(i) What is the minimum launch speed required to put a satellite into a circular orbit?

(ii) How many times higher is the energy required to launch a satellite into a polar orbit than that necessary to put it into an Equatorial one?

(iii) What initial speed must a space probe have if it is to leave the gravitational field of the Earth?

(iv) Which requires a higher initial energy for the space probe-leaving the solar system or hitting the Sun?

Question

(i) What is the minimum launch speed required to put a satellite into a circular orbit?

(ii) How many times higher is the energy required to launch a satellite into a polar orbit than that necessary to put it into an Equatorial one?

(iii) What initial speed must a space probe have if it is to leave the gravitational field of the Earth?

(iv) Which requires a higher initial energy for the space probe-leaving the solar system or hitting the Sun?

Accepted Answer

(i) The acceleration of a satellite moving at speed [latex] v [/latex] in a circular orbit of radius [latex] R [/latex] is[latex] g = v^2 / R [/latex] . If [latex] R [/latex] is the radius of the Earth (or more precisely, a slightly larger value), then [latex] g [/latex] has to be the gravitational acceleration at the Earth's surface; this defines the 'first cosmic speed', [latex] v_1=\sqrt{Rg}=7.9 \ km \ s^{-1} [/latex] for the speed of the satellite.

All of this speed is not strictly necessary for launching a satellite if the 'initial speed' provided by the rotation of the Earth is taken into account. This help is greatest at the Equator, approximately [latex] 0.5 \ km \ s^{-1} [/latex], and means that an initial speed of [latex] 7.4 \ km \ s^{-1} [/latex] with respect to the Earth is sufficient, but only if the satellite is launched eastward from on or near the Equator.

(ii) The angular momentum of satellites in polar orbits (passing over the poles) is zero with respect to the axis of rotation of the Earth. This condition has to be fulfilled right from the launch, since the angular momentum will not change later. The 'help' described above cannot therefore be utilised. Indeed, the rotation of the Earth is a drawback, since, not only must the satellite receive the speed of [latex] 7.9 \ km \ s^{-1} [/latex] in a north-south direction, but, in addition, the unacceptable west-east speed due to the rotation of the Earth has to be cancelled. The latter would not have to be taken into account if the satellite were launched from the neighbourhood of one of the poles; there are obvious technical difficulties in doing this!

The initial speed necessary to put a satellite into a polar orbit is therefore at least [latex]7.9/7.4 ≈ 1.06 [/latex] times greater, i.e. the necessary kinetic energy is at least [latex] 1.13 [/latex] times as great. This does not seem a big difference, but in reality the slightest increase of the initial speed requires an enormous effort. The reason for this is that the carrier rocket has to be accelerated as well as the satellite, and the mass to be launched increases exponentially with the intended final speed.

(iii) In order to escape the attraction of the Earth (i.e. to move far from the Earth), the probe has to acquire the escape or 'second cosmic' speed, [latex] v_2=\sqrt{2gR}=\sqrt{2}v_1\approx 11.2 \ km \ s^{-1} [/latex] . The rotation of the Earth can again be used. Launching eastward from the Equator, a launch speed of [latex] 10.7 \ km \ s^{-1} [/latex] relative to the Earth is sufficient.

(iv) The Earth revolves around the Sun at a speed of approximately [latex]30 \ km \ s^{-1} [/latex] . In order to reach the Sun, a space probe has to be launched at an initial speed of [latex]30 \ km \ s^{-1} [/latex] (or more exactly, [latex]29.5 \ km \ s^{-1} [/latex]; smaller by [latex]0.5 \ km \ s^{-1} [/latex] due to the rotation of the Earth). If the aim is to leave the solar system, the space probe has to reach [latex] \sqrt{2} [/latex] times the speed of [latex]30 \ km \ s^{-1} [/latex], but the 'initial speed' of the Earth in its orbit can be subtracted from this value if the launching is well timed and directed, i.e. a speed of [latex]12 \ km \ s^{-1} [/latex] is sufficient.

Thus, it is easier to make a space probe leave the solar system than to send it into the Sun. (The former has been successfully attempted, the latter we are still waiting for.) The situation is even more favourable if the possibilities offered by the outer planets (Mars, Jupiter, Saturn, ... ) are taken into account. A space probe launched at the right time and in the right direction can be significantly 'pushed by' (i.e. receive energy from) these planets, a phenomenon known as 'gravitational slingshot'. Designed in this way, the space probe does not have to propel itself very far, it only has to reach Mars or Jupiter and the rest happens 'automatically'.

Note. If the rocket driving the probe can be fired in two stages, one to induce an elliptical orbit and the other to bring the rocket to rest at the aphelion of the orbit, less total energy is needed to hit the Sun than that calculated above.

Question P.17: (i) What is the minimum launch speed required to put a satel...

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