(a) What velocity must an artificial Earth satellite have to…

(a) What velocity must an artificial Earth satellite have to move in a circular orbit at a height $H$ above the Earth's surface? (b) What is its period? (c) What is the first cosmic velocity (the orbital velocity at the Earth's surface), and what is the corresponding period?

Step 1: Idea

A satellite is a body in permanent free fall. It falls toward the Earth all the time, but moves sideways fast enough that the curved surface drops away beneath it just as fast. Mathematically: gravity supplies the centripetal force for a circle of radius $R + H$.

Step 2: (a) Velocity

\[ m\frac{v^2}{R + H} = \gamma\frac{Mm}{(R + H)^2}. \]

The satellite's mass cancels. Using $\gamma M = gR^2$, with $g$ the surface value: $v = \sqrt{\frac{\gamma M}{R + H}} = R\sqrt{\frac{g}{R + H}}$.

The higher the orbit, the slower the satellite.

Step 3: (b) Period

The period is the length of the orbit divided by the speed: $T = \frac{2\pi(R + H)}{v}$.

\[T = 2\pi\sqrt{\frac{(R + H)^3}{\gamma M}}\]

This is Kepler's third law: $T^2$ is proportional to the cube of the orbit radius.

Step 4: (c) At the surface ($H = 0$)

The first cosmic velocity is the speed of a (hypothetical) satellite skimming the surface, ignoring air and mountains: $v_1 = \sqrt{gR} = \sqrt{9.81 \cdot 6.37 \cdot 10^6}$.

\[ T_1 = 2\pi\sqrt{\frac{R}{g}}. \]

\[\begin{gathered} v_1 \approx 7.9\ \mathrm{km/s} \\ T_1 \approx 5060\ \mathrm{s} \approx 84\ \mathrm{min} \end{gathered}\]

This is the smallest speed with which any body can orbit the Earth. Real low satellites (e.g. at 400 km) move at about 7.7 km/s with periods of about 92 min.

Check

$ 2\pi R/v_1 $ $= 2\pi \cdot 6.37 \cdot 10^6/7905 $ $= 5063\ \mathrm{s}\ \checkmark $

Units: $ \sqrt{\mathrm{m/s^2} \cdot \mathrm{m}} $ $= \mathrm{m/s}\ \checkmark $