354 EINSTEINBESSO ON THE MERCURY PERIHELION
where
C is
a
constant.
Since
f
depends
on
t through
r,
it
follows that
Kepler's
second
law
does
not
hold
in this
case.
The Hamiltonian of
a
unit
point mass
moving
in
the
field
of
a
rotating
sun
is
used
to
obtain
an expression
for
q2
(see
[p.
21],
[eqs.
144145]). Writing
q2
and
xy

yx
in
polar coordinates,
one
obtains
two
equations relating
dr,
d(p,
and
dt
(see [p.
22],
[eqs.
146147];
[p.
23],
[eqs.
155156];
[p.
24],
[eqs.
164165];
[p. 32], [eqs.
189190]).
When
dt
is
eliminated from these
two equations,
an
equation
of
the
form of
eq.
(7)
is
obtained,
relating
d(j)
and dr
(see [p.
22],
[eq. 148]; [p. 23], [eq. 157];
[p.
24],
[eq.
166];
[p.
32],
[eq.
191]).
Using
eq.
(8)
to
integrate
this
equation
between
r1
and
r2,
the values
of
r
at
perihelion
and
aphelion,
one
finds
the
angle
between
perihelion
and
aphelion.
The
final
result
is given
in
[eq. 204] on [p. 34]
and in
[eq. 210] on [p.
35]:[46]
or,
/
dØ
_ _
2SoK2MT3
Jr1
a6(1
)
where
T is
the
period
of revolution of
the orbit. All
other
symbols
were
defined
above.
From
eq.
(18)
one sees
that the rotation of the
sun
produces
a
retrogression
of the
perihelion.
Some numerical calculations
for the effect
in
the
case
of
Mercury
are
found
on
[p.
29]
and
[pp.
3435]. First,
the
quantity
S
is
evaluated
making
the
simplifying assumption
that the
sun
has
a
uniform
mass
density (see
[p.
34],
[eq.
205]). As
in the
case
of
a
static
sun,
the number inserted for the
mass
of
the
sun,
and
thereby
the number inserted
for
S,
is
a
factor
10
too
large.
So,
the
end
result of
8.7
•
1010
given
on [p.
35]
for the
retrogression
of the
perihelion
in
fractions of
n per
half
a
revolution
is
a
factor
100
too
large.
This result
is
not
converted
to
seconds
of
arc per century
in
the
manuscript.
When the
erroneous
factor
100 is
corrected
for,
the
result
corresponds
to
something
in
the order of 103"
per
century.
This
is
of
the
same
order of
magnitude
as
the
effect of
the rotation
of
the
sun
predicted
by
the
general theory
of
relativity
in its final form.[47]
[46]"S0"
at
this
point
in
the
manuscript
should
be
"So"
(see
[p.
32],
note
141).
[47]See, e.g., Weinberg
1972,
p.
233. In
Lense and
Thirring 1918,
p.
161,
an
upper
limit of
0.01"
per century is given
for
the
total effect of
the
sun's rotation.
(18)