Y. B. PARK, I.-T. CHEON

1970

order to simplify the calculation, let us consider a

spherical galaxy with the mass G11

510

M

10R

100R

and

the radius G ly completely covered by the dark

matter (DM) with a thickness DMG

4

RR

1.01

ly,

i.e. the radius of the sphere including the DM is

MG

. The density of the dark matter is assumed

to be as large as 10 times that of the galaxy. (i) Without

any dark matter, the GLE of the corpuscle passing by the

galaxy surface is found as 26.11

RR

byqua-

tions (9), (24) and (33), provided

using E

and are

aced by G

R

repl

andG. (iihen the corpuscle

passes beside the surface of the dark matter, the GLE is

obtained as 27.8

R) W

9

, which

is calculated by replac-

ing

aR byGDM

nd MM and

M

R inua-

tion (9). (iii) The light can penetrate into the DM without

scattered and sneak by the surface of the galaxy. The

path of the corpuscle in the region outside the DM can be

calculated with Equations (9), (24) and (33) by assuming

that the total mass, GDM

MM the sphere of

the radius G

R. The orbits definitely a hyperbola. On

the other hand, the path of the corpuscle inside the DM is

able to obtain by a numerical method explained below.

Let us consider a

Eq

de

corpuscle which starts from the

ga

, is in

i

si

laxy surface, namely 0

and G

R, and dashes in

the dark matter until runnay frothe DM region.

Let the corpuscle be located at

r

ing awm

in a certain

maxy oment after it started from the galsurface, i.e. a

distance, r, from and an angle,

, around the galaxy

center. According to the Gaussian heorem, the gravita-

tional potential outside a sphere can be determined by the

total mass inside the sphere. Thus, the total mass inside a

sphere with the radius r participates in formation of a

hyperbolic orbit of the corpuscle at that moment. Remark

that the location of the corpuscle,

r

T

, is situated actu-

ally at the common point on the sphere surface and the

hyperbolic orbit. Since the corpuscle migrates in the DM,

it forms every moment a different sphere with a different

radius and, accordingly, the total mass inside the sphere

will change simultaneously. Namely, the hyperbola to

which the corpuscle on the surface of the sphere formed

in each moment belongs is not the same one anymore as

formed in the previous moment. The trace of all these

points which the corpuscle occupied every moment

yields a curved line. It may not be a simple hyperbola but

a slightly deviated one. Such a physical process might be

explicitly described with following equations. The total

mass of the system is 1

G

MM

with

3

1DM

GG

r

R

1,

DM

G

(35)

M

Rr

where

r

for GD

R

is the distance of

uscle of ththe e center e galaxy when it is

located at the angle

corpfrom th

around the focus. The eccentricity

at this position is

22

Rc 111,

22

GG

G

Rc

ee

GM GM

(36)

and, then, for

GDM

Rr R

,

1.

1cos11cos

G

ee

Re e

r

(37)

Notice that

is also a function of

G

r

R

. Generally,

it is possible to solve Equation (37) analytically with

respect to

G

R

r

. However, it is not easy and, thus, we

solve Equation (37) numerically with respect to the value

of

r

one-by-one for each angle

during the cor-

puscle passes through the DM regionThe orbit of the

corpuscle found in this manner shows slightly deviated

from a simple hyperbola, particularly in the region of the

dark matter. Actually, the region of the DM in which the

light is moving holds only within 16 degrees around the

galaxy center, i.e. 88

.

.

Finally, our result found for the GLE is 27.97

,

whi

5. Conclusions

that the GLE could be derived in the

hich is slightly larger than that of the case (ii). Ts

result is obviously understandable because of that the

closer the orbit of the corpuscle is to the gravitational

center, the larger the GLE becomes.

It has been shown

framework of Newtonian mechanics if the photon were

regarded as a corpuscle with the tiny effective mass,

2

c

h

m

, which was based on the quantum theory and

the Special T

optically

vi

informations of the dark

m

6. Acknowledgements

ut under the KAST Mentor

heory of Relativity. The factor 2 arising

from the GTR has been derived by introducing the

concept of the curved space due to the gravitational field

and its normal reaction. Of course, all these procedures

would be expediential. However, it would help senior

high school and undergraduate university students to

comprehend the physical structure of the GLE.

Since existence of the dark matter is not

sible because it does not interact with the light, the

GLE is known as a powerful method to observe it. The

present work investigates the GLE by a spherical galaxy

completely covered by the dark matter. The results are

definitely in detectable range.

It is true that one can obtain

atter by investigating the GLE.

This work has been carried o

Copyright © 2012 SciRes. JMP