Download Adiabatic evolution of mass-losing stars

MNRAS 434, 2940–2947 (2013)
doi:10.1093/mnras/stt1208
Advance Access publication 2013 August 7
Adiabatic evolution of mass-losing stars
Lixin Dai,1‹ Roger D. Blandford1‹ and Peter P. Eggleton2‹
1 Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University, Menlo Park, CA 94025, USA
Livermore National Laboratory, 7000 East Ave, Livermore, CA 94551, USA
2 Lawrence
Accepted 2013 July 1. Received 2013 May 21; in original form 2012 September 10
ABSTRACT
We investigate the evolution of the stellar structure, when a star fills and overflows its Roche
lobe in a circular, equatorial orbit around a supermassive black hole. The stellar mass-loss timescale is anticipated to be long compared with the dynamical time-scale and short compared
with the thermal time-scale of the star; so, the entropy as a function of enclosed mass is
conserved. For a representative set of stars, we calculate how the stellar entropy, pressure,
radius, density and orbital angular momentum vary when the star adiabatically loses mass. We
also provide interpolated formulae of the stellar mean density in terms of the remaining stellar
mass for different types of stars. As the stellar orbit changes with the stellar density, Sun-like
stars, upper main-sequence stars and red giants will spiral inwards and then outwards, while
lower main-sequence stars, brown dwarfs and white dwarfs will always spiral outwards. We
discuss the validity and limitation of the adiabatic mass-loss assumption and show that such
a mass-transfer process is always stable on dynamical time-scales when the mass ratio of the
two objects is large.
Key words: accretion, accretion discs – stars: evolution – stars: kinematics and dynamics –
stars: mass-loss – galaxies: active.
1 I N T RO D U C T I O N
When a star loses mass on a time-scale faster than the Kelvin–
Helmholtz (or thermal) time-scale but slower than the dynamical
time-scale, as can happen in extreme mass-ratio inspirals (EMRI),
the structure of the star will evolve adiabatically so that the entropy
as a function of enclosed mass S(m) is approximately conserved.
This is true for radiative and convective zones (except near the
surface) in most of the physical scenarios. Webbink (1985) gave a
general introduction to binary mass transfer on various time-scales.
Hjellming & Webbink (1987) used polytropic stellar models to explore the stability of this adiabatic process and listed several critical
mass ratios above which the donor stars are unstable on dynamical
time-scales. Soberman, Phinney & van den Heuvel (1997) further
discussed the stability of binary system mass-transfer processes on
the thermal and dynamical time-scales.
The previous works generally assumed comparable masses of
the two objects in the binary system and an unchanging distance
between them throughout the process. In this paper, we will use real
stellar models to study how stars respond under adiabatic mass-loss
on a time-scale slower than the dynamical time-scale yet faster than
E-mail: [email protected] (LD); [email protected] (RDB);
[email protected] (PPE)
the thermal time-scale. The stellar orbital radius from the hole is not
held as a constant, but evolves depending upon the mean density of
the stripped star. Nuclear reactions, which are highly temperaturesensitive (Woosley, Heger & Weaver 2002), will be shut off soon
after the mass-loss starts and central temperature drops. (The star
might continue to burn for a short time but this will not change the
evolution of the structure, as long as mass-loss time-scale is much
shorter than the thermal time-scale.)
In our EMRI binary system, the star is assumed to be in a circular, equatorial orbit around the central supermassive black hole
(SMBH). When the star just fills its Roche lobe, Roche mass transfer starts and materials flow out of the inner Lagrangian point L1.
During this mass-transfer phase, the stellar orbital radius may increase or decrease. However, we assume that the change is slow
enough so that the stellar orbit remains quasi-circular. If the orbit
shrinks to be smaller than the innermost stable circular orbit (ISCO)
of the SMBH, the star will plunge into the hole. For instance, a nonrotating black hole has an ISCO radius of 6Rg and a black hole with
a maximal prograde spin (i.e. spin parameter a = 0.998) has an
ISCO radius of 1.237Rg (Thorne 1974). Here, Rg , the gravitational
radius of the hole, defined as GMBH /c2 (G is the gravitational constant, MBH is the mass of the black hole and c is the speed of light)
measures the size of the hole.
In this paper, we investigate the stellar structure evolution under these conditions, for a representative set of stars and planets.
C 2013 The Authors
Published by Oxford University Press on behalf of the Royal Astronomical Society
Adiabatic evolution of mass-losing stars
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In Section 2, we introduce the methodology used to calculate the
evolution of a star composed of ideal gas under adiabatic mass-loss
assumption. In Section 3, we study in details the adiabatic evolution
of solar-type stars, lower main-sequence stars, upper main-sequence
stars, red giants, white dwarfs, brown dwarfs and planets. In Section
4, we discuss the various assumptions used in this paper and summarize the results. We will illustrate the use of these evolutionary
models in forthcoming papers.
After this work was completed, our attention was drawn to a
recent paper: Ge et al. (2010). Although the end application of their
paper is different, some of the calculations overlap (and agree with)
those presented below.
2 METHODOLOGY
For a star composed of ideal gas,we have:
1
dr
=
,
dm
4πr 2 ρ(m)
(1)
Figure 1. The effective local entropy of the Sun, defined as P3/5 ρ −1 , as a
function of the enclosed mass. The Sun has an almost constant entropy in
the convective zone near the surface.
where r is the radius, m is the enclosed mass and ρ is the density.
This gives how the gas pressure P changes with mass:
Gm
dP
=−
.
(2)
dm
4πr 4
Ignoring radiative contributions, Coulomb and degeneracy corrections, the entropy per particle is S/N ∼ Cv ln (P/ρ γ ), with γ =
5/3 for ideal monatomic gas (Cv is the heat capacity). We can define
an effective entropy S̃ = P 3/5 ρ −1 for such an ideal gas.
For a mass-losing star with constant local entropy, we can use the
following equations to calculate its new equilibrium property:
S̃(m)
dr
=
,
dm
4πr 2 P 3/5
Gm
dP
=−
.
(3)
dm
4πr 4
For a star with a known entropy profile, we can solve for its new
equation of state with boundary conditions r (0) = 0 and P (0) =
P0 . We solve for the remaining stellar mass M and the surface
radius R = r (M ), where P (M ) = 0. Then the volume and average
density of the star in this new model can be obtained.
When the star orbits in a circular equatorial orbit and loses mass
through Roche mass transfer, due to the large mass ratio between
the star and the hole, the stellar orbital period follows PR ∝ ρ̄−1/2 ,
where PR is the Roche orbital period and ρ̄ is the mean density
of the stripped star (e.g. Frank, King & Raine 2002). Therefore,
the evolution of stellar structure governs the evolution of the stellar
orbit. A direct calculation gives the angular momentum of the binary
1/3
system L ∼ m PR ∼ m ρ̄−1/6 . L should continue to decrease
through gravitational radiation during the mass transfer. Here, we
neglected the much smaller angular momentum carried away by
the hot stream flowing out of the inner Lagrangian point (L1).
Therefore, the changes of ρ̄ and L determine whether the mass
transfer is stable or catastrophic on dynamical time-scales. The exact
formulae of PR and L have been discussed in another paper (Dai
& Blandford 2013), where relativistic effects are also considered.
3 R E S U LT S
3.1 Solar model
Let us start with the Sun as a representative of main-sequence
stars. We adopted Guenther and Demarque’s standard solar model
Figure 2. Stellar pressure versus enclosed mass of a Sun-like
star that adiabatically loses mass. The four coloured curves represent the four cases when the stellar central pressure decreases to
0.7 P0 (blue), 0.4 P0 (green) and 0.1 P0 (red) from 1 P0 (purple).
(Guenther & Demarque 1997; Demarque et al. 2008) to derive S̃(m)
as in Fig. 1. Note that the entropy is almost constant for the outer
30 per cent of its radius, which corresponds to the convective zone
of the Sun.
Using this entropy profile, we can compute the radius of the star
r and the pressure of the star P as functions of enclosed mass
m , when the star adiabatically evolves into a new equilibrium. As
mass is stripped off the star, the pressure inside the star naturally
decreases, so the stellar material will expand. In Figs 2–4, we show
r (m ), P (m ) and P (r ) in four different phases of the evolution:
the original profile of the Sun and as the central pressure drops
to 70, 40 and 10 per cent of the original central pressure with the
mass-loss.
With the stripping of the stellar material and the expansion of the
star, the stellar volume and mean density will evolve accordingly.
For the solar model, the total volume decreases first and then remains
almost constant, and its mean density increases first and then drops.
We show the evolution of the stellar mean density in Fig. 5 and
interpolate it as
ρ̃ = 2.81 × m̃ + 22.25 × m̃2 − 32.27 × m̃3 + 8.18 × m̃4
(4)
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L. Dai, R. D. Blandford and P. P. Eggleton
Figure 3. Stellar radius versus enclosed mass of a Sun-like star that adiabatically loses mass. The four curves represent the same cases as in Fig. 2.
Figure 6. Dimensionless angular momentum of a Sun-like star that adiabat−1/6
−1/6
ically loses mass. L is defined as M ρ̄
and L is defined as M ρ̄ .
The red points represent the same phases as in Fig. 5.
with error <3 per cent, where ρ̃ is the dimensionless stellar mean
density in terms of the initial mean density of the star and m̃ is the
remaining stellar mass in terms of the solar mass.
The Sun becomes denser in the initial phase of mass transfer, so
its orbital period decreases, which means that the Sun continues to
inspiral in quasi-circular orbits. As the Sun loses ∼35 per cent of
the initial mass, its density starts to drop and the Sun will spiral out
for the rest of the Roche-mass-transfer phase.
Lastly, we check the dynamical stability of the Roche mass transfer. As shown in Fig. 6, the effective angular momentum L continues to decrease as the stellar mass is stripped, thus confirming that
the mass transfer is stable. For all types of stars that we will study
in the rest of the paper, this stability criterion is always fulfilled.
Figure 4. Stellar pressure versus radius of a Sun-like star that adiabatically
loses mass. The four solid curves still represent the cases when a Sun-like star
loses its mass until its new central pressure is P0 = 1, 0.7, 0.4 and 0.1 P0 ,
from the top (purple) to the bottom (red). The five blue dashed lines are
contours enclosing the same enclosed mass: 0.1, 0.3, 0.5, 0.7 and 0.9 M
from left to right, respectively. (The two rightmost lines are very hard to
see.) We can observe that as the star loses its mass, the surface pressure of
any shell decreases and that shell enclosing constant mass expands.
3.2 Other stars
3.2.1 Lower main-sequence stars
A lower main-sequence star below 0.4 M is entirely convective,
and thus it has a constant entropy throughout the star (e.g. Girardi
et al. 2000; Padmanabhan 2001). Here, we have chosen a star with
∼0.3 M as a representative. For the subsequent discussions on
lower main-sequence stars, upper main-sequence stars and red giants, we have adopted stellar models computed by one of us (PE).
All the zero-age main-sequence stars have metallicity Z = 0.01.
As an ∼0.3 M lower main-sequence star is stripped adiabatically, it should expand due to the decrease of pressure, similar to the
Sun. The major difference between this model and the solar model
is that the total volume of such a lower main-sequence star always
increases due to the convective nature of the star. Therefore, the
star will become less and less dense in the Roche phase (Fig. 7).
Therefore, as soon as a lower main-sequence star reaches the Roche
regime, it starts to spiral out immediately.
For such a star, its density evolution could be interpolated using
a simple formula:
ρ̃ = m̃2 ,
Figure 5. The evolution of the stellar mean density as a Sun-like star
adiabatically loses mass. The red points show the mean density computed
from the model at different mass-loss phases. The blue curve shows the
interpolated formula as in equation (4).
(5)
with ρ̃ being the dimensionless mean density and m̃ being the
dimensionless remaining mass in terms of the initial stellar mean
density and mass, respectively.
Adiabatic evolution of mass-losing stars
Figure 7. The change of stellar mean density as a lower main-sequence
star with initial mass ∼0.3 M adiabatically loses mass. The red points
are computed from the model at different phases. The blue curve represents
equation (5).
2943
Figure 9. The change of stellar mean density as a red giant with initial mass
∼7.1 M adiabatically loses mass. The red data points are the data points
computed using the adiabatic model. The mean density for M ∼ 1 M is
probably inaccurate due to the fragility of the red giant envelope.
3.2.2 Upper main-sequence stars
Upper main-sequence stars have convective cores for more efficient
energy transport (e.g. Woosley et al. 2002). This mixing of material
around the core removes the helium ash from the hydrogen burning
region, allowing more of the hydrogen in the star to be consumed
during the main-sequence lifetime. Therefore, entropy is almost
constant in the centre. The outer regions of a massive star are radiative. We investigate an upper main-sequence star with initial mass
∼7.9 M .
The mean density of an upper main-sequence star responds to the
mass-loss in a way similar to the Sun, as shown in Fig. 8. Therefore,
after mass transfer starts, such a star will also continue to spiral in,
reach a turning point and then spiral out. The stellar mean density
changes in a similar way as that of the Sun, and can be expressed
as
ρ̃ = 1.39 × m̃ − 5.71 × m̃2 + 20.8 × m̃3 − 15.41 × m̃4 ,
(6)
with ρ̃ being the dimensionless mean density and m̃ being the
dimensionless remaining mass in terms of the initial stellar mean
density and mass, respectively.
3.2.3 Red giants
Main-sequence stars above 0.5 M evolve into red giants in their
late phases (Reimers 1975; Sweigart & Gross 1978). Here, we study
a red giant with initial mass ∼7.1 M . This red giant has a dense
helium core of ∼1 M and an extended hydrogen burning envelope. Red giants have a much steeper pressure gradient compared
with main-sequence stars, especially at the transition between the
core and the envelope. Therefore, when the very dilute hydrogen
envelope is fed to the hole, the central pressure is not affected much.
We plot Fig. 9 to see how the mean density of the red giant
changes. As mass stripping starts, the red giant continues to spiral
in. The core of the red giant responds to the mass-loss adiabatically.
The envelope may have a faster thermal time-scale comparable
with the mass-loss time-scale; so, in this regime, the response of the
envelope is not strictly adiabatic. However, the structure change of
the envelope is still determined by the adiabatic expansion of the
core. After most of the envelope is stripped, the core expands, so its
orbit will also expand. A star is a red giant for only a small fraction
(10–25 per cent) of its fusion lifetime (e.g. Padmanabhan 2001).
Therefore, it would be rare to observe a red giant feeding an SMBH
accretion disc.
3.2.4 White dwarfs
White dwarfs are composed of a non-degenerate gas of ions and a
degenerate and at least partly relativistic gas of electrons (Kippenhahn & Weigert 1994). We need to redefine its effective entropy
since the entropy of a degenerate electron gas differs from that of
a non-degenerate ideal gas. However, an easier way of obtaining
the structure of the evolved white dwarf is to use its radius–mass
relation:
1/2
M −1/3
M 4/3
−1
(7)
1−
R (M ) 0.022μe R
Mch
Mch
Figure 8. The change of stellar mean density as an upper main-sequence
star with initial mass ∼7.9 M adiabatically loses mass. The red points
are computed from the model at different phases. The blue curve represents
equation (5).
(Nauenberg 1972). In this equation, R and M are the total radius and mass of the evolved white dwarf. μe is the average number of nucleons per electron and μe = 2 for He, C and O, which
is also a good approximation for most other options. Mch is the
Chandrasekhar mass, which is ∼1.459 M . The formula holds true
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L. Dai, R. D. Blandford and P. P. Eggleton
Figure 10. The stellar mean density of a white dwarf, as its total mass
adiabatically decreases from 1 M to zero. As its mass is stripped, its mean
density decreases.
except when the stellar mass is close to its upper limit Mch , where
the radius should approach a constant.
We consider the adiabatic mass-loss of a 1 M white dwarf
with complete degeneracy and zero temperature, and we show its
mean density–mass relation in Fig. 10. As the white dwarf is tidally
stripped, its total volume increases in this adiabatic evolution, ensuring that the stellar mean density decreases throughout the Roche
phase. Therefore, as a white dwarf loses its mass under the Roche
process, it will always spiral out. Its density change is similar to
that of a convective low-mass star with an extra correction factor:
3/2
0.65
.
(8)
ρ̃ = m̃2
1.65 − m̃
3.2.5 Brown dwarfs
Brown dwarfs never undergo hydrogen reactions. They are supported by the degeneracy pressure of their non-relativistic electrons
(Kippenhahn & Weigert 1994). Their ions are treated as ideal gases.
Brown dwarfs are also fully convective.
The total pressure of a brown dwarf, including the ideal ion gas
pressure and the electron degeneracy pressure, goes as
P Kρ 5/3 ,
(9)
where K is a constant close to 1013 dyn cm−2 (e.g. Burrows &
Liebert 1993; Burrows et al. 2001, 2002; Padmanabhan 2001 and the
references therein). The effective entropy, still defined as P3/5 ρ −1 ,
therefore would be a constant, confirming the convective structure.
Using the relation between P and ρ, we can solve the equations
of equilibrium (equation 3) for a brown dwarf with a certain new
central density and total mass. Here, we used a 1 Gyr brown dwarf
with total mass 0.05 M with central density 458.0 g cm−3 and total
radius 6.4 × 109 cm (Burrows & Liebert 1993). K is calculated to be
∼8.5 × 1012 dyn cm−2 , in order to fit the model. Using our routine,
we show how the stellar mean density varies with the remaining
mass in the same way as the lower main-sequence star, as shown in
Fig. 11. The brown dwarf, therefore, moves outwards from the hole
as its mass is tidally stripped.
3.2.6 Planets
Planets like Jupiter (Baraffe, Chabrier & Barman 2008; Nettelmann
et al. 2008) have complex structures. However, we can study two
Figure 11. The stellar mean density evolution as the mass of a brown dwarf
adiabatically changes from 0.05 M to zero. The red points are calculated
from the model. The blue solid curve is plotted using the function ρ̃ = m̃2 .
extreme cases, namely, completely solid planets and completely
liquid ones, in order to see how they react to tidal consumption by
a massive hole.
The densities of solid planets like Earth will remain unchanged
when the planets are tidally stripped. And these solid planets tend
to have relatively uniform densities from the centre to the surface.
Therefore, the mean density of such a planet does not change and
the stellar orbit remains the same during the mass transfer.
Liquid planets (even with a solid core) can be taken as selfgravitating spheres following polytropic models. For such a planet,
its pressure and density roughly follows P = Kρ 1 + 1/n , where K is a
constant and the polytropic index n = 1 in this case (Kippenhahn &
Weigert 1994). As the planet is stripped, it still follows the polytropic
model and satisfies the hydrostatic equations (1) and (2). Therefore,
we can do simple calculations to obtain the new total radius Rp of
the planet:
πK
.
(10)
Rp =
2G
In other words, the total radius of the planet remains the same
against mass-loss. Therefore, the mean density of the planet decreases linearly with mass, meaning that such a planet spirals out
as mass is stripped.
4 DISCUSSION
The primary motivation for this paper is to understand the response
of different types of stars and planets when they are tidally stripped
by a massive black hole in a stable manner. Such a situation can arise
when a star or a planet is formed in an accretion disc and then evolves
through quasi-circular orbits of diminishing radius under the action
of gravitational radiation. [Other capture scenarios are possible and
have been widely discussed in the literature (e.g. Novikov, Pethick
& Polnarev 1992; Sigurdsson & Rees 1997; Alexander & Livio
2004).]
We have constrained the orbit to be circular for simplicity, because of the rapid decay of the stellar orbital eccentricity as angular
momentum and energy are carried away by gravitational radiation.
Peters (1964) gave a detailed calculation of this and showed how
Adiabatic evolution of mass-losing stars
2945
Figure 12. Semimajor axis ar as a function of orbital eccentricity e in
the decay of a two-point mass system. Different colours represent different
initial conditions. We set the initial semimajor axis ar0 to be the same
(105 Rg ) and initial orbital eccentricity e0 to be different: (from bottom to
top) 0.9999 (red), 0.999 (blue), 0.99 (green) and 0.9 (orange).
the eccentricity changes as the orbit of a two-point mass system
shrinks:
c0 e12/19
121 2 870/2299
e
,
(11)
ar =
1
+
1 − e2
304
where ar is the semimajor axis, e is the eccentricity and c0 is a
constant determined by the initial condition ar = ar0 when e =
e0 . Fig. 12 shows how e varies with ar for a set of different initial
conditions. It can be seen that the eccentricity decreases very quickly
as the orbit shrinks.
As soon as the stellar orbit shrinks so that the star fills the Roche
lobe, the star starts to be tidally stripped and its orbit evolves due to
the combined effort of gravitational radiation and the mass transfer
between the star and the SMBH. We have discussed the details of
such mass transfer and stellar orbital evolution in another paper
(Dai & Blandford 2013). We assume that in this Roche accretion
process, the star responds adiabatically to the mass-loss. In order
to check the validity of this assumption, we compare the Kelvin–
Helmholtz time-scale and the mass-loss time-scale of stars. The
Kelvin–Helmholtz time-scale is the time for the star to lose its
gravitational energy through radiation. Therefore, we can estimate
this time-scale using
TKH ∼
Gm2
.
r L
(12)
Here, L is the luminosity of the star. In order to calculate this
time-scale, we adopt the data simulated by M. MacLeod using a
stellar evolution code called Modules for Experiments in Stellar
Astrophysics (MESA). For the mass-loss time-scale, we simply use
m
(13)
TML ∼
ṁ
and adopt mass-loss rate ṁ from the subsequent paper (Dai &
Blandford 2013). The mass-loss rate depends on both the remaining
stellar mass and the mass of the SMBH.
Figure 13. A comparison of thermal and mass-loss time-scales (in log
scale) of a lower main-sequence star with initial mass ∼0.3 M (panel a)
and an upper main-sequence star with initial mass ∼7.9 M (panel b). The
thermal time-scales are plotted using solid curves. The mass-loss time-scales
are plotted using dashed curves. The red, blue and green colours denote the
scenarios when the star is tidally stripped by a 106 , 107 and 108 M SMBH.
We show in Fig. 13 the comparison of time-scales for the lower
main-sequence star and the upper main-sequence star discussed
in Section 3.2, when they are stripped by SMBHs with different
masses. For a convective lower main-sequence star, the thermal
time-scale remains much longer than the mass-loss time-scale, until
the star loses most of its mass. Therefore, the star follows adiabatic
evolution until the two time-scales are comparable. For an upper
main-sequence star with a radiative envelope, if the SMBH has a
mass on the high end, the adiabatic assumption still holds for the
majority part of the mass-loss (except initially near the surface).
However, when such an upper main-sequence star is stripped by a
massive black hole with mass ≤106 M , the two time-scales are
comparable and adiabatic assumption is a poorer approximation.
Our study shows that different types of stars behave rather differently, under the adiabatic mass-loss assumption. However, for all
stars with convective structures, such as a lower main-sequence star,
a white dwarf or a brown dwarf, their density–mass relationships
all go with the same formula (with very small deviations):
ρ̃ = m̃2 .
(14)
For main-sequence stars above ∼1 M , their densities do not monotonically decrease during the mass-loss. As their radiative envelope
is not as dense as the compact core, they tend to get denser initially
2946
L. Dai, R. D. Blandford and P. P. Eggleton
with the mass-loss, then eventually behaves similarly as the convective stars when the mass transfer goes to the core. The case of the red
giant is more complicated. The Kelvin–Helmholtz time-scale of the
envelope is much faster than the dense core. Therefore, adiabatic
mass transfer may not be a valid assumption initially. However, as
the very sparse envelope needs to respond to the adiabatic expansion of the core, we can still use the same method to calculate the
overall change of the structure. These results imply that all stars
or planets will eventually spiral out during Roche mass transfer.
Further implications of this general result will be discussed in the
next paper.
For such a stable adiabatic mass-loss to happen, the star approaching the SMBH in quasi-circular orbits needs to fill the Roche lobe
outside the ISCO of the hole. Otherwise it will plunge into the hole
before Roche overflow happens. The ISCO of a black hole of mass
M and spin parameter a has radius RISCO that satisfies the equation:
RISCO 2
RISCO
RISCO
+ 8a
− 3a 2 = 0
−6
(15)
M
M
M
(Bardeen, Press & Teukolsky 1972). The ISCO resides between Rg
(prograde with maximal spin) and 9Rg (retrograde with maximal
spin).
When the Roche mass transfer happens, we shall show in the
next paper that ρ̄R = MBH /(0.683rR3 ) in the Newtonian limit, where
ρ̄R and rR are the stellar mean density and orbital radius of the star
fulfilling the Roche condition. In other words,
1/3 MBH
ρ̄ −1/3
13
cm.
(16)
rR ∼ 1.27 × 10
106 M
ρ̄
If we compare this Roche radius with the tidal disruption radius rT ,
which is (Rees 1988)
1/3
M −1/3
MBH
R
cm ,
(17)
rT = 5 × 1012
106 M
R M
we find that rR has the same order as rT , but is bigger by a factor of
∼2. This is because the tidal stripping near the surface of the star
can happen farther from the hole than the disruption of the whole
star.
A comparison of rR and RISCO indicates whether a star can go
through the Roche process outside the ISCO, and how close to the
hole that star starts to lose mass. Fig. 14 shows how the dimensionless Roche radius (in terms of black hole mass) varies with the mass
of the black hole and the type of the star. A denser star reaches the
Roche limit closer to a massive black hole. For the same star, it can
get closer (with respect to the black hole size) to a more massive
hole before the Roche transfer starts. The plot also demonstrates
that, for example, for white dwarfs such mass transfer can only happen if the black hole has mass ≤106 M ; for a main-sequence star
approaching an SMBH with mass ≥108 M , whether the Roche
mass transfer can happen depends on the size of the ISCO which is
determined by the SMBH spin.
If the mass transfer is initiated when the star is in a relativistic
orbit, the resulting periodic emission may be detectable by X-ray
telescopes, which we will discuss in forthcoming papers. Furthermore, the behaviour of the star could, in principle, be monitored as
an ‘EMRI’ source by a future space-borne gravitational radiation
detector.
AC K N OW L E D G E M E N T S
This work was supported by the US Department of Energy contract
to SLAC no. DE-AC02-76SF00515. We would like to acknowledge
Figure 14. Roche radius versus massive black hole mass for various stars.
From top to bottom: a red giant with a mass ∼7.1 M (red, dashed), an
upper main-sequence star with a mass ∼7.9 M (blue), a Sun-like star
(orange), an Earth-like planet (green) and a lower main-sequence star with
a mass ∼0.3 M (pink), a brown dwarf with a mass 0.05 M (brown) and
a solar mass white dwarf (purple). A Jupiter-like planet has a Roche radius
similar to that of the Sun-like stars, since their mean densities are similar.
Since the red giant has a very low density envelope, it is uncertain when the
mass transfer starts.
R. Wagoner, J. Faulkner and S. Phinney for helpful discussions. In
particular, we would like to thank M. MacLeod and E. Ramirez-ruiz
for the help in providing MESA simulation data.
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