IGCSE Physics 0625: Orbital Speed, v = 2πr/T and Elliptical Orbits

Astronomers studying a distant star, Talara, have identified two planets in its system: Vesa, the closer of the two, and Corim, which orbits much further out.

(a) Vesa orbits Talara at an average orbital radius of $2.00\times10^{11}\text{ m}$ , completing one full orbit every $3.00\times10^7\text{ s}$ . Determine the average orbital speed of Vesa. [3]

(b) Corim orbits Talara at a much greater average orbital radius than Vesa. Without doing any further calculation, state how the average orbital speed of Corim compares with that of Vesa, and explain your answer in terms of the strength of Talara's gravitational field. [2]

(c) Vesa's orbit is noticeably elliptical rather than perfectly circular, so its distance from Talara varies during one orbit. State and explain, using the idea of conservation of energy, how Vesa's orbital speed changes as it moves from the point in its orbit closest to Talara to the point furthest away. [2]

Part (a): Average orbital speed of Vesa

The average orbital speed of a body in a roughly circular orbit is found from:

$v = \frac{2\pi r}{T}$

where $r$ is the average orbital radius and $T$ is the orbital period.

The distance travelled in one orbit is the circumference of the orbit:

$2\pi r = 2\pi \times 2.00\times10^{11}\text{ m} = 1.2566\times10^{12}\text{ m}$

Dividing by the orbital period:

$v = \frac{1.2566\times10^{12}\text{ m}}{3.00\times10^7\text{ s}}$

$v = \boxed{4.19\times10^4\text{ m/s}}\ \text{(3 s.f.)}$

Part (b): Comparing Corim’s orbital speed with Vesa’s

Corim orbits more slowly than Vesa.

The strength of Talara’s gravitational field decreases with increasing distance from the star. Since Corim orbits at a much greater distance from Talara than Vesa does, it experiences a weaker gravitational pull, and only needs a smaller orbital speed to remain in a stable orbit. This is why planets further from their star have smaller average orbital speeds than planets closer in.

Part (c): Speed changes in Vesa’s elliptical orbit

Because Vesa’s orbit is elliptical, its distance from Talara is not constant, so (unlike in a perfectly circular orbit) its orbital speed is not constant either.

Vesa moves fastest at the point in its orbit closest to Talara, and slowest at the point furthest away.

This can be explained using conservation of energy. The total energy of Vesa (its kinetic energy plus its gravitational potential energy) stays constant throughout the orbit.

As Vesa moves closer to Talara, its gravitational potential energy decreases, and this energy is transferred to kinetic energy, so its speed increases.
As Vesa moves further from Talara, its kinetic energy decreases as it is transferred back into gravitational potential energy, so its speed decreases.

Final answers

(a) Average orbital speed of Vesa $=$ $4.19\times10^4\text{ m/s}$ (3 s.f.)
(b) Corim orbits more slowly than Vesa, because Talara’s gravitational field is weaker at Corim’s greater distance
(c) Vesa is fastest closest to Talara and slowest furthest away, as gravitational potential energy and kinetic energy are exchanged while the total energy stays constant