precession of the perihelion, whirlpool galaxies and binary pulsars

Ih
OTT
'I rd."fl· l' r. U. in I
de clip
b
h. E
orr In 0 c rn 15 . h COl
. 'TR")D
'r] 11 gala.x. il18ry pul aI.
RI
m 31ld
p ar UI
del
put al I r.

Pl!J i 'I III
9
,"The ire
( .ill. I II q d
I
I , ffi..... en' a I) } arhmt nih\: Fl]J
h
:7: .. F
lnn
pq .1. "
• lum'" 1. '
1 n. 1 ")7 1"i
i..:iJ··
\llumt: ·
pici . I)' p' p'r
II.
ynami d
in t n

∗
†
m
m(r)
m(r → ∞) = 1
m
m
m(r)
∗
†

→ ∞
lim
r→∞ m(r) = −a2 b2 ɛ2 x2 − b2 x2 − α2 α2 b2 .
ɛ =
√ a 2 b 2 x 2 − α 2 b 2 + a 2 α 2
a b x
2 2 2 2
2 a r − a α x − α r
m(r) = −
.
α r 2 + a 2 α
m ɛ
b a, α
x x
x > 1 θ x < 1
m
x
x = α √ b 2 − a 2
a b
m
2 2 2 b ɛ − b
m(r) =
r2 + 2 α b2 − 2 a2 α r − α2 b2 + a2 α2 (b2 ɛ2 − b2 ) r2 + a2 b2 ɛ2 − a2 b2 .
ɛ
r
m
m
m
1 ζ2
m(r) = −
b2 r2
1 1
+
a2 r2 −1 .
a2
lim m(r) =
r→∞ b2
a ≈ b
r
ζ ζ
ζ = 0
m

m
m
lim
r→∞ m(r) = a2 e−2 θ ζ α2 e2 θ ζ − b2 ɛ2 x2 + b2 x2 α2 b2
√ −α 2 b 2 e 2 θ ζ + a 2 α 2 e 2 θ ζ + a 2 b 2 x 2
ɛ =
a b x
m(r) = −
θ ζ
2 a r ζ e
·
1
α b (r2 + a2 θ ζ
e−2
)
(a 2 α 2 − α 2 b 2 ) e 2 θ ζ + a 2 b 2 x 2
(α 2 b 2 − a 2 α 2 ) r 2 e 2 θ ζ + (a 2 α 2 b 2 − 2 a 2 α b 2 r) x 2
(α 2 b 2 − a 2 α 2 ) r 2 e 2 θ ζ − a 2 b 2 r 2 x 2
+ a 2 α b ζ 2 − α b r 2 e 2 θ ζ + 2 a 2 b r − a 2 α b x 2
.
θ
θ m
r
x
x = α √ b2 − a2 θ ζ e
.
a b
m(r) = −
1
a2 2 2 2 2
b ɛ − a b ζ 2
(b 2 ɛ 2 − b 2 ) r 2 + a 2 b 2 ɛ 2 − a 2 b 2
+ 2 a b ɛ 2
− 2 a b − b2 − a2 ɛ2
(ɛ2 − 1) r2 + 2 α r − α2 ζ
− 1
+ b 2 − b 2 ɛ 2 r 2 + 2 a 2 α − 2 α b 2 r + α 2 b 2 − a 2 α 2
.
θ
r m
a > b
m

m(r) a = 1, α = 1
m(r) a = 1.05, b = 1, alpha = 3

m(r) a = b = 1
m(r) a = 1.01, b =
1, α = 1, ζ = −1, θ = π/4

m(r) a = 1.01, b =
1, α = 1, ɛ = 0.3

m(r) a = 1.01, b =
1, α = 1, ɛ = 0.3

m(r) a = 1.01, b =
1, α = 1, ɛ = 0.3