Astronomy Principles and Practice Fourth Edition.pdf

The astronomical unit 183
For an exact balance, F G = F RP and, therefore,
GM ⊙ ( 4 3 πa3 )ρ
r 2 = L ⊙ πa 2
4πr 2 c
and, hence,
a =
L ⊙ 3
GM ⊙ c 16πρ .
Substituting for the various values and constants and by assuming a typical density of
3000 kg m −3 , the balance of forces is achieved for a particle with radius
3·8 × 10 26
3
a =
6·67 × 10 −11 × 1·99 × 10 30 × 3 × 10 8 ≈ 0·2 µm.
16π × 3 × 103 Thus, for such tiny dust particles emanating from a comet, no matter the distance from the Sun, they
would continue to travel with their velocity at ejection and travel in a straight line without a force acting
on them. Measurement of the movement of some parts of the cometary dust tail verifies this. The main
body of the comet continues in its orbit about the Sun.
There are proposals and design studies for utilizing radiation pressure as a means of propulsion
within the solar system. By making very large ‘sails’ of light-weight material with high area-to-mass
ratio, it is feasible to propel light-weight space vehicles using solar radiation as the ‘fuel’. By altering
the attack of the sail at any time, the orbit of the vehicle can be readily modified at any time. The
principles behind the manoeuvres are referred to as solar sailing.
13.8 The astronomical unit
In section 12.7 it was shown how an accurate scale model of the Solar System could be obtained, all
the planetary distances being expressed in terms of the Earth’s distance from the Sun. Astronomers
have found it convenient to use data connected with the Earth’s orbit and the Sun as their units of time,
distance and mass. Newton’s precise statement of Kepler’s third law for a planet of mass m 2 revolving
about the Sun of mass m 1 may be written in the form (see equation (13.31))
k 2 (m 1 + m 2 )T 2 = 4π 2 a 3 .
Taking the solar mass, the mean solar day and the Earth’s mean distance from the Sun as the units
of mass, time and distance respectively, the equation becomes
k 2 (1 + m 2 )T 2 = 4π 2 a 3
where k 2 is written for G, the gravitational constant, and m 2 , T and a are all in the units defined earlier.
The quantity k is called the Gaussian constant of gravitation.
If, as was done by Gauss, the planet is taken to be the Earth and T given the value of 365·256 383 5
mean solar days (the length of the sidereal day adopted by Gauss), while m 2 is taken to be 1/354 710
solar masses, k is found to have the value 0·017 202 098 95, the value of a being, of course, unity. This
distance, the semi-major axis of the Earth’s orbit, was called the astronomical unit (AU).
The concept has since been refined. From time to time, various quantities have been determined
more accurately but to avoid having to re-compute k and other related quantities every time,
astronomers retain the original value of k as absolutely correct. This means that the Earth is treated like
any other planet. The unit of time is now the ephemeris day. The Earth’s mean distance from the Sun is
now taken as 1·000 000 03 astronomical units while the Earth–Moon system’s mass is 1/328 912 solar
masses.