Mathematical Principles of Theoretical Physics

7.6. THEORY OF DARK MATTER AND DARK ENERGY 487

1.Dark matter: the additional attraction.LetMrbe the total mass in the ball with radius

rof galaxy, andVbe the constant galactic rotational velocity. By the force equilibrium, we

infer from (7.6.63) that

(7.6.64)

V^2

r

=MrG

(

1

r^2

+

k 0

r

−k 1 r

)

,

which implies that

(7.6.65) Mr=

V^2

G

r

1 +k 0 r−k 1 r^3

.

The mass distribution (7.6.65) is derived based on both the Rubin rotational curve and the

revised formula (7.6.63). In the following we show that the mass distributionMrfits the

observed data.

We know that the theoretic rotational curve given by Figure7.15(b) is derived by using

the observed massMoband the Newton formula

F=−

mMobG

r^2

.

Hence, to show thatMr=Mob, we only need to calculate the rotational curvevrby the Newton

formula from the massMr, and to verify thatvris consistent with the theoretic curve. To this

end, we have

v^2 r

r

=

MrG

r^2

,

which, by (7.6.65), leads to

vr=

V

√

1 −k 0 r−k 1 r^2

.

Ask 1 ≪k 0 ≪ 1 ,vrcan be approximatively written as

(7.6.66) vr=V( 1 −

1

2

k 0 r+

1

4

k^20 r^2 ).

It is clear that the rotational curve described by (7.6.66) is consistent with the theoretic rota-

tional curve as illustrated by Figure7.15(b). Therefore, it implies that

(7.6.67) Mr=Mob.

The facts of (7.6.64) and (7.6.67) are strong evidence to show that the revised formula

(7.6.63) is in agreement with the astronomical observations.

We now determine the constantk 0 in (7.6.63). According to astronomical data, the aver-

age massMr 1 and radiusr 1 of galaxies are about

(7.6.68)

Mr 1 = 1011 M⊙∼= 2 × 1041 Kg,
r 1 = 104 ∼ 105 pc∼= 1018 Km.
The observations show that the constant velocityVin the Rubin rotational curve is about
V= 300 km/s. Then we have
V^2
G

= 1024 kg/km