Download Kepler`s Laws

Problem Set – Kepler’s Laws and Binary Stars
Nov. 2015
For Science Olympiad, you can generally assume that orbits are circular, but in real life,
orbits are usually elliptical.
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 a 3
G (m1  m 2 )
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.
For exoplanet hosts that are single-lined spectroscopic binaries, where the system is
inclined at angle i, we have the “binary mass function” form of Kepler’s law. If we know
the mass of the host star and ( astar sini) we can get the mass of the planet.
If the planet transits the star, we also know i=90 and sin i = 1.
1. HAT-P-11b was discovered by the transit method. The planet orbits its host star at a
semi-major axis of 0.053 AU and a period of 4.89 days. What is the mass of the host
star in solar masses, assuming its mass is much larger than the mass of the planet?
2. 51 Pegasi b was discovered from radial velocity variations. The host star has a mass
of 1.05 solar masses. The planet orbits with a period of 4.23 days with a semi-major
axis of 0.053 AU. What is the minimum mass the planet can have? How does it
compare to the mass of Jupiter?
3. WASP-43b orbits a KV7 star with a mass of 0.71 times the mass of the Sun. The
orbit of the planet is only 0.8 days. What is the semi-major axis of its orbit in AU?
4. The host star of WASP-18b is an F6 star with a mass of 1.25 solar masses. The
planet orbits with a semi-major axis of only 0.020 AU and a period of 0.94 days.
What is the estimated mass of the planet and how does it compare to Jupiter?
5. HD 106906b is an F5V star with a mass of about 1.5 solar masses. A planet with a
mass of about 11 times the mass of Jupiter has been detected at a minimum projected
separation from the star of 650 AU. If its current distance is its semi-major access,
what would its orbital period be? The period could be longer or shorter depending on
the actual orbit of HD 106906b.
6. 55 Cancri is a binary star at a distance of about 41 light years. It consists of a
G8Vstar with a mass of 0.95 times the mass of the sun, orbited by a smaller, M3.5V
red dwarf with a mass of 0.13 solar masses at a distance of 1065 AU. The G star
hosts a family of at least five planets, the inners most of which, planet e, transits.
a. What is the orbital period of the red dwarf?
b. Planet e has a mass of 8 times the mass of the Earth, and an orbital period of
0.78 days. What is its semi-major axis?
7. Kepler-186 is an M1V star at a distance of 500 light years, with a mass of 0.48 solar
masses. It is orbited by at least 5 planets. Planet f has an orbital period of 130 days.
What is its semi-major axis?
8. HR 8799, a young star with a mass of 1.47 solar masses, hosts four exoplanets that
have been detected in images. The planets have semi-major axes of 14.5, 24, 38, and
38 AU. What are their orbital periods?
More Practice Questions for Kepler’s Laws
1. Two stars orbit in a binary system with a period of 40 years. The orbits are circular,
and the stars are separated by 20 astronomical units. What is the total mass of the two
stars?
2. An exoplanet has been detected orbiting a star similar to the Sun. The period of the
brown dwarf’s orbit is 4.6 days, and its semi-major axis is 0.05 AU. What is the mass
of the star?
3. A star has been observed to orbit the black hole at the center of the Milky Way
Galaxy with an orbital period of 12 years. The semi-major axis of the orbit is 800
AU. What is the mass of the black hole?
4. The orbital period of Sirius A and B is 50 years, and the semi-major axis of the
relative orbit is 20 AU. The ratio of the distances of Sirius A and Sirius B from the
center of mass is aA/aB=0.466. Calculate the masses of both stars.
5. 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 the orbit?
6. 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 semi-major axis of 2 AU and a
period of 10 months.
a. What is the combined mass of the system in solar masses?
b. What is the mass of Star Y in solar masses?
7. 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.