356
EINSTEIN-BESSO ON
THE
MERCURY PERIHELION
With the
help
of Newton's second
law,
Fz
and
Fy
in
eq. (21) can
be
replaced
by z
and
y,
respectively.
For
z
and
y one can
then substitute
eq.
(15),
the
Euler-Lagrange equations
for
a
unit
point
mass
in
the metric
field
of
a
rotating
sun
(see
[p.
47],
[eq.
322]).
In
this
way,
the
integrand
in
eq.
(21)
can
be
expressed
in
(x, y,
z)
and
(x, y,
z).
These
coordinates and their time derivatives
are
then written
in terms
of
r, i,
0, 0,
0,
and
0
• •
(see
[p.
46],
[pp.
48-49]). Assuming
that
0 is
negligible compared
to 0,
one can
then
evaluate the
integral
in
eq. (21).
The end result
is
(see [p.
49],
[eq. 329])
Sok
8LX
=
sin
i
cos
0
n.
(22)
r
From
eqs.
(20)
and
(22)
it
follows that the
precession
80
of
the
nodes
per
revolution
is
given by
([p.
49],
[eq.
331])
86
=-J (23)
2
fr
From
eq. (23) one sees
that the rotation of the
sun
produces
a
retrogression
of the
nodes.
On
[pp.
41-42],
[p.
47],
and
[p.
49],
the
expression
for
50
in
eq.
(23)
is
evaluated
for
Mercury, Venus,
and Mars. The end results for
Mercury
and
Venus
given
on [p.
41]
and
[p.
49] are
-2.3"
and
-0.9"
per century, respectively.
Due
to
some
trivial
arithmetical
errors,
these results
are a
factor
in
the order of
103
too
large.[53]
When this
is
corrected
for,
the
effect of
the
sun's rotation
on
the nodes of
a
planetary
orbit
is
seen
to be
of the
same
order of
magnitude
as
its
effect
on
the orbit's
perihelion
calculated
earlier
in
the
manuscript.
Both effects
are
completely negligible.
On
[p.
41],
the results
for
Mercury
and
Venus
are
listed
along
with
the numbers
given
by
Newcomb[54]
for
the
discrepancy
between
(Newtonian)
theory
and observation for the motion of
nodes.
The
discrepancies
given by
Newcomb
are
5.0" and
10.2"
per century
for
Mercury
and
Venus, respectively.
So,
the effect of the sun's rotation
in
the "Entwurf"
theory only
seems
to
make the
discrepancies slightly
greater.
III
With the
exception
of
one
page
on
the
Nordström
theory
([p.
53]),
one
page
on
special
relativity
([p.
39]),
and
some
auxiliary
calculations
in
Newtonian
mechanics, all
calcu-
lations
in
the
manuscript
are
done
in
the
context
of the "Entwurf"
theory.
This
argues
for
dating
the
manuscript
to
a
period
between
May 1913,
when Einstein
and
Grossmann
finished
the "Entwurf"
paper,[55]
and November
1915,
when Einstein abandoned the
[53]See
[p.
49],
note
231.
[54]Newcomb 1895,
p.
109.
For
more
details,
see [p.
41],
note
188.
[55]A
reference
to
the
"Entwurf"
separatum
occurs
in
what
appears to
be the
earliest
part
of
the
manuscript,
on [p.
8],
thus
providing
clear evidence that
it
was
written after the
paper was
finished.