general theory: how to
determine the mass of an invisible object in a "binary" system
a worked example:
determination
of the mass, orbital radius, temperature, and composition of the planet
accompanying the star HD 209458
black
hole data for determining the mass of an invisible black hole in a
binary system
general theory: how to determine the mass of an invisible object in a "binary" system
Singleline spectroscopic binaries are special
cases of binary
stars in which
only one star is visible, yet the binary nature of
the system is apparent because
the single star that is visible shows periodic
doppler variation in its spectral lines.
For simplicity, let's assume that
1) the orbits are circular, in which case
the radial velocity curves (determined from the
Doppler effect) have
a sinusoidal shape. The spectroscopic
binary applet will help
give you a visual
picture of how the shape of the radial velocity variations is
determined by the shape
of the orbits, the size of the orbits, the masses of the
orbiting objects in the system,
and the orientation of the orbits relative to Earth.
(Once the applet opens, set the
parameters to the following values:
e =
0 which means the orbits are circular
i
= 90 which means that the orbits are in the
plane of Earth's line of sight
(see below),
and then
click on Enter button to update the changes that you have made.
2) that the plane of the two orbits is
inclined
to the plane of the sky by an inclination
angle of 90 degrees. [This
means that the plane of the
orbits
is
inclined by (90 
i) to the earth's line of sight to the system.]
In general, the
actual value of this inclination angle unobservable
and unknown.
{diagram showing the angle i goes here}
The two equations that describe the physics of binary systems are
1) the center of mass condition:
where
a = radius of the circular orbit
M = mass of the orbiting object
and the subscripts i and v
stand for
invisible
and visible object, respectively.
2) Kepler's 3rd law
defines the relation between the common orbital period P of the
two
objects around the center of mass, the orbit sizes ( a_{i} and a_{v }),
and the masses of the two objects ( M_{i} and M_{v })
(From
here on, special units will be used: solar masses
for
M; years for P;
astronomical units for a . In these units, 4p^{2}/G
= 1):
For circular orbits, the orbital speed of
each
object is constant throughout
the orbit and given
by
The only things that can be determined from
observations of a singleline spectroscopic
binary are
a) the orbital
period
P of the radial velocity of the visible object
(this is
actually the orbit period of both the invisible and visible object)
b) the orbital
speed
v_{v} sin i of the visible object
(the amplitude
of the radial velocity variation, determined from the Doppler
effect formula)
c) the mass M_{v}
of the visible object (inferred from its spectral type)
Therefore, any information derived about
the
invisible object must use only these
3 determinable quantities.
Combining equations (1)  (3) above gives:
[hint: first combine equations (1) and (2)
to eliminate a_{i }
: i.e., solve equation (1) for a_{i}
and
use the result to replace a_{i}
in equation (2); then use equation (3) with your new equation
to eliminate a_{v} :
i.e., solve for a_{v} in
equation (3) and use the result to to replace a_{v}
with
the variable P and v_{v}
]
Extrasolar Planets
determining the planet's mass Let's
use equation (4) along with the observed
properties of a planet's parent star
to determine the mass and the orbit size
of the invisible planet.
The number of detected extrasolar planets now
exceeds 130;
a complete list can
be found at either
the quantity on the
lefthand
side of equation (4) is called the mass function; it doesn't seem,
at first glance, to
be particularly useful, since it is a function of 3 things (1 known: M_{v}
and
2 unknowns: M_{i}
and i ) and involves an inequality rather than an equality.
It turns out, however, that a useful lower limit
to
Mi can be found.
(remember that the period should be in years;
the orbit speed, in au/yr; and the masses,
in solar units; a table of
stellar
masses according to
spectral type will be needed)
The star HD 209458
was the first to have its planet detected both by spectroscopic
and photometric methods. The radial velocity
of the star varies with time over a regular
period of 3.52 days.
The observed quantities for this system are
star
with extrasolar planet 
star's radial velocity amplitude 
period of radial velocity variation 
relative to the sun, use L_{star}/L_{sun} 2.512^{(4.7  M) } don't forget to convert to actual luminosity of the sun in watts by multiplying by L_{sun} M ( L_{star}/L_{sun})_{ } 
and SC 
HD209458 
86.5
m/s = .0182 au/yr 
3.52 days = .00965 yr 
4.6 
G0 V (1.05) 
MF = 2.4 x 10^{10} (solar masses is the unit, assuming you used the units above)
MF =
M_{i}^{3 }sin^{3}i / (M_{i} + M_{v})^{2
}= 2.4 x 10^{10} M_{sun}
Because sin i <
1,
Furthermore,
we know the mass M_{v }of
the visible star (from the observed spectral class
of the star; it is 1.05 solar masses. The
previous equation then becomes
M_{i}^{3} / (M_{i} + 1.05 M_{sun})^{2 }> 2.4 x 10^{10} M_{sun}
We now have
an equation in a single
unknown;
although it cannot be solved analytically,
it can be easily solved by trial and error
(guessing values) or by using a graphing calculator.
Can you find the solution to this inequality?
determining the planet's orbital radius
Once
an upper limit to the planet's mass is known, its orbit
radius
(or distance from
its parent star) can
be found from equation (2) above. The planet's mass is very much
smaller than its parent star's mass; therefore,
the M_{i} term on the lefthand side can be
ignored. Similarly [because of the center of
mass condition, equation (1)], the star's
orbit size around the system center of mass is much
smaller than the planet's orbit size.
Therefore, the a_{v}
term on the
righthand side can be ignored, and
Using the values of M_{v} and P above, we find a_{i} = 0.046 au. This
is about 9x smaller
than Mercury's orbit about the sun.
determining the planet's temperature
Next, let's calculate
the
equilibrium
blackbody temperature of a planet. We assume that
thermal equilibirium (i.e., constant
temperature)
applies, and consequently that the power
( = energy/time) emitted by the
planet is the power absorbed
from its parent star:
P_{absorbed }= P_{emitted}
_{ }The left hand side
is
found from geometry, corrected by a coefficient that takes into account
reflected light; the right hand side is given
by the StefanBoltzmann law:
L_{star} (1  A) (p R_{p}/4 p d_{p})^{2} = 4 p R_{p}^{2}s T_{p}^{4}
^{ }L_{star}
=
luminosity (power) of the parent star
A = planet's albedo
= (light reflected)/(light incidnet)
R_{p} = planet's radius
T_{p} = planet's temperature
d_{p} = distance of planet from
parent star
s =
StefanBoltzmann constant
Solving for T_{p } gives
T_{p}^{4 } = L_{star}(1  A)/(16 p s d_{p}^{2})
Notice that the equilibrium temperature
depends
on the "guessed" albedo of the planet; the
ratio of the temperature derived with albedo
= 0.95 to the temperature derived with an
albedo of 0.05 is approximately 2.
Albedos of planets in our solar system are listed in the
table below. The lowest albedo is around 0.05
(Earth's moon); the highest, around 0.7 (Venus).
This calculation doesn't take into account
thermal energy released from the planet's interior,
tidal energy released via a starplanet
interaction,
the greenhouse effect in the atmosphere,
etc. It would be a good idea to have
students calculate the equilibrium temperature of the
solar system's planets first to have students
have some sense of how close this calculation
comes to actual temperatures.
data for solar system objects:

(a.u.) 
(albedo) 
_{(Kelvin)} 
_{(Kelvin)} 























































determining the
planet's
composition
with the termperature, the physical status
(solid, liquid, or gas) of various chemicals in the
planet's makeup can be determined... the
types of materials that make up (typical solar
system) planets include

(gas, liquid > solid) temperature (at ~ one atmosphere) 
uncompressed density (g/cm^{3}) 
in solar nebula 
Ca, Ti, Al silicates  1400  1800 K  3 
.00001 
Fe/Ni 
1300  1500 K  7 
.00001 
Mg
silicates 
1300
K 
3 

other metalic silicates & sulfides 
900  1200 K  3 

hydrous silicates 
500600 K 
2 

ices (water, ammonia, methane) 
150  300 K  1 
.0001 
hydrogen, helium  < 20 K  1 
Warning: the makeup of a planet may
more likely represent the composition (and physical
state) at birth rather than at present.
In general, stellar luminosities were higher during their
premain sequence phase (when the planets
formed) than at present (when the star is in the
main sequence phase).
Unfortunately, the size of the planet can
be
determined (unless the planet can actually be
imaged or until it eclipses or is eclipsed
by another object), no further information
(gravitational field, escape velocity,
atmospheric
retention, density) can be determined.
However, the planet around HD209458 happens to
be one of those that transits its star.
The transit
data is here. From the transit data, it can be determined
that the radius of this
planet is 1.27 R_{Jupiter}. Can you
figure out 2 different methods of determining the planet's
radius from the lightcurve data given in the
link? Because both the mass and radius of
the planet, the average density of this planet can
be determined: 0.41 g/cm^{3} .
What type of
substances in the table above match a density such
as this?
Black Holes
The number of strong black hole candidates of stellarmass
size
is still small;
the observed properties of some of the black hole's companion are
listed
below
(click on link for radial velocity curve) 
CM radial velocity 
velocity amplitude 
radial velocity curve 

orbit inclination (degrees) 
invisible black hole 
Cygnus
X1, the first candidate 
72 km/s 
5.6
days 
O9/B0
supergiant (M/M_{sun} = 33) (see Universe, section 245) 
7.0 M_{sun}  
V404
Cygni 
0.4 km/s 
211 km/s 
6.473
days 
late
G or early K (K0IV?) 
55 + 4 
6.3 M_{sun} 
A062000,
(V616 Mon) 
10 km/s 
433 km/s 
7.75 hours  K3.5 V  37 + 5 
3.2 M_{sun} 
QZ Vul GS2000+25 
19 km/s 
518 km/s 
8.3
hours 
K3
 K5 V (M/M_{sun} = 0.55) 
10 + 4 
5.9 M_{sun} 
Nova
Scorpii 1994 (GRO J165540) 
142 km/s 
216 km/s 
2.92 days (or 2.4 or 2.62 days) 
G0
V or F6 IV (M/M_{sun} = 1.1) 
67 + 3 
5.5 M_{sun} 
GU Muscae (Nova Mus 1991) GS1124683 
16 km/s 
409 km/s 
10.4 hours 
K0
 K4 V (M/M_{sun} = 0.70) 
54 + 20 
4.2 M_{sun} 
4U154347 
87
km/s 
124
km/s 
1.12
days 
A2V 
21
+ 2 

Nova
Ophiuchi 1977 H1705250 
54 km/s 
420 km/s 
12.51 hours 
K5V 
60  80 

Nova
Velorum 1993 
475
km/s 
0.285
days 
K6
V  M0 V 
3.6  4.7 M_{sun}  
GRS1915+105 
140
km/s 
33.5
days 
K
 M III (M/M_{sun} = 1.2 ) 
14 M_{sun} 

XTE 1650500 
3.8
M_{sun} lowest mass; found by quasi periodic oscillations 
for another list of black hole candidates, go here
How do we know that these objects are black
holes and not something else like planets, or
neutron stars, or white dwarfs, or brown
dwarfs,
or just some type of star? Notice that all the
black holes candidates are at least 3 solar
masses... the upper limit to neutron star masses is
approximately 3 solar masses; the upper limit
to white dwarf masses is 1.4 solar masses; the
upper limit to brown dwarf masses is 0.8 solar
masses; the upper limit to planetary masses is
even smaller.... therefore each of these
candidates
is ruled out as a possible companion to
these stars...
A different argument is required to rule
out
another star as the possible "invisible"
companion. Here is a
table of stellar
mass objects arranged by temperature and
brightness.
Notice that all of the objects in the stellar
table that have mass more than this minimum
3 solar masses are bright (in most
cases, in fact, brighter than the visible star in the
system) and
therefore
should be visible  unless it is a black hole
.