EINSTEIN-BESSO ON
THE MERCURY
PERIHELION
347
terms
in the small deviations from the Minkowski values
representing
the weak
grav-
itational
field
are
put
into
yjv,
which
is
the solution of
the
"Entwurf"
field
equations
in first
approximation.
A
quantity
of the order of
magnitude
of
yjv
is
now
chosen
as
the
expansion parameter
for the
power
series
expansion
of
y^v.
So,
y^v
will be
of
nth
order
in
yjv.
These
terms, beginning
with
y^v,
are
found
as
follows. All terms in the
field
equations
that contain
one
factor
y^v
or
two
factors
yjv
and
nothing
smaller
are
collected,
y°Jv
and
yjv are
inserted, and
y2Jv
is
solved
for.
The
higher-order
terms
are
arrived
at in
the
same way.
For the
most
part,
the
manuscript
deals with
two
cases,
the
field
of
a
static
spherical
mass
distribution and the
field
of
a slowly,
uniformly,
and
rigidly rotating spherical
mass
distribution. These metric
fields
represent
the
gravitational
fields
of
a
static and
a
rotating sun, respectively.
In
the
case
of
a
static
sun,
one
has to
go
to
second order
to
find
a
contribution
to the field
that
gives
rise to
a
precession
of
planetary
orbits; in
the
case
of
a
rotating sun,
the first-order contribution
already
does.
The effects
in the
latter
case,
however,
are
much smaller than
in
the
former.[21]
The effect of the
field
of
a
static
sun on a
planetary
orbit
is
that
it
produces
an
advance of
the
perihelion, i.e.,
the
point
of the orbit closest
to the
sun,
in
the
plane
of the orbit. The effect of the sun's rotation
is
more complicated.[22]
The
manuscript
contains calculations for
two
effects:[23] the
retrogression
of
the
perihelion
in
the
plane
of
the orbit, and
the rotation of
the
plane
of the orbit around the
same
axis
as
the
axis
of rotation of the
sun
but
in
the
opposite
direction. On the
assumption
made
in
the
manuscript
that
this axis is
perpendicular
to
the
ecliptic,
this
second effect
is
nothing
but
a
retrogression
of
the
nodes of
the orbit,
i.e.,
the
points
at
which the orbit intersects
the
ecliptic.
The motion of
the
nodes
is
a
component
of
the
perihelion
motion because
of
the
way
the
longitude
of
the
perihelion
of
a
planetary
orbit
is
measured,
from vernal
[21]The
expansion parameter
in the
case
of the
rotating
sun
is
much smaller than
in
the
case
of
a
static
sun.
Strictly speaking,
therefore,
one
cannot
calculate the effects of the first-order
contributions
to
the field
of
the
rotating
sun
without
taking
into account the
effects of
the second–
order contributions
to
the
static
field at
the
same
time. Since
all
effects
are very
small, however,
one can,
to
a
very good approximation,
calculate the effects
separately
and
add
them (for
a more
detailed discussion of
this
point
in
the
context
of
the
general theory
of
relativity
in its final form,
see
Lense and
Thirring
1918,
pp.
158-159).
[22]The effect
is analogous
to
the
gravitational
effect
produced
inside
a
rotating
shell (see
[pp.
36-37];
see
also
Thirring
1918 for
a
thorough
discussion of
this
latter effect
in the
context
of
general relativity
in its final form).
[23]A
much
more
systematic
treatment
(in
the
context
of
the
general theory
of
relativity
in its
final form)
of secular variations
in
orbital elements due
to
the rotation of the central
attracting
body
is
given
in
Lense and
Thirring
1918. They
conclude that the effect of rotation of the
sun
on
planetary
orbits
will
never
exceed 0.01"
per century
(ibid.,
p.
161), although
the effects of
the rotation of
planets
in
the solar
system
on
the orbits of their satellites
can
be
as
much
as a
few
seconds of
arc
per century
(ibid.,
p.
162).
For
a
modern discussion of secular
perturbations
in
the
context
of
the
Parametrized Post-Newtonian
(PPN) Formalism,
see
Will
1993, especially
pp.
176-183.
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