CALCULATING THE MASS AND DENSITY OF EARTH
Kepler's Third Law (Newton's Form):
4! 2
=
a 3 MG
P2
The period of the moon's orbit is 27.32 days, or 2,361,000 seconds (roughly). The semimajor axis (a) is 384,400,000 m. G, the gravitational constant, is presently calculated to
be 6.673·10-11 m3kg-1s-2. Doing the math:
P 2 4! 2
=
a 3 MG
(
)
3
4! 2 3.844 #10 8 m
4! 2 a 3
" M=
=
3 (
%
GP 2
$11 m
6.673#10
2.361#10 6 s
'
2*
kg
#s
&
)
(
)
2
+ 6 #10 24 kg
The value from NASA is 5.9736·1024 kg.
Taking an average radius of Earth of 6371 km, the density can be calculated:
M 6 "10 24 kg
6 "10 27 g
!=
=
=
4 3
4
V
#r
# 6.371"10 8 cm
3
3
(
)
3
$ 5.54 g " cm %3
The density for a spherical shell 300 km thick or 1400 km thick:
M
6 "10 24 kg
6 "10 27 g
!=
=
=
+ 40.9 g " cm $3
3
3
4
4 %
3
V
# ( ro $ ri )
# ' 6.371"10 8 cm $ 6.071"10 8 cm (*
)
3
3 &
(
) (
)
M
6 "10 24 kg
6 "10 27 g
!=
=
=
+ 10.5 g " cm $3
3
3
4
4 %
3
V
# ( ro $ ri )
# ' 6.371"10 8 cm $ 4.971"10 8 cm (*
)
3
3 &
(
) (
)