vis viva
DESCRIPTION
sedTRANSCRIPT
The Vis-Viva EquationThe potential energy for a satellite of mass m in a planetary gravitational field is
EGMm
r
m
rpotential = − = −µ(1)
where M is the mass of the planet and µ = GM . The kinetic energy is
E mVkinetic = 12
2 (2)
where V is the speed of the satellite, i.e., V r r2 = ′ ⋅ ′r r or V r= ′r
(3)
By the law of conservation of total energy, the sum of the kinetic and potential energiesdoes not change as the satellite moves about its orbit:
E E mVm
rConstantpotential kinetic+ = − =1
22 µ
(4)
What this means is that if r1 and V1 are the position and speed of the satellite at one point ofthe orbit, and if r2 and V2 are the position and speed at a second point of the orbit, then
12
121
2
122
2mV
m
rmV
m
r− = −µ µ
(5)
Canceling out the common factor of m,
12
121
2
122
2V
rV
r− = −µ µ
(6)
This equation is true for any two points we choose along the orbit. If we let the positionand speed at point 1 be r and V (instead of r1 and V1), and if we choose point 2 to beperigee, then equation (6) becomes
12
12
2 2Vr
Vrperigeeperigee
− = −µ µ(7)
But we know already that
r a eperigee = −( )1 and Va
e
eperigee = +−
µ 11
(8), (9)
Substituting equations (8) and (9) into equation (7),
12
12
11 1
2Vr a
e
e a e− = +
−−
−µ µ µ
( )(10)
Factoring out the common µ / ( )a e1 − on the right hand side of (10)
12 1
12
1 11
12 2
22 2
2Vr a e
ea e
e
a− =
−+ −
=−
+ −
= −µ µ µ µ( )
( )( )
(11)
Solving for V2 we find that
Vr a
2 2 1= −
µ (12)
Equation (12) is called the Vis-Viva Equation.