EINSTEIN-BESSO
ON THE MERCURY PERIHELION
355
3.
Precession
of
the
nodes
of
an
orbit
in the
field
of
a
rotating
sun
to first
order
([pp.
41-42],
[pp. 45-49]).[48]
A
Cartesian coordinate
system
is
chosen
at rest
with
respect
to
the fixed
stars
and with its
origin at
the center
of
the
sun
(see
the
figure
above
which
is adapted
from
the
figure
in the
upper
left
corner
of
[p.
46]).
The z-axis
is
chosen
along
the axis
of
rotation
of
the
sun.
It
is
assumed that
the
plane
of
rotation
coincides
with the
ecliptic,
which
is
taken
to be the
xy-plane.
The
position
of
the
planet
on
the
celestial
sphere
can
be
specified
via three Eulerian
angles.[49]
In the
manuscript
they are
called
i, 6,
and 0. The
angle i
is
the
inclination of
the
planetary
orbit
to the
ecliptic.
This
is
a
small
angle
and it is
assumed
to
remain
constant. The
angle
0 is the
longitude
of
the
ascending
node.[50] It is
assumed
to
vary slowly over
the
course
of
a century.[51]
Apart
from
the
precession
of
the
nodes of
its orbit, the
planet is
assumed
to be in
uniform
circular motion. This
means
that
the
angle 0
between
the line
of
the
nodes
and the line
connecting
the
sun
and the
planet
increases
linearly
with time
(0
=
0).
In
other
words,
the
area
velocity
f
is
assumed
to have
a
constant
magnitude.
In
Newtonian
mechanics, the
relation between
the
area
velocity
and the
angular
momentum
L
for
a
unit
mass
point moving
on
the
orbit
shown
in the
figure
is
given by
L
=
2
f
(sini
sin0,
sini
cos0, cos/).
(19)
A small
change
80 in 0
corresponds
to
a
small
change
in the
direction of
the
angular
momentum.
The
change
in the
x-component,
for
instance, would
be[52]
8LX
=
2
/
sin
i
cos
0 80.
(20)
The
change
in
Lx during one period
of revolution
T
can
be
calculated
in
Newtonian
mechanics with
the
help
of
the
equation
dL/dt
=
x x
F, where
F
is
the
force
on
the
unit
point mass
in
orbit around
the
sun (see
[p.
49],
[eq. 326]):
1T
zF~)
fdØ(YFz
zF~).
(21)
In the
manuscript,
this
relation
from Newtonian mechanics
is
used
in
combination
with
the
"Entwurf"
theory
to derive
an
expression
for
the
precession
80 in
one
revolution.
[48]See
sec.
II.2a above for
the
metric
field
of
a
rotating
sun
to
first order. This
calculation
is
only
concerned
with the
precession
of
nodes in this field.
[49]See
Einstein's "Lecture Notes for
Introductory
Course
on
Mechanics"
(Vol. 3,
Doc.
1,
[p.
97])
for Einstein's discussion of Eulerian
angles;
see
also Smart
1953,
p.
348,
for their
use
in
astronomy.
The
phrase
"Eulerian
angles"
is
not
used in the
manuscript.
It
is
unclear what
relation,
if
any,
there
is
between the
usage
of these
angles on
[pp.
45-49] and the
designation
"Eulerscher
Fall"
on [pp.
43-44].
[50]The symbols i
and 0 have the
same
meaning
in Newcomb 1895
(see,
e.g.,
p.
113).
[51]In
this
calculation
the simplifying assumption is
made
that 0 is the
only
orbital element
subject
to
secular variations.
See
Lense and
Thirring
1918
for
a more
complete
discussion
in
the
context
of
the
general theory
of
relativity
in its final form.
[52]An
argument analogous to
the
one given
below
can
be
given
starting
from the
y-component.
Both
arguments
lead
to the
same
result.