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.