Astroparticle Physics

33717 Solutions“The precise statement of any problem is the most important stepin its solution.”Edwin Bliss17.1 Chapter 11. a) The centrifugal force is balanced by the gravitational force between the Earth andthe satellite:mv 2= G mM ♁R ♁ R♁2 ≈ mg ,where it has been taken into account that the altitude is low and R ≈ R ♁ .Asaresult,√v ≈ gR ♁ ≈ 7.9km/s .b) The escape velocity is found from the condition that at infinity the total energyequals zero. SinceE tot = 1 2 mv2 + E potand the potential energy is given byE pot =−G mM ♁R ♁,one obtains12 mv2 = G mM √♁⇒ v = 2MR ♁ G/R ♁ ≈ 11.2km/s .♁c) The centrifugal force is balanced by the gravitational force between Earth and thesatellite. From the equality of the magnitudes of these two forces one gets:v 2r = GM ♁r 2 .From the relation between the velocity v and revolution frequency ω one obtains: