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.