Astronomy Principles and Practice Fourth Edition.pdf

184 Celestial mechanics: the two-body problem
We may note that the precise definition of the astronomical unit is given by Kepler’s third law,
k 2 (1 + m)T 2 = 4π 2 a 3
with the Sun’s mass taken to be unity, k = 0·017 202 098 95, and the unit of time taken to be one
ephemeris day. It is the radius of a circular orbit in which a body of negligible mass, undisturbed by
the gravitational attractions of all other bodies, will revolve about the Sun in one Gaussian year of
2π/k ephemeris days. It has a value of 1·496 00 × 10 11 m.
Problems—Chapter 13
Take the length of the year to be 365·25 days.
1. An asteroid’s orbit has a semi-major axis of 4 AU. Calculate its sidereal period.
2. Given that the semi-major axis of the orbit of Venus is 0·7233 AU, calculate its sidereal period.
3. The synodic period of Mars is 780 days. Calculate its mean distance from the Sun in astronomical units.
4. The orbital period of Jupiter’s fifth satellite about the planet is 0·4982 days and its orbital semi-major axis is
0·001 207 AU. The orbital period and semi-major axis of Jupiter are 11·86 years and 5·203 AU respectively.
Find the ratio of the mass of Jupiter to that of the Sun.
5. Suppose a planet existed in a circular orbit about the Sun of radius 0·1 AU. What would be its period of
revolution
6. A communications satellite in a circular equatorial orbit always remains above a point of fixed longitude.
Given that the sidereal day is 23 h 56 m long, the year 365 1 4
days in length and the distances of the satellite
from Earth’s centre and Earth from the Sun are 41 800 km and 149·5 × 10 6 km respectively, calculate the
ratio of the masses of the Sun and Earth.
7. The minimum and maximum heights of an artificial Venusian satellite above the solid surface of the planet (as
measured by radar) are 696 and 2601 km respectively. The satellite’s period of revolution is observed to be
104 minutes. If the semi-major axis and sidereal period of the Venusian orbit are 0·723 AU and 0·615 years
respectively, calculate the radius of Venus, given that 1 AU is 149·5 × 10 6 km and the mass of Venus is
1/403 500 times the Sun’s mass.
8. The period of revolution of an artificial satellite of the Moon is 2 h 20 m . Its minimum and maximum distances
above the lunar surface are, respectively, 80 and 600 km. If the radii of the Moon and of the Moon’s orbit are
1738 and 384 400 km respectively and the lunar sidereal period is 27 1 3
days, calculate the semi-major axis
and eccentricity of the lunar artificial satellite’s orbit and the ratio of the masses of the Earth and Moon.
9. The period of Jupiter is 11·86 years and the masses of the Sun and Jupiter are 330 000 and 318 times that of
the Earth respectively. Calculate the change in Jupiter’s orbital period if its mass suddenly became the same
as that of the Earth.
10. The semi-major axis of the orbit of Mars is 1·524 AU and the orbital eccentricity is 0·093. Assuming the
Earth’s orbit to be circular and coplanar with that of Mars, calculate (i) the distance of Mars from the Earth
at closest approach, (ii) the ratio of the speeds of Mars in its orbit at perihelion and aphelion and (iii) the
speed of Mars at perihelion in AU per year.
11. Halley’s comet moves in an elliptical orbit of eccentricity 0·9673. Calculate the ratio of (i) its linear
velocities, (ii) its angular velocities, at perihelion and aphelion.
12. Two artificial satellites are in elliptical orbits about the Earth and they both have the same value for their semimajor
axes. The ratio of the linear velocities at perigee of the two satellites is 3 2 and the orbital eccentricity
of the satellite with the greater perigee velocity is 0·5. Find the orbital eccentricity of the other satellite and
the ratio of the apogee velocities of the two satellites.
13. An artificial satellite moves in a circular orbit inclined at 30 ◦ to the equator in a period of 2 hr at a height of
1689 km. What is its horizontal parallax Neglecting the Earth’s rotation, find the maximum time for which
the satellite can remain above the horizon (a) at a station where the satellite passes directly overhead, (b) at
Winnipeg (latitude 50 ◦ N). (Earth’s radius = 6378 km.)