WD+BD systems derived in Nelson et
al. (2018) was $\simeq 40$ min.

Photometric surveys like Kepler, K2, TESS, and the Zwicky Transient Facility
(ZTF; Bellm & Kulkarni 2017); are discovering and will continue to discover
interesting short-period binaries containing a WD (see, e.g., G. Murawski’s
website111https://sites.google.com/view/mgab-astronomy/eclipsing-white-
dwarfs). There are a number of new WD+WD binaries (see, e.g., Burdge et al.
2020b) as well as some likely WD+BD binary candidates, but none in the latter
category we are aware of with periods shorter than 68 minutes.

For a number of years now, researchers have been interested in whether planets
might be found orbiting WD dwarfs (e.g., Agol 2011; Lund et al. 2018; Bell
2019; Cortes & Kipping 2019). Such planets would have to have survived the
giant phase of the WD progenitor. There have been a number of tantalizing
suggestions in this regard. Gänsicke et al. (2019) proposed that there is a
disintegrating planet orbiting WDJ0914+1914 with a $\sim$ 9-d period. More
recently Vanderburg et al. (2020) reported the first intact transiting planet
orbiting WD 1856+534 in a 1.4-day orbit. This is a gas giant planet with
$M\lesssim 14\,M_{J}$.

There is also evidence for dust-emitting bodies orbiting WDs. WD 1145+017
exhibits deep dips with a characteristic period of 4.5-hour (Vanderburg et al.
2015; Gänsicke et al. 2016; Rappaport et al. 2017a). There is also ZTF
J013906.17+524536.89 with dips recurring at a $\sim$107-day period
(Vanderbosch et al., 2019). Manser et al. (2019) reported the discovery of a
123 minute periodicity in the motion of gas in the disk orbiting WD SDSS
J122859.93+104032.9, which they attributed to an orbiting massive rocky body.
In all these cases, the dust and gas probably originate from orbiting
asteroids or planetesimals which are not the subject of this paper, and we do
not consider them further.

We show in Fig. 1 an illustrative plot of equatorial eclipse durations for
objects of various masses transiting a white dwarf. Gas giant planets to brown
dwarfs, spanning 2.5 orders of magnitude in mass, and orbital periods less
than a day, have a typical transit duration of $\sim$8 minutes (within a
factor of a few) since all these objects can have the same radii to within
$\pm 15\%$. In this work we are interested in how knowledge of the orbital
period can inform us about the nature of the transiting body.

Specifically, the goal of this paper is to identify the minimum allowed
orbital period, $P_{\rm min}$, of H-rich bodies orbiting a host star as a
function of their mass. The host star is assumed sufficiently dense so as not
to limit $P_{\rm min}$, i.e., to prevent the body from reaching its Roche
limit (see, e.g., Roche 1849; Davidsson 1999; Holsapple & Michel 2006;
Rappaport et al. 2013). Orbiting bodies with masses ranging from Saturn’s mass
to solar-mass stars on the main sequence are considered. In Section 2 we
derive analytic relations for $P_{\rm min}$ as a function of the mass and
radius of the orbiting H-rich body, as well as a function of the body’s mean
density. In Section 3 we give a general radius-mass relation for cold H-rich
material based on Eggleton’s (Eggleton, 2006) analytic fitting formula. In
Section 4 we present results for $P_{\rm min}$ as functions of the mass and of
the density of the H-rich body, both in the form of graphs and analytic
expressions. We return in Sect 5 to discuss the effect on our results of
H-rich objects that are not highly centrally concentrated. Section 6 contains
a summary and our conclusions.

Figure 1: Equatorial transit duration vs. orbital period for objects of
various sizes that are occulting a 1.4 $R_{\oplus}$ white dwarf. The radii of
the occulting bodies, $R_{p}$, are written next to each curve in units of the
Earth’s radius ($R_{E}$). Objects covering a wide range in mass from Saturn
through brown dwarfs, to stars at the bottom of the main-sequence, may all
have comparable radii. In this work we explore the constraints that are set on
the masses of the transiting bodies by their orbital periods.

##  2 Dependence of $P_{\rm min}$ on Mass and Radius

We start by asking two questions: (1) how close can a H-rich body be to its
host star before it starts to overflow its Roche lobe, and (2) how applicable
is the Roche potential to bodies like Jupiter, super-Jupiters, and brown
dwarfs?

Regarding the first question, we start by writing an expression for the size
of the Roche lobe, $R_{\rm L}$, as a function of the mass ratio, $q$, of the
two stars and the orbital separation, $a$, assuming a circular orbit. This
takes the form

$R_{\rm L}=f(q)\,a~{},$ (1)

where we consider two well-used functions to represent $f(q)$. Note that $4\pi
R_{\rm L}^{3}/3$ is defined to be the volume of the Roche lobe, and we
therefore we can call $R_{\rm L}$ the ‘volumetric radius’ of the Roche lobe.
Inserting this expression into Kepler’s third law, we have:

$\frac{G(M_{\rm host}+M_{\rm com})f^{3}(q)}{R_{\rm
L}^{3}}=\left(\frac{2\pi}{P}\right)^{2}$ (2)

where $M_{\rm host}$ and $M_{\rm com}$ are the masses of the host star and
H-rich companion, respectively, $P$ is the orbital period, and $q\equiv M_{\rm
com}/M_{\rm host}$.

An analytically convenient and reasonably accurate approximation to $R_{\rm
L}$, normalized to the orbital separation, was given by Kopal (1959) for
$q<1$:

$f_{\rm K}=\frac{2}{3^{4/3}}\left(\frac{q}{1+q}\right)^{1/3},$ (3)

where the numerical value of the leading factor is 0.4622. A more accurate
expression, covering a much larger range in $q$, was derived by Eggleton
(1983) and is based on an elegant fitting formula applied to the results of
numerical integrations of the Roche-lobe volume:

$f_{\rm E}=\frac{0.49\,q^{2/3}}{0.6\,q^{2/3}+\ln(1+q^{1/3})}.$ (4)

For an extensive discussion of other formulations for the size of the Roche
lobe see Leahy & Leahy (2015).

We start with the simpler, but more insightful, of the two expressions to
derive the minimum period before Roche-lobe overflow commences. Inserting the
expression $a=R_{\rm L}/f_{\rm K}(q)$ from Eqns. (1) and (3) into Eqn. (2), we
find

$P=\frac{2\pi}{G^{1/2}}\sqrt{\frac{81}{8}}R_{\rm L}^{3/2}M_{\rm com}^{-1/2}$
(5)

which is independent of $M_{\rm host}$, and this is the motivation behind
using the Kopal (1959) formulation of $f_{\rm K}(q)$.

The minimum orbital period will be attained when the orbit shrinks to the
point where the companion radius equals $R_{\rm L}$, in which case we have

$\displaystyle P_{\rm min}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{G^{1/2}}\sqrt{\frac{81}{8}}R_{\rm com}^{3/2}M_{\rm
com}^{-1/2}$ (6) $\displaystyle\simeq$ $\displaystyle 8.85\left(\frac{R_{\rm
com}}{R_{\odot}}\right)^{3/2}\left(\frac{M_{\rm
com}}{M_{\odot}}\right)^{-1/2}~{}{\rm hr}$

When we do the calculations in this work, we will actually utilize the more
accurate $f_{\rm E}(q)$ expression for the Roche lobe dependence. However, we
can still cast the expression for $P_{\rm min}$ explicitly as a function only
of $R_{\rm comp}$ and $M_{\rm com}$, multiplied by a correction factor that is
a very weakly dependent function of $q$:

$\displaystyle P_{\rm min}$ $\displaystyle\simeq$ $\displaystyle
8.85\,\xi(q)\left(\frac{R_{\rm com}}{R_{\odot}}\right)^{3/2}\left(\frac{M_{\rm
com}}{M_{\odot}}\right)^{-1/2}~{}{\rm hr}$ $\displaystyle{\rm
with}~{}~{}\xi(q)$ $\displaystyle\equiv$ $\displaystyle\left[\frac{f_{\rm
K}(q)}{f_{\rm E}(q)}\right]^{3/2}$ (7)

A plot of the slowly varying function $\xi(q)$ is shown explicitly in Fig. 3
of Nelson et al. (2018).

Finally, the expression for $P_{\rm min}$, Eqn. (6) can also be cast as a
function of the density of the companion only. The right hand side of that
equation is manifestly in the form the inverse square root of the density of
the companion star. It can therefore be rewritten as

$\displaystyle P_{\rm min}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{243\pi}{8G}}\,\bar{\rho}_{\rm com}^{~{}-1/2}$ (8)
$\displaystyle=$ $\displaystyle 10.45\,\left(\frac{{\rm
g~{}cm}^{-3}}{\bar{\rho}_{\rm com}}\right)^{1/2}~{}{\rm hr}$

where $\bar{\rho}$ is the mean density.

##  3 Mass-Radius Relation for Degenerate H-Rich Bodies

In order to estimate the radius of cold H-rich bodies we made use of the zero-
temperature models of Zapolsky & Salpeter (1969). These models represent the
lower limit to the radius of brown dwarfs and planets for a specific mass and
(homogeneous) composition. We utilize an analytic expression devised by
Eggleton (2006) for the dependence of the radius on the mass and chemical
composition of these objects:

$R_{\rm com}\simeq 0.0128(1+X)^{5/3}M_{\rm com}^{-1/3}\,g(M_{\rm com};X)$ (9)

with

$\displaystyle g$ $\displaystyle=$ $\displaystyle g_{1}\cdot g_{2}$ (10)
$\displaystyle g_{1}$ $\displaystyle=$ $\displaystyle\sqrt{(1-(M_{\rm
com}/M_{\rm ch})^{4/3}}$ (11) $\displaystyle g_{2}$ $\displaystyle=$
$\displaystyle\left(1+3.5(M_{\rm com}/M_{\rm p})^{-2/3}+M_{\rm p}/M_{\rm
com}\right)^{-2/3}$ (12) $\displaystyle M_{\rm ch}$ $\displaystyle=$
$\displaystyle 1.44\,(1+X)^{2}~{}M_{\odot}$ (13) $\displaystyle M_{\rm p}$
$\displaystyle=$ $\displaystyle 0.000566\,(1+X)^{3/2}~{}M_{\odot}$ (14)

where $X$ is the H-mass fraction. In the expressions for $M_{\rm ch}$ and
$M_{\rm p}$ we have simplified the original expressions of Eggleton (2006) for
the case of objects composed of H and He only.222The original expressions of
Eggleton (2006) were in terms of $Z_{N}$ and $A$, the atomic number and atomic
weight, respectively, of each of the chemical constituents of the star. For
objects composed solely of H and He we derived an approximate weighting based
on the value of $X$ only (see, Nelson & Rappaport 2003). We plot in Fig. 2 the
expression given by Eqn. (9) with $X=0.7$ for masses between that of Saturn
and the bottom of the zero-age main sequence ($M\simeq 0.074\,M_{\odot}$).

For masses above the bottom of the ZAMS, we use a simple $R(M)$ relation

$R_{\rm com}(M_{\rm com})=0.85\,\left(M_{\rm
com}/M_{\odot}\right)^{0.85}~{}R_{\odot}$ (15)

which is derived from a regression analysis applied to the lower main-sequence
models of Dorman et al. (1989) for the FGVH EOS (Fontaine et al., 1977) down
to 0.085 M⊙.

Also plotted in Fig. 2 are a sampling of planets, brown dwarfs, and lower
main-sequence stars taken from a compilation of Chen & Kipping (2017); and we
augment this with our own compilation of brown dwarfs listed in Table
LABEL:tbl:BDs. Because the distinction between gas-giant planets and brown
dwarfs has been subject to considerable debate, we have included an extended
discussion of this issue in the Appendix.

Figure 2: Mass-radius relation for H-rich bodies spanning masses from Saturn
to stars on the ZAMS up to 1 $M_{\odot}$ (red curve). For the region between
Saturn’s mass and the end of the brown-dwarf range (at $\sim$0.074
$M_{\odot}$) we used equation (9) with X = 0.7. For stars on the ZAMS we use
the simple expression given by equation (15). The blue and cyan dots are an
empirical sample of planets and main-sequence stars from Chen & Kipping
(2017). The green points are our compilation of brown dwarfs taken from the
literature (see Table LABEL:tbl:BDs and Sect. 3). The black curve is an
approximation to the red curve which has been smoothly blended near the
transition between the ZAMS stars and brown dwarfs. Note how both the red and
black curves hug the lower locus of measured objects – as desired. Figure 3:
Mean density-mass relation for H-rich bodies spanning masses from Saturn to
stars on the ZAMS up to 1 $M_{\odot}$. The rest of the descriptors are the
same as in Fig. 2.

In Fig. 3 we employ the same $R(M)$ relations used to construct the
theoretical curve in Fig. 2 to compute the mean density of the orbiting body
as a function of its mass. Here we also superpose the same collection of
planets, brown dwarfs, and lower main-sequence stars compiled by Chen &
Kipping (2017) and augmented with our list of BDs given in Table
LABEL:tbl:BDs. Note that, as desired, the curve forms a near upper boundary to
the empirically observed systems.

Table 1: Mass-Radius Pairs for Brown Dwarfs†

Name | Mass | Radius | Reference
---|---|---|---
| $M_{J}$ | $R_{J}$ |
SDSS J0857+0342 | $91.1\pm 12.6$ | $1.07\pm 0.04$ | 1
Kepler-503 | $78.6\pm 3.1$ | $0.96^{+0.06}_{-0.04}$ | 2
WD1032+011 b | $69.6\pm 6.4$ | $1.0\pm 0.1$ | 3
EPIC 201702477 | $66.9\pm 1.7$ | $0.757\pm 0.065$ | 4
TOI-569 | $64.1\pm 1.9$ | $0.75\pm 0.02$ | 5
WD 1202-024 b | $63.9\pm 10.5$ | $0.88\pm 0.05$ | 6
CoRoT-15 b | $63.3^{+4.3}_{-4.1}$ | $1.12^{+0.30}_{-0.15}$ | 2
WASP-030 | $62.5\pm 1.2$ | $0.951^{+0.028}_{-0.024}$ | 7
KOI-415 b | $62.1\pm 2.69$ | $0.79^{+0.12}_{-0.07}$ | 2
V* V2384 Ori a | $56.7\pm 4.8$ | $6.52\pm 0.33$ | 8
TOI-811 b | $55.3\pm 3.2$ | $1.35\pm 0.09$ | 2
TOI-852 b | $53.7\pm 1.3$ | $0.75\pm 0.03$ | 2
TOI-503 b | $53.6\pm 1.1$ | $1.29\pm 0.30$ | 9
SDSS J1411+2009 | $52.4\pm 2.1$ | $0.70\pm 0.04$ | 10
EPIC 212036875 b | $52.3\pm 1.9$ | $0.874\pm 0.017$ | 11
TOI-1406 b | $46.0\pm 2.7$ | $0.86\pm 0.03$ | 2
Kepler-492 b | $40.8^{+1.1}_{-1.5}$ | $0.82\pm 0.02$ | 12
WASP-128 b | $37.19^{+0.83}_{-0.85}$ | $0.94^{+0.22}_{-0.18}$ | 2
EPIC 219388192 b | $36.0\pm 1.6$ | $0.846\pm 0.021$ | 2
V* V2384 Ori b | $35.6\pm 2.8$ | $4.81\pm 0.24$ | 8
NLTT 41135 b | $33.7\pm 2.8$ | $1.13^{+0.27}_{-0.17}$ | 1
KELT-1 b | $27.4\pm 0.93$ | $1.116^{+0.038}_{-0.029}$ | 13
CoRoT-3 b | $21.96\pm 0.70$ | $1.037\pm 0.069$ | 14
GPX-1 b | $19.7\pm 1.6$ | $1.47\pm 0.16$ | 2
Kepler-39 b | $19.1\pm 1.0$ | $1.11\pm 0.03$ | 12
HATS-70 b | $12.9^{+1.8}_{-1.6}$ | $1.384^{+0.079}_{-0.074}$ | 2

Notes. †Brown dwarfs taken from the literature with masses measured to
$\lesssim 15\%$ and radii with lower limits of $\lesssim 20\%$. (1) Parsons et
al. (2012) (2) Schneider et al. (2011); (3) ; (4) Bayliss et al. (2017); (5)
Carmichael et al. (2020); (6) Rappaport et al. (2017b); (7) Triaud et al.
(2013); (8) Stassun et al. (2006); (9) Šubjak et al. (2020); (10) Littlefair
et al. (2014); (11) Carmichael et al. (2019); (12) Bonomo et al. (2015); (13)
Siverd et al. (2012); (14) Southworth (2011).

##  4 Minimum Orbital Periods

We have used the $R(M)$ relations displayed in Fig. 2 and described by Eqns.
(9) and (15), in conjunction with Eqn. (7) to derive the minimum allowed
orbital period vs. the body’s mass. The results are shown in Fig. 4.

As we can see, there is a general trend of decreasing $P_{\rm min}$ from 620
min (10.3 hr) for Saturn-mass objects (red circle in Fig. 4), to 430 min (7.2
hr) for Jupiters (orange circle), to 104 min (1.7 hr) for objects on the
boundary between super-Jupiters and brown dwarfs (blue circle), all the way
down to 37 min (0.62 hr) for the coldest and most massive brown dwarfs (purple
circle; see also Nelson et al. 2018). These values are summarized in Table
LABEL:tbl:pofm. A simple fitting formula which is applicable for masses over
the range $3\times 10^{-4}-0.074\,M_{\odot}$ is:

$\ln P_{\rm min}\simeq 1.01-1.085\ln m_{\rm com}-0.052\ln^{2}m_{\rm com}$ (16)

where $P_{\rm min}$ is in minutes, and $m_{\rm com}\equiv M_{\rm
com}/M_{\odot}$.

Figure 4: Minimum allowed orbital period of H-rich bodies as a function of
their mass. We have used the $R(M)$ relations displayed in Fig. 2 and
described by Eqns. (9) and (15), in conjunction with Eqn. (6). The various
closely spaced colored curves (red, orange, … blue, purple) are for different
masses of the host star ranging from 0.3 $M_{\odot}$ to 1.4 $M_{\odot}$,
respectively. With the use of the approximate Eqn. (6) all the curves would
merge; but, not quite so with the more exact expression given by Eqn. (7). The
dotted red curve is the limit obtained for incompressible fluid bodies (see
text). Heavy, filled, colored circles refer to fiducial-mass objects detailed
in Table LABEL:tbl:pofm. Figure 5: Minimum allowed orbital period of H-rich
bodies as a function of their mean density. Here we have used Eqn. (8) which
is derived from the approximate expression Eqn. (6), but supplemented with the
slowly varying function $\xi(q)$ defined in Eqn. (7). The various closely
spaced colored curves (red through purple) are for different masses of the
host star ranging from 0.3 $M_{\odot}$ to 1.4 $M_{\odot}$, respectively. With
the use of the approximate Eqn. (8) all the curves would merge; but, not quite
so with the more exact expression given by Eqn. (7). The appearance of the
plot results from the fact that the curves start at the upper left (planets),
decrease to the minimum period (brown dwarfs), and finally nearly retrace the
same path back up toward the upper left (lower MS stars).

Finally, we calculate $P_{\rm min}$ as a function of the mean density of the
H-rich body. For this, we use Eqn. (8) multiplied by the function $\xi(q)$
given in Eqn. (7). The latter factor makes use of the more accurate Eggleton
(1983) expression for the Roche-lobe radius.

The results for $P_{\rm min}(\bar{\rho}_{\rm com})$ are plotted in Fig. 5.
Here we see a nearly linear relation in the $\log(P_{\rm
min})-\log\bar{\rho}_{\rm com}$ plane. The light dashed red line is a
reference slope $-1/2$, as would be expected from Eqn. (8). The expression for
that dashed line is given by:

$\displaystyle P_{\rm min}\simeq 9.9\,\left(\frac{{\rm
g~{}cm}^{-3}}{\bar{\rho}_{\rm com}}\right)^{1/2}~{}{\rm hr}$ (17)

We note that for main-sequence stars with mean densities between 2 and 10 g
cm-3 (corresponding to masses of 1.0 to 0.28 $M_{\odot}$) there is a small
dispersion in $P_{\rm min}$ of $\pm 10\%$. This is due to the fact that the
Kopal (1959) Roche lobe formula, which leads to the simple $\bar{\rho}_{\rm
com}^{~{}-1/2}$ dependence, is not really applicable when the H-rich companion
star is more massive than the host star (which we took to range from 0.3 to
1.4 $M_{\odot}$. However, when we utilized the more exact Roche-lobe
expression of Eggleton (1983) we see the deviation from the simple
$\bar{\rho}_{\rm com}^{~{}-1/2}$ dependence for the extremely low mass host
stars. Nonetheless, the overall $\bar{\rho}_{\rm com}^{~{}-1/2}$ dependence
does an excellent job of representing $P_{\rm min}$ over two orders of
magnitude in H-rich companion star mass and density.

Table 2: Minimum Orbital Periods of H-Rich Bodies

Object | $P_{\rm min}$ | $\langle\rho\rangle$
---|---|---
| (hr) | g cm-3
Saturn | 10.3 | 0.69
Jupiter | 7.2 | 1.33
5 $M_{J}$ | 3.3 | 8.69
15 $M_{J}$ | 1.7 | 33.9
Max brown dwarf | 0.62 | 280

Notes. Illustrative points taken from Fig. 4.

##  5 Central Concentration of the H-Rich Body

Figure 6: Internal models for massive gas giant planets generated with MESA.
The models span the range from 0.1 $M_{J}$ to 10 $M_{J}$. Most of the curves
overlap to within the widths of the curves. The solid and dashed line segments
denote logarithmic slopes of +2 and +1, respectively.

Thus far, we have been working under the assumption that the orbiting H-rich
bodies are substantially centrally concentrated. Bodies that might be roughly
represented as $n=3$ or even $n=3/2$ polytropes would qualify. They have
ratios of central to mean densities of 54 and 6, respectively. For such
centrally concentrated objects, we assume that the Roche-lobe formula, given
by Eqn. (4), is quite appropriate and accurate in terms of describing the size
of the critical potential surface.

However, we know that massive gas giant planets and brown dwarfs are less
centrally concentrated than an $n=3/2$ polytrope. To make this more
quantitative, we show in Fig. 6 pressure-density curves, $P(\rho)$, for the
interiors of cold H-rich bodies of masses of 0.1 $M_{J}$ to 10 $M_{J}$, in 5
logarithmic steps. These models were generated with MESA (Paxton et al., 2011,
2013, 2015, 2018, 2019) version r13573 using the included ‘make_planets’ test
case. The models assume a metallicity $Z=0.02$, no solid core, and are shown
at an age of 10 Gyr. It is impressive that the curves nearly overlap, at least
on this broad logarithmic scale. The heavy black solid and dashed lines mark
logarithmic slopes of 2 and 1, respectively. And since most of the mass is
represented fairly well by a $P\propto\rho^{2}$ relation, we conclude that
these objects are more nearly represented by $n=1$ polytropes and less
centrally concentrated than an $n=3/2$ polytrope.

Figure 7: Ratio of central ($\rho_{c}$) to mean ($\bar{\rho}$) density of the
gas-giant models shown in Fig. 6 plus one additional model at 0.2 $M_{J}$.
Ratios of 3-4 are typical for these gas giants.

In Fig. 7 we show in more detail the ratio of central density to mean density
for six different models over a wide mass range. The geometric mean ratio of
$\rho_{c}/\bar{\rho}$ is $\simeq 3.6$. That ratio for an $n=1$ polytrope is
3.13. Hence, we conclude the bulk of the non-main-sequence objects we are
considering are well represented by $n=1$ polytropes.

Unfortunately, to our knowledge, there are no equivalent expressions to Eqn.
(4) for $n=1$ polytropes with $\rho_{c}/\bar{\rho}$ is $\simeq 3.1$ that are
filling their critical potential lobes. This would be a good exercise for a
self-consistent field calculation (see, e.g., Hachisu 1986). But, that is
beyond the scope of this paper. Thus, for now, we jump to a model that is of
uniform density—and with a known solution. This is the Roche limit for a
uniform density, incompressible fluid body. The Roche limit is usually
expressed as $a_{\rm crit}=2.44R_{\rm host}({\bar{\rho}}_{\rm
host}/\bar{\rho}_{p})^{1/3}$ where $a_{\rm crit}$ is the critical (i.e.,
minimum) orbital separation to avoid mass transfer, $R_{\rm host}$ is the
radius of the host star, and the $\rho$’s are the mean densities of the host
star and orbiting ‘planet’ – in this case the H-rich body.

For our purposes in this paper, we can rewrite the Roche limit in the
following form:

$P_{\rm min}\simeq\sqrt{\frac{3\pi(2.44)^{3}}{G\rho_{\rm
com}}}=12.6\,\left(\frac{{\rm g~{}cm}^{-3}}{\bar{\rho}_{\rm
com}}\right)^{1/2}~{}\,{\rm hr}.\vspace{0.1cm}$ (18)

This is directly analogous to Eqn. (8), except that the leading coefficient
here is somewhat larger. This reflects the fact that the material in an
incompressible configuration is less tightly bound than a compressible fluid
of the same mean density.

This relation for $P_{\rm min}$, using the results for an incompressible fluid
($n=0$), is shown as a dashed red curve in Fig. 4. As we can see, it only
raises the minimum allowed period by some 30%. We take this to be a firm upper
limit on what the $P_{\rm min}(M_{\rm com})$ might be. For most of the H-rich
objects we are considering we surmise that the true answer lies somewhere
between the two limiting curves (red and black in Fig. 4).

##  6 Summary and Conclusions

In this work we have examined how the minimum orbital period of cold H-rich
bodies depends on the mass of the object, or alternatively, its density. The
basic conclusions are as follows. (1) For any gas-giant planet or brown dwarf
the minimum period is $\sim$37 minutes. (2) For objects with orbital periods
$\lesssim 100$ minutes we can conclude that we are observing a brown dwarf (or
second WD) rather than a gas-giant planet. More generally, (3) we give an
approximate analytic expression (see Eqn. 16) for the minimum period as a
function of the mass of the gas-giant planet or brown dwarf. We can use this
information to eliminate gas giant planet candidates with transit durations of
2-15 minutes and periods of $\lesssim 100$ minutes. The same is true for brown
dwarf candidates with periods $\lesssim 40$ minutes.

Our work makes use of the radius-mass relation, $R(M)$, for cold H-rich bodies
based on the zero-temperature models of Zapolsky & Salpeter (1969). These
models represent the lower limit to the radius of brown dwarfs and planets for
a specific mass and (homogeneous) composition. For purposes of comparing this
theoretical $R(M)$ relation with the empirical data, we have utllized the Chen
& Kipping (2017) compilation of planets, brown dwarfs, and stars that has been
augmented by our own compilation of a more complete list of 26 brown dwarfs.

J.S. is supported by the A.F. Morrison Fellowship in Lick Observatory and by
the National Science Foundation through grant ACI-1663688. L.N. thanks the
Natural Sciences and Engineering Research Council (NSERC Canada) for financial
support through the Discovery Grants program. This research has made use of
NASA’s Astrophysics Data System.

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Commentary on the Mass-Radius Relationship for Brown Dwarfs and Gas Giant
Planets

The issue of distinguishing brown dwarfs (BDs) from gas giant planets (GGPs)
is very complex. Because of the importance of delineating these two
populations, several attempts have been made to construct suitable
definitions. In 2003, the Working Group on ExtraSolar Planets (WGESP) of the
IAU proposed a working definition whereby the distinction was based solely on
the mass of the object333IAU’s working definition (immediately after
2003):http://www.astro.iag.usp.br/~dinamica/WGEP.html. Specifically, the mass
defining the boundary between GGPs and BDs was proposed as 13 $M_{J}$
regardless of how the object was formed or its location (Boss, 2007). This
definition has ostensibly arisen because a cloud of hydrogen-rich, solar-
metallicity gas with a mass of 13 $M_{J}$ (or higher) that undergoes
gravitational collapse will fuse most of its primordial deuterium over an
interval of $\lesssim 30$ Myrs (Nelson et al. 1985, 1986; Spiegel et al. 2011,
and references therein). This phase is referred to as the deuterium-burning
main sequence (DBMS). The upper end of the BD mass range is defined as the
mass for which hydrogen-burning cannot achieve thermal quasi-equilibrium
within a Hubble time. Higher mass objects that collapse as a result of the
fragmentation of a hydrogen-rich gas cloud will be able to sustain nuclear
fusion and thus become hydrogen-burning main sequence (HBMS) stars.

The formulation of precise definitions in astronomy (or any other natural
science) can be a challenging task. For example, there is no absolutely
precise definition of when a star is first on the zero-age main sequence
(ZAMS). The reason for this is that stars never achieve complete thermal
equilibrium (i.e., their gravo-thermal luminosity is never zero). This type of
difficulty also arises in trying to establish the BD-MS star ‘boundary’
because, although there is complete agreement that stars achieve approximate
thermal equilibrium via sustained hydrogen burning, high-mass BDs can attain a
substantial degree of thermal equilibrium within a Hubble time. Moreover, the
treatment of the input physics (e.g., opacities, EOS) and the assumed
metallicity can have a profound effect in establishing the boundary between
BDs and MS stars when mass is taken to be the only determining criterion. For
a solar metallicity, the uncertainty in the input physics places the upper
limit on the BD mass to be between $\simeq 0.072$ to 0.080 $M_{\odot}$.

The difficulty in establishing whether an object is a GGP or a BD is much more
problematic. According to conventional wisdom, it is believed that GGPs form
as a result of (cold) core accretion in a circumstellar dusty disk (Burgasser,
2008). Assuming that there is sufficient mass in the disk for accretion, it is
thus possible for the mass of the object to grow much larger than 13 $M_{J}$
without undergoing deuterium burning (i.e., no DBMS phase). However, it is
also possible that relatively massive objects can form in circumstellar disks
by direct fragmentation of the disk. This process is very similar to the way
in which stars and, by extension, BDs form444Such a pathway has been suggested
for the giant planet GJ 3512B (Morales et al., 2019).. Knowing the formation
process would help in reaching a definitive conclusion, but given the absence
of such knowledge, we must rely on observables such as the mass ($M$), radius
($R$), and $T_{\rm eff}$. Higher order observables that derive from multiband
spectra and direct imaging of the atmosphere could provide the detailed
atmospheric information (e.g., cloud structures, temperature profile, species
differentiation, presence of grains, etc.) that is needed to make a more
robust determination.

All of these issues have led to considerable debate on how GGPs should be
distinguished from BDs or even if such a delineation should be made. Hatzes &
Rauer (2015) (hereafter HR) make the claim that: “objects with masses in the
range $0.3\,M_{J}-60\,M_{J}$ follow a tight linear relationship [in the $\log
M-\log\rho$ plane] with no distinguishing feature to separate the low-mass end
(giant planets) from the high-mass end (brown dwarfs).” They propose that all
objects with masses in the range of $0.3<M/M_{J}<60$ should be viewed as GGPs.
This definition is based on the linearity (and continuity) of the slope in the
mass-density relationship (see their Figure 1). This is a purely
phenomenological definition and thus does not address the underlying physics
of these objects. Moreover, the paucity of data in the range of
$20\,M_{J}-40\,M_{J}$ makes this type of analysis particularly challenging.
Our Figure 2 (see also Table 1) contains significantly more data than was
available to HR and reveals that the assumption of a linear mass-radius (MR)
relationship (in the logs) should be re-examined. Our red curve (zero
temperature hydrogen-rich models) shows that the slope of the MR relationship
changes considerably between masses of 0.3 $M_{J}$ to $\simeq
0.072\,M_{\odot}$ (i.e., just below the hydrogen-burning minimum mass [HBMM].
Near the HBMM, we would naively expect the $M-R$ exponent ($R\sim M^{\xi}$) to
be approximately -1/3 because the EOS is dominated by nonrelativistic electron
degeneracy. However, even at this upper end of the mass range for BDs, the
effects of Coulombic interactions start to become significant and this has the
effect of making $\xi$ more positive (i.e., flattening the $M-R$
relationship). For even lower masses, other effects such as Thomas-Fermi
corrections that more accurately account for electron-nucleus interactions, in
addition to exchange effects and correlation energies (see, e.g., Seager et
al. 2008) further increase the value of $\xi$.

For chemically homogeneous, zero-temperature objects, the Zapolsky-Salpeter
(1969) models show that $\xi$ changes from approximately -1/3 to +1/3 as the
mass decreases to terrestrial values. In examining the observational data in
Figure 2, we see that the smallest GGPs and BDs (in radius) tend to follow the
theoretical predictions reasonably closely (red curve). There is considerable
scatter in the BD mass range but this is to be expected because these BDs are
discovered at various stages in their contraction (BDs have very long Kelvin-
Helmholtz times and thus their radii tend to be reasonably sensitive to their
ages). Nonetheless, there is a reasonably pronounced dip in the radii
(compared to the radii of MS stars) at $M\simeq 0.07M_{\odot}$. This feature
is completely consistent with the HBMM inferred from theoretical models of
Population I objects. Moreover, the lower envelope of $R-M$ observations is
not inconsistent with the zero-temperature theoretical models that predict the
lower limit for the radii of objects of a specific mass (and chemical
composition). We emphasize that in the absence of any other information
concerning the correlation between radius and mass, it is reasonable to carry
out a linear regression (i.e., to assume a constant $\xi$); however, the
changing ‘physics’ of these objects (with mass) requires that $\xi$ be treated
as variable. Thus we believe that the claim of Hatzes & Rauer that: (1) BDs
should be subsumed into “the upper end of the giant planet sequence”; and, (2)
the boundary between GGPs and MS stars should be set equal to 0.06 $M_{\odot}$
appears to be an oversimplification.

Chen & Kipping (2017) have re-examined/refined the work of HR with respect to
the mass limits of ‘Jovian worlds’. Based on their analysis, this limit
extends from $0.41\,M_{J}$ to 80 $M_{J}$ (higher-mass objects are defined to
be stars). They claim that: “There is no discernible change in the $M-R$
relation from Jupiter to brown dwarfs. Brown dwarfs are merely high-mass
planets, when classified using their size and mass”. This conclusion is
largely based on a linear regression analysis carried out on GGP and BD data
(see their Figure 3). They find a relatively flat dependence of radius on mass
($R\sim M^{0.04}$). Again, the challenge with this type of purely empirical
analysis is that brown dwarfs are rare; the Chen & Kipping (2017) analysis
includes 150 objects in the mass range corresponding to giant planets, and
only 5 brown dwarfs (two of which have such high masses that their status as
brown dwarfs becomes questionable). Therefore any regression over the entire
range of mass will be dominated by the properties of the giant planets.
Moreover, because the mass limits of 0.41 $M_{J}$ to 13 $M_{J}$ straddle the
mass corresponding to the extremum in the radius of the zero-temperature
models (that mass is approximately the geometric mean of the limits), we would
naturally expect any regression to show a relatively flat $M-R$ relationship.
The fact that a power law can provide an acceptable fit to the $M-R$
relationship over this mass range does not imply that objects in this mass
range are indistinguishable.

The bottom line is that the theoretical models (for a given age and ignoring
the DBMS) imply that there should be no abrupt change in the $M-R$ relation
exponent ($\xi$) across any reasonable boundary that is chosen to delineate
GPPs from BDs (with the proviso that the chemical structure of the objects is
similar). This continuity in $\xi$ does not, however, imply that BDs should
subsumed into the class of GGPs nor does it imply that an extension of a
linear fit to the MR relationship determined for GGPs can be reasonably
applied to BDs. For example, the $M-R$ data shown in Figure 2 are not
inconsistent with the simple Zapolsky & Salpeter (1969) models (red curve),
and those models show that $\xi$ can change significantly over the nominal BD
mass range. Moreover, if objects in this mass range are not formed by core
accretion and undergo a DBMS phase (even if it only lasts for $<$ 100 Myr),
they are clearly a separate class from GGPs because they formed similarly to
stars and achieved approximate thermal equilibrium via nuclear fusion.