An asteroid moves around the Sun in an elliptical orbit with semi-major axis and…

PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

An asteroid moves around the Sun in an elliptical orbit with semi-major axis

a = 4.00\,\text{AU}

and eccentricity

e = 0.50

. The orbital period is

8.00\,\text{years}

. The perihelion distance is therefore

2.00\,\text{AU}

and the aphelion distance is

6.00\,\text{AU}

. Assume that, during short intervals, the swept area can be taken as

\frac12 r^{2}\Delta\theta

, where

\Delta\theta

is in radians. Answer the following. (i) Calculate the areal velocity of the asteroid in

\text{AU}^2\,\text{year}^{-1}

. (ii) Find the time taken (in days) for the asteroid to advance through an angular displacement of

10^{\circ}

when it is near perihelion (

r = 2.00\,\text{AU}

). (iii) Find the corresponding time taken for the same

10^{\circ}

displacement when the asteroid is near aphelion (

r = 6.00\,\text{AU}

). (iv) Comment briefly on how the results illustrate the variation of orbital speed with distance from the Sun.