Question

Consider the Oort clound to consist of a large number of comets at a distance from the Sun where the period of a circular orbit around the Sun would be a million years.

A. What is the radius of this orbit?

B. If a comet in this orbit is perturbed, so that its new orbit has an aphelion at its original orbital radius, but now a perihelion at 0.3AU, how long will it take to reach its perihelion?

Answer 1

Time period T = 1 milion years = 10⁷ years = 365*24*3600*10⁷ seconds

Orbital radius r =?

Use kepler's 3rd law: T² / r³ = (4*pi*pi)/(GM)

Mass of sun M = 1.99 * 10³⁰ kg

r³ = T² * GM / (4*pi*pi) = (365*24*3600*10⁷)² * 6.67 * 10^-11 * 1.99 * 10³⁰ / (4*3.14*3.14) = 3.35 * 10⁴⁷ m³

Or,

A. Orbital radius r = cube root (3.35 * 10⁴⁷ m³) = 6.95 * 10¹⁵ m = 6.95 * 10¹⁵ / (1.496 * 10¹¹) = 4.65 * 10⁴ AU

B.

comet in this orbit is perturbed, so that its new orbit has an aphelion at its original orbital radius, but now a perihelion at 0.3AU, how long will it take to reach its perihelion?

Aphelion r' = 46500 AU

perihelion r'' = 0.3 AU

First we find orbital speed of the oort cloud/ comets when it was not yet perturbed:

v = square root (GM/r) = square root (6.67 * 10^-11 * 1.99 * 10³⁰ / 6.95 * 10¹⁵ ) = 138.2 m/s

Semi major axis a = distance between perihelion and aphelion = r' -r'' = 46500 - 0.3 AU = 46499.7 AU

Which can be approximated to be same as Aphelion or the original radius.

Thus time taken to go from aphelion to perihelion will be same as time for covering half the ellipse or half the original orbital radius = 0.5 billion years.