Integrating Analytic and Wave Solutions of the Solar System We would like to see how
our wave solution for the solar system figures into the classical analytic theory of the formation
of our solar system.The protoplanetary disc that evolves into the planets has two forces that
balance its pressure, the centripetal force of the gas disc due to its rotation around the protostar
and the inward gravitational force on the disc from the protostar , and these are
related by the density of the gas that makes up the disc. The pressure gradient of the disc in
radial equilibrium balancing the inward gravity and outward centripetal force is
7.1.
We can solve this for pressure in the protoplanetary disc as a function of r, distance from the
star, as follows: Assume the gas is isothermal, meaning the temperature T is constant so we can
relate pressure and density with
Where is the speed of sound in the gas which depends on its temperature. We take the gas to
be in nearly Keplerian rotation. That is the rotation is given by Newtonian gravity:
And we take into account that the rotational velocity is slowed down by gas pressure using the
the parameter which is less than one:
We can say for a protoplanetary disc like that from which our solar system originated that its
density varies with radius as a power law:
is the reference density at and s is the power law exponent. We can write
.
We have from 7.1:
7.2.
Since , we have that which gives from 2:
v
2
ϕ
/r
GM
⋆
/r
2
ρ
d P
dr
= − ρ
(
GM
⋆
r
2
−
v
2
ϕ
r
)
P = c
2
s
ρ
c
s
v
K
=
GM
⋆
r
η
ρ(r) = ρ
0
(
r
r
0
)
−s
ρ
0
r
0
v
2
ϕ
= v
2
K
(1 − η)
2
≈
GM
⋆
r
(1 − 2η)
d P
dr
= − ρ
(
GM
⋆
r
2
⋅ 2η
)
P = c
2
s
ρ
d P/dr = c
2
s
dρ /dr
We integrate both sides:
And, we have
7.3
We take
as small because is small and r is large so we can make the approximation . We
have
7.4.
What we can get out of this is since the deviation parameter, , is given by
7.5. and
7.6.
Where, is the Boltzmann constant, is the molecular weight of
hydrogen, and is the mass of hydrogen is basically the mass of a proton is 1.67E-27kg. Since
for a protoplanetary cloud at Earth orbit T is around 280 degrees Kelvin we have
dρ
ρ
= −
2ηGM
⋆
c
2
s
r
2
dr
∫
ρ
ρ
0
dρ
ρ
= −
2ηGM
⋆
c
2
s
r
2
∫
r
r
0
dr
ln
(
ρ
ρ
0
)
=
2ηGM
⋆
c
2
s
(
1
r
0
−
1
r
)
ρ(r) = ρ
0
exp
2ηGM
⋆
c
2
s
(
1
r
0
−
1
r
)
P
0
= c
2
s
ρ
0
exp
2ηGM
⋆
c
2
s
(
1
r
0
−
1
r
)
2ηGM
⋆
c
2
s
(
1
r
0
−
1
r
)
η
e
x
≈ 1 + x
P
r
≈ P
0
1 +
2ηGM
⋆
c
2
s
(
1
r
0
−
1
r
)
P
0
= c
2
s
ρ
0
η
η = −
1
2
(
c
s
v
K
)
2
dlnP
dln R
c
s
=
k
B
T
μm
H
k
B
= 1.38E − 23J/K
μ ≈ 2.3
m
H
Typically in discs the pressure decreases with radius as a power law
Where , so
7.7.
So, essentially, by the chain rule
to clarify things. The reason 7.7 is significant is that equation
Where
c
s
= 1k m /s
P(R) ∝ R
−q
q ≈ 2.5
dlnP
dln R
≈ − 2.5
η = −
1
2
(
1k m /s
30k m /s
)
2
(−2.5) = 1.5E − 3
dlnP
dln R
=
dlnP
d R
⋅
d R
dln R
=
1
P
d P
d R
⋅ R =
R
P
d P
d R
L
eart h
ℏ
⊙
=
7.05E 33
2.8314E 33
= 2.4899 ≈ 2.5 = 2
1
2
L
eart h
=
4
5
π M
e
f
e
R
2
e
ℏ
⊙
= 2.8314E 33J ⋅ s
λ
moon
c
=
ℏ
2
⊙
GM
3
m
⋅
1
c
= 1.0secon d s