9.2 Orbit of a Planet
131
Our next manipulation is to use the inverse of the radius, u = 1/r , rather than the
radius r. Then
r
= −
u
u 2 ,
(9.25)
and the radial equation (9.24) becomes
u
2
+ u
2
=
c
2
2
− 1
h 2
+
2mu
h 2 + 2mu
3
.
(9.26)
Next we perform another trick and differentiate (9.26) to get a second order equation;
we do this because the second order equation is a close analog of the classical equation
and easy to solve for the orbit. Thus
2u
u
+ 2uu
=
2mu
h 2 + 6mu
2 u
, thus u
+ u =
m
h 2 + 3mu
2
.
(9.27)
We will see that the first term on the right gives the classical solution, an elliptic
orbit, and the second term gives a small relativistic correction.
Let us pause to consider the special case of circular orbits, which is a fair approximation for planets in the solar system. Take the radius to be a constant r = r c , so
that (9.27) becomes
1
r c
=
m
h 2 +
3m
r c
1
r c
, circular orbit.
(9.28)
For the sun the geometric mass m is of order 1 km, which is very much smaller than
any planetary orbit, so 3m/r c is a small dimensionless quantity; moreover it is thus
clear that the first term m/ h
2 on the right side of (9.28) must be about 1/r c .
Having established the approximate relative size of terms let us return to the
general equation (9.27) and rewrite it as
u
+ u = A +
ε
A
u
2
, A ≡
m
h 2 , ε ≡ 3m A 1.
(9.29)
where A has the dimension of inverse distance, and ε is small and dimensionless.
Solution of (9.29) is a nice exercise in perturbation theory. We expand the solution
as a power series in the small parameter ε and work to first order in ε. This gives
immediately the zeroth and first order equations
u = u 0 + εu 1 , u
0 + u 0 = A, u
1 + u 1 =
u
2
0
A
.
(9.30)
131
Our next manipulation is to use the inverse of the radius, u = 1/r , rather than the
radius r. Then
r
= −
u
u 2 ,
(9.25)
and the radial equation (9.24) becomes
u
2
+ u
2
=
c
2
2
− 1
h 2
+
2mu
h 2 + 2mu
3
.
(9.26)
Next we perform another trick and differentiate (9.26) to get a second order equation;
we do this because the second order equation is a close analog of the classical equation
and easy to solve for the orbit. Thus
2u
u
+ 2uu
=
2mu
h 2 + 6mu
2 u
, thus u
+ u =
m
h 2 + 3mu
2
.
(9.27)
We will see that the first term on the right gives the classical solution, an elliptic
orbit, and the second term gives a small relativistic correction.
Let us pause to consider the special case of circular orbits, which is a fair approximation for planets in the solar system. Take the radius to be a constant r = r c , so
that (9.27) becomes
1
r c
=
m
h 2 +
3m
r c
1
r c
, circular orbit.
(9.28)
For the sun the geometric mass m is of order 1 km, which is very much smaller than
any planetary orbit, so 3m/r c is a small dimensionless quantity; moreover it is thus
clear that the first term m/ h
2 on the right side of (9.28) must be about 1/r c .
Having established the approximate relative size of terms let us return to the
general equation (9.27) and rewrite it as
u
+ u = A +
ε
A
u
2
, A ≡
m
h 2 , ε ≡ 3m A 1.
(9.29)
where A has the dimension of inverse distance, and ε is small and dimensionless.
Solution of (9.29) is a nice exercise in perturbation theory. We expand the solution
as a power series in the small parameter ε and work to first order in ε. This gives
immediately the zeroth and first order equations
u = u 0 + εu 1 , u
0 + u 0 = A, u
1 + u 1 =
u
2
0
A
.
(9.30)