Solutions To Angular Momentum Applied To Orbital Motion Problems - Practical Astronomy Focus

The Problems:

1) Find the velocity of said geosynchronous satellite which matches Earth's rotation rate.

2) Verify the conservation of angular momentum applies for a spacecraft in orbit around the Earth if its velocity at perigee is  10.7 km/ sec, its distance from Earth at perigee is  6.6 x 10 ³ km, its velocity at apogee is  0.75  km/ sec  and its distance at apogee is: 9.3 x 10 ⁴ km.  Find the period of the spacecraft. 

Solutions:

1) To find the velocity:

v = w r  =  (2p/ T) r

= { 2p x 42.4 x 10⁶ m} / 86400 s.

v = 3100 m/s

2) For conservation of angular momentum L in an elliptical orbit (with r_a  the radius vector at aphelion,  and  with r_p the radius vector at perihelion)

 L = mv_a r_a =  mv_p r_p

Or: v_a r_a =  v_p r_p

v_a r_a =    (0.75  km/ sec) (9.3 x 10 ⁴ km) » 7.0 x 10 ⁴ km²/sec

v_p r_p =    (10.7 km/ sec) (6.6 x 10 ³ km)

 » 7.0 x 10 ⁴ km²/sec

Given:  v_a r_a »  v_p r_p

To within the limits of errors, the principle of angular momentum conservation  is validated.

The period is obtained from: T = 2π (a³/m)^½

Where m  = G (m1 + m2) 

And:  a = [r_a +  r_p ] /2 =  (9.3 x 10 ⁴ km + 6.6 x 10 ³ km)/2 »

= (9.3 x 10 ⁷ m + 6.6 x 10 ⁶ m)/2 » 4.9 x 10 ⁷ m

T »

2π [(4.9 x 10⁷ m)³/[6.7 x 10^-11 Nm²/kg²) (6.0 x 10²⁴ kg)]^½

T » 4.3 x 10 ⁴ s » 12.2 hrs.