Gravity: Circular Orbits and Escape Energy

Compare circular speed, orbital energy and escape speed in an ideal model.

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For a small satellite outside a spherical body, set mu = GM. At circular radius r, v = sqrt(mu/r), specific total energy E/m = -mu/(2r), and period T = 2pi sqrt(r^3/mu). Escape speed at that radius is sqrt(2mu/r), neglecting other bodies and drag.

Worked example

In consistent model units, mu = 100 and r = 4 give circular speed 5 and specific total energy -12.5. Escape speed is 5 sqrt(2). Negative total energy indicates a bound orbit in this isolated model.

With potential energy zero at infinity, U equals minus GMm divided by r. At center distances r0, 2r0 and 4r0, relative energies are minus one, minus one half and minus one quarter.

Would you believe it?

Earth is in orbit too: it moves around the Sun under gravity. An orbit combines motion with a continually changing direction.

NASA: gravity and mechanics

Question 1 With mu = 100 and r = 4, circular speed is what? 5 25 50

Question 2 At the same radius, escape speed divided by circular speed is what? 2 1/2 sqrt(2)

Question 3 With mu = 100 and r = 4, specific total energy is what? -25 -12.5 +25

Question 4 If circular radius quadruples with mu fixed, period changes by what factor? 8 4 2

Question 5 Zero total energy has what meaning here for an escaping trajectory? It must stay at rest at every radius Gravity is zero at the starting point It just reaches infinity with speed approaching zero

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