DOC.
14
EINSTEIN AND BESSO MANUSCRIPT
421
[p.
30]
(Einstein)
[134]At
the
head
of
[p.
30],
above
the first
horizontal
line,
a
value for the
Newtonian
gravitational
constant
K
is
computed
by
considering
the orbit of
the
Earth around
the
sun as
a
circle and
setting
the
gravitational
force
on
the Earth
equal
to
the
centripetal
force. What
is
interesting
about the
calculation
is
that
in
the
course
of
it,
the mistake of
a
factor
10
in
the value used
on [p.
26]
and
[pp.
28-29]
for the
mass
of
the
sun
is
discovered. The calculation
proceeds as
follows.
[Eq.
184]
for K
is
obtained from
the
condition that the
gravitational
and
centripetal
forces
on
the Earth
balance each other
(on
[p.
35],
[eq.
214], and
[p.
42]
[see
note
195],
the
same equation
is
used
to
determine
K). In
the three lines below
[eq.
184],
the various factors
in this
expression
for
K
are
evaluated. For the
interpretation
of
[eq.
185]
and the numbers
to
the
right
of
[eq.
184], see
note 120.
An
error
occurred
in
computing
r3: 4.8
•
1039
should be 3.2
•
1039.
The results
are
inserted into
[eq. 184].
When the value for M
is not
corrected,
one
finds
K
=
9.6
•
10-9
m3/gs2,
which
is
a
factor
in
the order of
10 too
small. This
may
have
prompted
the correction of
the
erroneous
factor 10 in the
value for
M.
On
the
other hand, if
obtaining
a
wrong
value for
K
made
Einstein realize there had
to be
an error
somewhere,
it is
puzzling
that
the
remaining discrepancy
of
some
50% due
to
the
error
in r3
does
not
seem to
have bothered him. The
value is
used
in
the
calculation farther down
on
the
page (see
note
135).
[135]In the middle of
[p. 30],
between the
two
horizontal
lines,
another
attempt is
made
to
find
the
perihelion
advance of
Mercury
in
the
field
of
a
static
sun.
The end result of
this calculation,
in
fractions of
tt
per
half
a
revolution, is 1.65
•
10-8. The
calculation,
which contains several
errors, proceeds as
follows. The left-hand
side
of
[eq. 186]
is
equivalent
to
1/4(Ac0/F)2
(see
[eqs.
177-179]
on [p. 26];
C
=
F). In
[eq. 187],
equivalent to
[eq. 178],
C is computed.
Note that
the
eccentricity
e
of
Mercury's
orbit
is neglected,
and
that the
semi-major
axis
a
is
computed
via
a
=
(rm/re)re (see
note
122).
The
value
found for
C,
2.75
•
1019 cm2/s,
is too
high. Using
the numbers
given on
[p.
26]
and
taking
the
orbit's
eccentricity into
account,
one
arrives
at
C
=
2.67
•
1019
cm2/s.
The
erroneous
value for
C is
used
together
with
the
values
for
K and
M
in
[eqs.
184-185] to
compute
KM/cC,
giving
2.3
•
10-4. The
right-hand
side of
[eq. 186]
is
found
by
squaring
this
number.
Using
the numbers
given
on
[p. 26],
correcting
the
error
of
a
factor
10
in
the
mass
of the
sun,
one
obtains the value
1.65
•
10-4
rather than 2.3
•
10-4
for
KM/cC. In
[eq.
188],
[eq. 186]
is
multiplied
by
5/16.
In
fact,
in order to find the
perihelion
advance,
the
equation
should
be
multiplied
by
an
extra
factor
4
(see
[p.
26],
[eq. 177]; [p. 39],
[eq.
253]).
When this
is done,
using
the corrected value for
KM/cC,
the end result becomes
3.4.
10-8.
[136]After
discovering
the
error
in the value for the
mass
of the
sun
in
[eq. 185],
Einstein
presumably expected
to find
a
perihelion
advance
that
would
be
a
factor
100
smaller than the
advance he
had found
on [p.
28].
In fractions of
n
per
half
a
revolution, this
would be 3.4
•
10-8
(see note
128),
which
translates into
18" per century.
Presumably,
this
is
why
Einstein
wrote
this
number
immediately
below
[eq.
188].
When the various
errors
in
the calculation
on
[p. 30]
are
corrected,
this
is
indeed the result
one
finds
(see note
135).