## Detecting Invisible Objects: a guide to the discovery of Extrasolar Planets and Black Holes

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

Single-line 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:

(1)
Mi ai  =  Mv a

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 (
ai  and av ),
and the masses of the two objects ( Mi  and Mv )

(From here on, special units will be used:  solar masses for M; years for P;
astronomical units for a .  In these units, 4p2/G = 1):

(2)
(Mi +  Mv) P2  =  (a + av)3

For circular orbits, the orbital speed of each object is constant throughout
the orbit and given
by

(3)
v  =  2 p a /P

The only things that can be determined from observations of a single-line 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   vv sin i   of the visible object
(the amplitude of the radial velocity variation, determined from the Doppler
effect formula)
c) the mass Mv 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:

(4)
Mi3 sin3i / (Mi + Mv)=   P (vv sin i)3/(2 p)3

[hint: first combine equations (1) and (2) to eliminate  a :  i.e., solve equation (1) for ai and
use the result to replace
ai  in equation (2); then use equation (3) with your new equation
to eliminate
av :  i.e., solve for av  in equation (3) and use the result to to replace  av  with
the variable
P and vv  ]

equation (4) is useful because only quantities that are able to be determined from observations

are present on the right-hand side of this equation.

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 left-hand 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: Mv and
2 unknowns: Mi  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 star's absolute magnitude  (to get the star's luminosity  relative to the sun, use Lstar/Lsun   2.512(4.7 - M)  don't forget to convert to actual luminosity of the sun  in watts by multiplying by Lsun M ( Lstar/Lsun) star's spectral class and  mass  (solar units)   SC   ( M/Msun) HD209458 86.5 m/s = .0182 au/yr 3.52 days = .00965 yr 4.6 G0 V (1.05)

Entering the observed quantities for the symbols on the right side of equation (4)
results in a value of the mass function MF of

MF  =  2.4 x 10-10  (solar masses is the unit, assuming you used the units above)

Therefore,
(5)

MF  =     Mi3 sin3i / (Mi + Mv)=   2.4 x 10-10  Msun

Because   sin i  <  1,

(6)

Mi3 / (Mi + Mv)>    2.4 x 10-10   Msun

Furthermore, we know the mass Mv of the visible star (from the observed spectral class
of the star; it is 1.05 solar masses.   The
previous equation then becomes

(7)

Mi3 / (Mi + 1.05 Msun)>   2.4  x 10-10 Msun

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?

(answer:  approximately  M > 0.00064 Msun  or   0.67 MJupiter)

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
Mi  term on the left-hand 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
av term on the right-hand side can be ignored, and

(8)
Mv P2  =  (ai)3

Using the values of  Mv  and  P above, we find  ai   =  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:

(9)

Pabsorbed  =   Pemitted

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 Stefan-Boltzmann law:

(10)

Lstar  (1  -  A)  (p Rp/4 p dp)2  =   4 p Rp2s Tp4

Lstar  =  luminosity (power) of the parent star
A  =  planet's albedo  =   (light reflected)/(light incidnet)
Tp =  planet's temperature
dp =  distance of planet from parent star
=   Stefan-Boltzmann constant

Solving for T gives

(11)

Tp4   =   Lstar(1 - A)/(16 p s dp2)

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 star-planet 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:

 planet orbit size dp  (a.u.) A (albedo) calculated  Tp (Kelvin) actual  Tp (Kelvin) Mercury 0.387 0.1 440 100-620 Venus 0.723 0.7 250 750 Earth 1.0 0.4 250 290 (equatorial) Moon 1.0 0.05 270 90-400 Mars 1,52 0,2 220 130-290 Asteroid Ceres 2.77 0.1 160 Jupiter 5.22 0.5 100 160 (cloudtops) Saturn 9.55 0.5 75 95 (cloudtops) Uranus 19.9 0.5 53 55 (cloudtops) Neptune 30.1 0.5 43 55 (cloudtops) Pluto 39.5 0.5 37 50 (cloudtops)

determining the planet's composition

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

 material condensation (gas, liquid  --> solid) temperature (at ~ one atmosphere) uncompressed density (g/cm3) relative abundance 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 500-600 K 2 ices (water, ammonia, methane) 150 - 300 K 1 .0001 hydrogen, helium < 20 K 1

Warning:  the make-up 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
pre-main 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 RJupiter.  Can you figure out 2 different methods of determining the planet's
the planet, the average density of this planet can be determined:  0.41
g/cm3 .  What type of
substances in the table above match a density such as this?

Black Holes

The number of strong black hole candidates of stellar-mass size is still small;
the observed properties of some of the black hole's companion are listed below

 black hole candidate (click on link for radial velocity curve) CM radial velocity velocity amplitude period of radial velocity curve visible star information orbit inclination (degrees) derived minimum mass of the invisible black hole Cygnus X-1, the first candidate 72 km/s 5.6 days O9/B0 supergiant  (M/Msun  =   33) (see  Universe, section 24-5) 7.0  Msun V404 Cygni -0.4 km/s 211 km/s 6.473 days late G or early K (K0IV?) 55 + 4 6.3  Msun A0620-00,  (V616 Mon) 10 km/s 433 km/s 7.75 hours K3.5 V 37 + 5 3.2  Msun QZ Vul GS2000+25 19 km/s 518 km/s 8.3 hours K3 - K5 V (M/Msun  =   0.55) 10 + 4 5.9 Msun Nova Scorpii 1994 (GRO J1655-40) -142 km/s 216 km/s 2.92 days (or 2.4 or 2.62 days) G0 V or F6 IV (M/Msun  =   1.1) 67 + 3 5.5 Msun GU Muscae (Nova Mus 1991) GS1124-683 16 km/s 409 km/s 10.4 hours K0 - K4 V (M/Msun  =   0.70) 54 + 20 4.2 Msun 4U1543-47 -87 km/s 124 km/s 1.12 days A2V 21 + 2 Nova Ophiuchi 1977 H1705-250 -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 Msun GRS1915+105 140 km/s 33.5 days K - M III (M/Msun  =   1.2 ) 14 Msun XTE  1650-500 3.8 Msun 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 .