Problem Set – Kepler’s Laws and Binary Stars
Nov. 2011
Kepler's Laws
 Both objects in a binary orbit move about the center of mass in ellipses, with
the center of mass occupying one focus of each ellipse.
 A line connecting one star in a binary orbit with the other star will sweep out
an equal area in an equal time
 P2 = a3 (period in years squared = semi-major axis in AU cubed, for planets in
the solar system); the more general form is:
P2 
4 2
a3
G (m1  m2 )
P is the period, G is the gravitational constant, m1 and m2 are the masses of the two stars
in the binary system, and a is the average distance between the stars. This equation is
simple to use if we use units of years, AU, and solar masses:
a3
P 
m1  m2
2
Determining masses from visual binaries. The ratio of masses can be determined from
the ratio of the angular separations of the stars from the center of mass:
m1 a 2  2


m2 a1  1
Here, a is the separation in kilometers, and is the angular separation on the sky. If the
distance is known, then a = d, with  in radians. If the distance is known, then the
masses of the individual stars can be determined from the two equations above.
Eclipsing binaries. In the case of eclipsing binaries, we know we are looking at the
binary in the plane of its orbit (inclination i = 90 degrees, sin i = 1). If we can measure
the radial velocity of both stars, we can also get the masses of both stars:
m1 v2

m2 v1
and m  m2 
P (v1  v 2 ) 3
2G sin 3 i
If we don't know the inclination, and only have the radial velocity of one star, we can
rearrange the terms to give the "mass function:"
m23
P 3
sin 3 i 
v1
2
2G
(m1  m2 )
The mass function depends only on two observable quantities, the period and the radial
velocity. If we can estimate the mass of one component in some other way, we can
calculate the mass of the other component. If either m1 or the inclination is unknown, the
mass function sets a lower limit for the mass of m2.
Orbital eccentricity: an elliptical orbit is defined by the equation:
r + r' = 2a, where a is the semi-major axis (2a is the longest
dimension of the ellipse).
One can derive the relationship between the semi-major
axis, the semi-minor axis, and the eccentricity of the ellipse
(how squished it is):
b2 = a2 (1-e2)
Remember that a is the semi-major axis of the ellipse of the orbit, and is equal to the
average separation distance.
Perihelion is the closest approach between the stars, and the perihelion distance is given
by
rp = a (1-e)
Aphelion is the largest separation between the stars and the aphelion distance is given by:
ra = a (1+e)
Kepler's Laws/Binary Star Questions
1. A binary star system contains two stars, Star X and Star Y. Star X has a mass of 2
solar masses. The binary system is observed to have a separation currently of 2 AU
and a period of 10 months. Mass transfer is also observed from Star Y to Star X.
a. What is the combined mass of the system in solar masses? (J45)
b. What is the mass of Star Y in solar masses? (J46)
2. In a binary star system, each star has a mass of one solar mass, and the semi-major
axis is 6 AU. What is the period of revolution? (Q23)
3. The globular cluster M13 contains a binary star system with an orbital period of 26.7
years. The radius of one orbit is seen to be 1.68 times as large as the other and the
stars are 2.45 x 109 km apart. Calculate the masses of these two stars. (R33)
4. An eclipsing binary star system contains two stars, Star Q and Star R. Star R has a
mass of 0.75 solar units, a radius of 1,000,000 km, and an apparent magnitude of
+11.6. When Star Q passes in front of Star R, it completely occults the light of star
R. (See the figure below for the stars’ light curve.)
i. What is the luminosity of Star Q in solar units? (J51)
ii. What is the luminosity of Star R in solar units? (J52)
iii. What is the apparent magnitude of star Q (J53)
iv. How many times more energy per unit surface area is emitted by Star
Q? (J54)
v. What is the separation of the two stars in km? (J55)
vi. What is the distance to the star system in parsecs? (J56)
vii. What is the mass of Star Q? (J57)
viii. What is the absolute magnitude of Star R? (J58)
5. Two binary star systems are observed with separations of 2.0 AU and each contains a
1.0 solar-mass main sequence star and a fainter companion. System I has a period of
1.7 year and System II has a period of 1.9 years.
i. What is the mass of the heavier member of System I in solar masses?
(K61)
ii. What type of object is the heavier member of System I? (K62)
iii. What will happen to System I as the lighter member evolves off the
main sequence? (K63)
iv. What is the mass of the heavier member of System II in solar masses?
(K64)
v. What type of object is the heavier object in System II? (K65)
vi. What will happen to System II as the lighter member evolves off the
main sequence? (K66)
6. Image 22 shows a nearby binary system called Kruger 60. The central star, Kruger
60A, is an M-class star of mass 5.6E29 kg, and the years listed around the orbital
page indicate the associated position of Kruger 60B. The image shown is 4.26E12 m
wide and 4.68E12m high. (see image 22, Juniata)
i. What is the orbital period (in years) of Kruger 60B? (L67)
ii. What is the semi-major axis (in AU) of Kruger 60B's orbit? (L68)
iii. What is the mass of Kruger 60A in solar masses? (L69)
iv. What is the mass of Kruger 60B in solar masses? (L70)
7. What is the sum of the stellar masses in a visual binary of period 40 years, maximum
separation of 5.0", and parallax of 0.3"? Assume an orbital inclination of zero and a
circular orbit. (P17)
8. A binary system has an orbital period of 16.5 days with a total mass of 5 solar masses
and an eccentricity of 0.7. What is the distance between the two objects at periastron
and apastron in kilometers? (Q32)
Answer Key – Binary Star Questions
Kepler’s Laws/Binary Questions
1a. 12 solar masses (J45)
1b. 10 solar masses (J46)
2. 10.4 years (Q23)
3. 2.3 and 3.86 solar masses (R33)
4i. 800 solar luminosities (J51)
4ii. 5 solar luminosities (J52)
4 iii. About 6th mag (J53)
4iv 160 (J54) (if R and Q have similar radii)
4v. 7-8 x 109 km (J55)
4vi. ~500 pc (J56)
4vii. ~3 solar masses (J57)
4vii. 2.8-3.4 (J58)
5i. 1.8 solar masses (K61)
5ii. Neutron Star (K62)
5iii X-ray binary, Cataclysmic Variable (K63)
5iv. 1.2 solar masses (K64)
5v. White dwarf (K65)
5vi. Type Ia supernova, recurrent nova (K66)
6i. 45 years (L67)
6ii. 9.5 - 10 AU (L68)
6iii. 0.28 Msun (L69)
6iv. 0.14-0.2 Msun (L70)
7. *2.9 solar masses (115 solar masses P17)
8 *(periastron = 1 x 107 and apastron =5.4 x 107 (Q32)