Answer to Question #195707 in Astronomy | Astrophysics for Pranali Patil

Question

Comets in the Solar System orbit the Sun in the same way the planets do but they are highly elliptical orbits.  A partial list of comets can be found here - <span style="font-size: 10.5pt; color: rgb(17, 85, 204); background-color: transparent; font-variant-numeric: normal; font-variant-east-asian: normal; text-decoration-line: underline; text-decoration-skip-ink: none; vertical-align: baseline; white-space: pre-wrap;">https://nssdc.gsfc.nasa.gov/planetary/factsheet/cometfact.html</span>.  From this list, choose two comets and calculate the Kepler ratio for each of these comets orbiting the Sun.  Compare the ratios and determine whether or not Kepler’s third law applies for comets around the Sun in a galaxy.

Answer 1

Let us consider the 2P Encke comet with semi-major axis of orbit $a_1 = 2.21$ a.u. and period $T_1 = 3.30$ years and comet 21P Giacobini-Zinner with semi-major axis of orbit $a_2 = 3.52$ a.u. and period $T_2 = 6.52$ years.

The Kepler ratio for the first comet is

$\dfrac{T_1^2}{a_1^3} = \dfrac{3.30^2}{2.21^3} \approx 1.01.$

The Kepler ratio for the second comet is

$\dfrac{T_2^2}{a_2^3} = \dfrac{6.52^2}{3.52^3} \approx 0.97.$

We can see both the ratios are very close to 1 as it should be for the objects of the Solar system. The discrepancy may be due to the uncertainty in the initial data or due to the existence of another massive objects like Jupiter or Saturn, so the solution of two-body problem is not very accurate and correct.

Nevertheless, Kepler’s third law may be applied for comets if we want to obtain an approximate estimate.