by k.h.k. geerasee wijesuriyagalati,[email protected], 040722875933, romania dr. arfan...
TRANSCRIPT
ISSN (Print) 2313-4410 & ISSN (Online) 2313-4402Volume 26, 2018
© ASRJETS THESIS PUBLICATION http://asrjetsjournal.org/
By K.H.K. Geerasee Wijesuriya
How Does A Super Massive Black Hole Interact with Its Host Galaxy?
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How Does A Super Massive Black Hole Interact with Its Host Galaxy?
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1
How Does A Super Massive Black Hole Interact with Its Host Galaxy?
Doctor of Philosophy in Physics
Short Thesis number 05
Carlfield University
United States of America
February 2018
K.H.K. Geerasee Wijesuriya
Carlfield University Student ID: 2017-10-195618
Email address: [email protected] , [email protected]
I
2
Table of Content
Abstract ………………………………………………………..…………………….…..…...03
Notations ………………………………………………………….………………….......…..03
Acknowledgement ……………………………………………….……………………...…...04
Keywords ……………………………………………………………..…………….........…..04
Introduction…………………………………………………………………………..…..…...05
Literature review …………………………………………………………….…………..…...06
Conceptual Theoretical Framework ……………………………………………………...…..07
Research Content……………………………………………………………………….….….07
Section 1……………………………………………………………………08
Section 2……………………………………………………………………17
Result Evaluation ……………………………………………….……………….……….…...21
Discussion ……………………………………………………………………………..….…..21
Conclusion ……………………………………………………………………….……....…....21
References ……………………………………………………………………….……..…......22
Appendix ………………………………………………………………………………………22
II
3
Abstract Black hole is an object that is frequently appearing at the center of most of the galaxies in the
distant universe. Supermassive Black hole is a black hole with a high mass density in the core of
the black hole inside.
But astronomers have already dealt with the high massive black holes (in another words,
supermassive black holes). But the attempt of this research is to provide detailed innovative
arguments regarding the real nature of supermassive black holes.
Notations
Ψ1 = Wave function of the particle P1
Bi = the total magnetic field that influences on ‘i’ th particle
qi = the charge of the ‘i’th particle interact with the considering charged particle P1
mi = the mass of the ‘i’th particle interact with the considering charged particle P1.
M = the mass of the black hole
c = speed of light
rs = the Schwarzschild radius of the massive body
rs = 2GM / c2, where G is the gravitational constant.
III
4
Acknowledgement
I would like to thank and acknowledge my parents who are behind in my all achievements
Keywords
Schwarzschild radius ; Electromagnetic wave ; Black hole ; Supermassive ; charged particles;
Solar wind ; Particle
IV
5
Introduction
Black hole is an object that is frequently appearing at the center of most of the galaxies in the
distant universe. Supermassive Black hole is a black hole with a high mass density in the core of
the black hole inside.
If some object has arrived the radius of Schwarzschild radius, any Electromagnetic wave (EM)
that propagates towards the black hole never goes out of the black hole. But usually identified
black holes are the objects those have radius less than the Schwarzschild radius. Therefore any EM
wave that propagates towards a black hole never goes out. Since no any EM wave goes out of the
black hole, it appears to be dark. That’s why it called as a ‘black hole’.
But according to the General Theory of Relativity, the space-time near to a high massive object
bends due to the influence of the ‘high gravity’ of it. Since the black holes are much high gravity
objects in the universe, the space-time near to a black hole bends with a large angle. According to
such EM bending near to a black hole, there is an observation in Astronomy called as ‘gravitational
lensing’. When a light ray propagates near to a black hole, due to the space-time curvature near to
the black hole, the same star (that emitted the considering EM waves) appears at several positions
in the image of it.
But at the center of a black hole there is a place called as a ‘Singularity’. At the singularity, the
curvature of space-time tends to infinity. And all the laws of physics break at the singularity.
Therefore the discoveries regarding the black hole singularity have not been much developed.
But astronomers have already dealt with the high massive black holes (in another words,
supermassive black holes). But the attempt of this research is to provide detailed innovative
arguments regarding the real nature of supermassive black holes.
6
Literature Review
Although Newton’s Universal theory of Gravitation was acceptable before several centuries, it has
been limited by recent and present investigations. The most recent theory of gravity is Einstein’s
general theory of relativity. Instead of using the gravitational attractions to explain the nature of
gravity, Einstein used the curvature of the space-time to explain the nature of the gravity. Since
General theory of relativity has involved in all the areas of astrophysics all most, the applications
of the theory are useful indeed.
According to Einstein’s General theory of relativity, there is a radius limit away from a black hole
identified as ‘Schwarzschild radius’. Once the radius of an object in the universe arrived this radius
limit, no any electromagnetic wave or a particle can go away from the influence of the black hole.
Usually at the center of a supermassive black hole, there is a place called as ‘singularity’. At the
singularity position, all the physics laws break down and no one did explain the nature of the
singularity in detail.
We know that our sun and almost other highly energized stars in the distant universe eject a soup
of charged particles to the outer space with high speeds. Those charged particles are much
energized and capable to interact with outer environment of space. But when a charged particle of
the solar wind arrives the force acting area of a supermassive black hole, there may be special
situations under that particular status. First part of this research will explore regarding that
situation.
Generally, at the center of a high massive galaxy there is a supermassive black hole and its density
is much greater than the mass density of other black holes. Supermassive black holes are much
energized and it is capable to control the host stars, planets and other host objects which are nearby
to the supermassive black hole.
7
Conceptual Theoretical Framework
Supermassive Black hole is a black hole with a high mass density in the core of the black hole
inside.
Although astronomers have already dealt with the supermassive black holes, the attempt of this
research is to provide detailed innovative arguments regarding the real nature of supermassive
black holes and the nature of the interactions of the supermassive black hole with the host of its
galaxy.
Research Content
With this research, I intend to explore how the supermassive black hole controls the solar winds
of the nearby stars and how the supermassive black hole controls the orbiting velocity of nearby
stars/planets (we know that most of the black holes create the whole galaxy to rotate around the
center of the galaxy). That means with this research, I will explore the supermassive black holes
in two sections.
1st section is : how the supermassive black hole controls the solar winds of the nearby stars
2nd section is: how the supermassive black hole controls the orbiting velocity of nearby
stars/planets
8
Section 01: How the supermassive black hole controls the solar winds of the nearby stars
We know that in our solar system, the main power source is the Sun. While the sun is radiating
energy, it emits solar winds also towards the atmosphere. Our Earth also influenced by that wind
which generates by the Sun. The solar wind contains highly energized charged particles those are
harmful for the life.
Since the solar wind consists of charged particles, a high gravitational object (near to the exiting
and propagating area of solar wind) can influence on the solar wind. In this case, I will study how
the high gravitational force of the supermassive black hole influences on the charged particles of
the solar wind of the nearby stars.
Since the charged particles those emitted by the solar wind move though the free space, there is a
magnetic field that induces near to those charged particles. Moreover rather than the magnetic
fields induce due to the rest of the charged particles of the solar wind, there is another magnetic
field that influences on the solar wind’ particles. That is the magnetic field due to the black hole.
Let’s consider the solar wind that emits by a star near to the black hole. But due to the gravitational
field of the black hole, the potential energy of the solar wind particles changes with time. As well
as the kinetic energy also varies. Moreover, there are electric and magnetic potential energy
variations with the solar wind particles. Thus when the solar wind particles oscillate, high energy
photons emit and those emitting photons are influenced by the high gravity field of the black hole.
Let’s find the wave function for a charged particle (P1) in the solar wind as below:
(-ħ2 /2M1). ∇2 Ψ1 + {Σ [q 1 .qi ] / 4.π. ri – Σ GM1 mi /ri - GM1.M / R1 - Σ μ. Bi .cos θi} Ψ1 = E1 .Ψ1
Where M1 is the mass of the considering charged particle of the solar wind, q1 is the charge of
the considering particle P1 of the solar wind, Ψ1 is the wave function of P1, E1 is the eigenvalue
for the charged particle, Bi is the total magnetic field that influences on ‘i’ th particle (which
interacts with the considering charged particle P1 ), qi is the charge of the ‘i’th particle interact
with the considering charged particle P1 , mi is the mass of the ‘i’th particle interact with the
considering charged particle P1. ri is the distance between P1 and ‘i’th charged particle in the solar
wind. μ is the Bohr Magneton. M is the mass of the supermassive black hole, R1 is the distance
between the center of the black hole and P1.
9
Then (-ħ2 / 2M1). ∇2 Ψ1 = E’1 Ψ1 . Thus , ∇2 Ψ1 = [ -2M1 E’1 / ħ2 ]. Ψ1
where E’1 = E1 - Σ [q 1 . qi ] / 4.π. ri + Σ GM1 mi / ri + GM1.M / R1 + Σ μ. Bi .cos θi …..(1.0)
Then
∇2 Ψ1 + [2.E’1 .M1 / ħ2 ] Ψ1 = 0
Then Ψ1 = A.cos (x1 .r1) + B. sin (x1.r1) where x1 = [2.E’1 .M1 / ħ2 ] , r1 is the radial coordinate of
the wave function Ψ of P1. But we consider the center of the black hole as the origin. Then r1 = R1
………………………………….(01)
And A, B are complex constants.
But when r1 = 0, Ψ1 = 0 by (01) (since at the singularity, there is no any particle as P1)
Thus A = 0 . Thus Ψ1 = B. sin (x1.r1). But at r1 = rs , Ψ1 = 0 (at the event horizon also there is no a
physical particle). Thus B.sin (x1.rs) = 0. Thus sin (x1.rs) = 0. Thus (x1.rs) = nπ .
Thus x1 |r1 = rs = nπ / rs ; where n is an integer (positive or negative).
Thus [2.E’1 .M1 / ħ2 ] | r1 = rs = nπ / rs
E’1 | r1 = rs = [nħ2π / 2.M1 rs ] ………………………….(02)
Thus Ψ1 = B. sin ( [2.E’1 .M1 / ħ2 ]. r1 ) ………………………..(03)
By (02): [ E1 - Σ [q 1 . qi ] / 4.π. ri + Σ GM1 mi / ri + GM1.M / rs + Σ μ. Bi .cos θi ] =
[nħ2π / 2.M1 rs ] ……………………………..(04)
But Time dependent wave function can be written as: Ψ1 . e-i [ (E1’/ ħ) ] t = Ψ(t)
1
Thus ∂ Ψ(t)1 / ∂t = (-i.E’1 / ħ). Ψ(t)
1 .
Then ∂2 Ψ(t)1 / ∂t2 = (-i.E’1 / ħ)2. Ψ1 = - (E’1 / ħ)2. Ψ(t)
1 ……………………..(05)
By (04):
[ E1 - Σ [q 1 . qi ] / 4.π. ri + Σ GM1 mi / ri + Σ μ. Bi .cos θi ] = [nħ2π / 2.M1 rs ] - GM1.M / rs
By (1.0): E’1 = E1 - Σ [q 1 . qi ] / 4.π. ri + Σ GM1 mi / ri + GM1.M / R1 + Σ μ. Bi .cos θi
10
Thus E’1 = [nħ2π / 2.M1 rs ] - GM1.M / rs + GM1.M / R1
= [nħ2π / 2.M1 rs ] - GM1.M / R’1 , where R’1 = r1 . rs / (r1 - rs)
Thus E’1 = [nħ2π / 2.M1 rs ] - GM1.M / R’1 ……………………………(06)
By (05) and (06):
∂2 Ψ(t)1 / ∂t2 = {[ GM1.M / ħ2 R’1 ] - [nπ / 2.M1 rs ] }
2. Ψ(t)1 = a(t) = the linear acceleration of P1...(07)
Equation (07) indicates the acceleration of P1 (a particle from the solar wind) towards the
supermassive black hole.
But the particle P1 attracts towards the black hole in a spiral manner.
But the angular acceleration of P1 at time t (time t is the moment which the particle completes one
orbit along the spiral motion path. i.e [ ti = t – 0] is the orbital time of P1) = a(t) / d (t) = Δω / Δti
. But the acceleration of P1 towards the black hole changes with time. And d(t) is the radius of one
circle of motion of spiral motion (towards the black hole) at time ‘t’.
Δti = [ d(t) . Δω ] / a(t) = [ (d(t). vt2 / d(t2) ) – (d(t). vt1 / d(t1) ) ] / a(t)
= [ d(t2) / a(t2) ] * [vt2 / d(t2) ) – vt1 / d(t1) ] ……………………(07)
Here Δti ( = [t2 – 0] ) is an arbitrary time duration starting from the beginning of time (i.e.
starting from t = 0. i.e. t1 = 0
But v(t1) = ∂ Ψ(t)1 / ∂t |t = t1 = (-i.E’1 / ħ). Ψ(t)
1 | t= t1 =
(-i.E’1 / ħ). B0.i. sin ( [2.E’1 .M1 / ħ2 ]. r1 ). e
-i [ (E1’/ ħ) ] t1 (Where (B0 .i ) = B ).
and v(t2) = ∂ Ψ(t)1 / ∂t |t = t2 = (-i.E’1 / ħ). Ψ(t)
1 | t= t2 =
(-i.E’1 / ħ). B0.i. sin ( [2.E’1 .M1 / ħ2 ]. r1 ). e
-i [ (E1’/ ħ) ] t2 (Where (B0 .i ) = B ).
Thus v (t1) = (E’1 / ħ). B0.sin ( [2.E’1 .M1 / ħ2 ]. r1 ). e
-i [ (E1’/ ħ) ] t1 ……………………(08)
and
v(t2) = (E’1 / ħ). B0. sin ( [2.E’1 .M1 / ħ2 ]. r1 ). e
-i [ (E1’/ ħ) ] t2 ……………………….…(09)
11
By (07):
Δti = [ d(t2) / a(t2) ] * [vt2 / d(t2) ) – vt1 / d(t1) ] = [ v(t2) / a(t2) ] – [v(t1). d(t2) / ( d(t1).a(t2) ) ]…..(10)
But t1 is consider as the beginning of the motion of the particle P1 towards the black hole. Therefore
I consider t1 as 0. Thus by (08):
v (t1) |t1 = 0 = (E’1 / ħ). B0.sin ( [2.E’1 .M1 / ħ2 ]. r1 )…………………………….(11)
By (08), (09) and (10)
Δti = [ v(t2) / a(t2) ] – [v(t1). d(t2) / ( d(t1).a(t2) ) ]
But [ v(t2) / a(t2) ] =
(E’1 / ħ). B0. sin ( [2.E’1 .M1 / ħ2 ]. r1 ). e
-i [ (E1’/ ħ) ] t2 / {[ GM1.M / ħ2 R’1 ] - [nπ / 2.M1 rs ] }2. Ψ(t)
1
= (E’1 / ħ). B0. sin ( [2.E’1 .M1 / ħ2 ]. r1 ). e
-i [ (E1’/ ħ) ] t2 / {[ GM1.M / ħ2 R’1 ] - [nπ / 2.M1 rs ] }2. B.
.sin ( [2.E’1 .M1 / ħ2 ]. r1 ). e
-i [ (E1’/ ħ) ] t2
= (E’1 / i.ħ). / { [ GM1.M / ħ2 R’1 ] - [nπ / 2.M1 rs ] }2………………………….(12)
[v(t1) / a(t2) ) ] =
(E’1 / ħ). B0.sin ( [2.E’1 .M1 / ħ2 ]. r1 ) / { [ GM1.M / ħ2 R’1 ] - [nπ / 2.M1 rs ] }
2.
B. sin ( [2.E’1 .M1 / ħ2 ]. r1 ). e
-i [ (E1’/ ħ) ] t2
= (E’1 / iħ) / { [ GM1.M / ħ2 R’1 ] - [nπ / 2.M1 rs ] }2. e-i [ (E1’/ ħ) ] t2 …………………..(13)
Thus Δti = (E’1 / i.ħ). / { [ GM1.M / ħ2 R’1 ] - [nπ / 2.M1 rs ] }2 –
(k. E’1 / iħ) / { [ GM1.M / ħ2 R’1 ] - [nπ / 2.M1 rs ] }2. e-i [ (E1’/ ħ) ] t2 ; where k = d(t2) / d(t1)
Δti = (E’1 / i.ħ). { [ 1 / { [ GM1.M / ħ2 R’1 ] - [nπ / 2.M1 rs ] }2 -
k ei [ (E1’/ ħ) ] t2 / [ GM1.M / ħ2 R’1 ] - [nπ / 2.M1 rs ] }2. }
Δti = (E’1 / i.ħ). [ ( k ei [ (E1’/ ħ) ] t2 - 1) / { [ - GM1.M / ħ2 R’1 ] + [nπ / 2.M1 rs ] }2 ]
12
Δti = E’1 . ( k ei [ (E1’/ ħ) ] t2 - 1) / [ (iħ). { (nπ / 2.M1 rs ) – (GM1.M / ħ2 R’1 ) }2 ]
*** DO NOT confuse on ‘i’ term in Δti and ‘i’ term in (iħ). ‘i’ term in (iħ) is √(-1) . And ‘i’
term in Δti is just the order number.
But for some k0 complex number , I can write k.ei [ (E1’/ ħ) ] t2 as e [ ( i [ (E1’/ ħ) ]. k0. t2 ] .
Δti = E’1 . ( e i. (E1’/ ħ) . k0. t2 - 1) / [ (iħ). { (nπ / 2.M1 rs ) – (GM1.M / ħ2 R’1 ) }
2 ]
Δti = E’1 .(1 + [ i. (E1’/ ħ) .k0 .t2] + [ i. (E1’/ ħ) .k0. t2]2 / 2! +… + [ i.(E1’/ ħ) . k0 .t2 ]
n / n! +..-1)
/ [ (iħ). { (nπ / 2.M1 rs ) – (GM1.M / ħ2 R’1 ) }2 ]
Thus Δti = (E’1 / ħ) * [ 1 / { (nπ / 2.M1 rs) – (GM1.M / ħ2 R’1 ) }2 ] * ∑ (
E′1 . k0.t2
ħ)
𝑗
. ( i𝑗−1
𝑗! ∞
𝑗=1 )
= (E’1 / ħ) * [ 1 / { (nπ / 2.M1 rs) – (GM1.M / ħ2 R’1 ) }2 ] * 𝑒(𝐸′1.𝑘0.
𝑡2
ħ) * δj ………………(14)
Where δj = 1, -1, i or -i ;
(δj ) = 1 ; if j-1 = 4. b , b is a natural number
(δj ) = -1 ; if j -1 = 2.b , b is a natural number
(δj ) = i ; if j-1 = 1+4b , b is 0 or a natural number
(δj ) = -i ; if j-1 = 3+4b , b is 0 or a natural number
By (06): E’1 = [nħ2π / 2.M1 rs ] - GM1.M / R’1 = ħ2 [nπ / 2.M1 rs ] - GM1.M / ħ2 .R’1 ]
Thus { [nπ / 2.M1 rs ] - GM1.M / ħ2 .R’1 ] }2 = [ E’1 / ħ
2 ]2
By (14): Δti = (E’1 / ħ)* [ ħ4 / ( E’1 )2 ] * 𝑒(𝐸′1.𝑘0.
𝑡2
ħ) * δj
Δti = [ ħ3 / E’1 ] * 𝑒(𝐸′1.𝑘0.𝑡2
ħ) * δj
But for some k’ positive real number, [ ħ3 / E’1 ] * 𝑒(𝐸′1.𝑘0.𝑡2
ħ) = 𝑒(𝐸′1.𝑘0.𝑘′
𝑡2
ħ) = 𝑒(𝐸′1. 𝑘′′
𝑡2
ħ)
Thus Δti = 𝑒(𝐸′1.𝑘′′ .𝑡2
ħ) * δj …………………..(15) where k’’ = k’. k0
13
But [ ħ3 / E’1 ] = 𝑒(𝐸′1.𝑘0.𝑘′ 𝑡2
ħ) / 𝑒(𝐸′1.𝑘0.
𝑡2
ħ) = 𝑒(𝐸′1.𝑘0.𝑘′
𝑡2
ħ)− (𝐸′1.𝑘0.
𝑡2
ħ ) = 𝑒(𝐸′1.𝑘0.
𝑡2
ħ ) [𝑘′−1]
Thus E’1 > 0…………………………………..(15.1)
Thus; [ ħ / ( E’1 . k0 . t2 ) ] * [ ln (ħ3 / E’1) ] + 1 = k’ ……………………….(16)
And ln (ħ3 / E’1)ħ / (E’1. k0. t2) + 1 = k’ But k.ei [ (E1’/ ħ) ] t2 = e [ ( i [ (E1’/ ħ) ]. k0. t2 ]
Thus k = e [ ( i [ (E1’/ ħ) ]. k0. t2 ] – i . [ (E1’/ ħ) ] t2 = e i . [ (E1’/ ħ) ] t2 [k0 - 1]
Thus i . [ (E’1 / ħ) ] t2 .ln k + 1 = k0 ………………………..(17)
By (16) and (17): k’ = [ ħ / [ ( E’1 . { i . [ (E’1 / ħ) ] t2 .ln k + 1 } . t2 ) ] * [ ln (ħ3 / E’1) ] + 1
But k’’ = k’. k0 =
{ [ ħ / [ ( E’1 . { i .[ (E’1 / ħ) ] t2 .ln k + 1 } . t2 ) ] * [ ln (ħ3 / E’1) ] + 1 } *{ i . [ (E’1 / ħ) ] t2 .ln k + 1 }
= [ ħ / (t2 E’1 ) ] * [ ln (ħ3 / E’1) ] + [ ( i . [ (E’1 / ħ) ] t2 .ln k ) + 1 ]
Thus k’’ = [ ħ / (t2 E’1 ) ] * [ ln (ħ3 / E’1) ] + [ ( i . [ (E’1 / ħ) ] t2 .ln k ) + 1 ]………………..(18)
By (15): Δti = 𝑒(𝐸′1.[ ħ / (t2 E’1 ) ] ∗ [ ln (ħ3 / E’1) ] .𝑡2
ħ) * 𝑒(𝐸′1.[ ( i .[ (E’1 / ħ) ] t2 .ln k ) + 1 ] .
𝑡2
ħ) * δj
Δti = 𝑒( [ ln (ħ3 / E’1) ] ) * 𝑒 i .[ E1′. t2 / ħ)]^2 (ln k ) + E′1 .𝑡2
ħ * δj
= (ħ3 / E’1 ) * eE1’. t2 / ħ * δj * ( eln k )i. [E’1. t2 / ħ]^2
Thus Δti = (ħ3 / E’1 ) * eE1’. t2 / ħ * δj * k i. [E’1. t2 / ħ]^2 here E’1 = [nħ2π / 2.M1 rs ] - GM1.M / R’1
real (Δti ) = (ħ3 / E’1 ) * eE1’. t2 / ħ * (-1)κ ; κ is a natural number.
But E’1 = [nħ2π / 2.M1 rs ]*R’1 – [ GM1.M / R’1 ]
e - E1’. t2 / ħ . Δti = e - E1’. Δti / ħ . Δti = (ħ3 / E’1 ). (-1)κ
t = T t = T
Then ∫ e - E1’. Δti / ħ . (Δti ) dt = (-1)κ * ħ3 ∫ ( 1 / E’1 ) dt
t = 0 t = 0
14
Where we consider T as the orbital time period of the particle P1 (arbitrary taken from the solar
wind particle soup) one complete oscillation, while moving towards the Black hole.
(- ħ / E’1 ). ( e - E1’. T / ħ – 1 ). [T – 0] - ∫1. (- ħ / E’1 ). ( e - E1’. t2 / ħ ). dt = (-1)κ * [ ħ3 / E’1 ] * T
Thus (- ħ / E’1 ). ( e - E1’. T / ħ – 1 ). [T – 0] - (ħ / E’1)2.[ e - E1’. T / ħ – 1 ] = (-1)κ * [ ħ3 / E’1 ] * T
But (ħ / E’1)2.[ e - E1’. T / ħ – 1 ] > 0. Thus
(- ħ / E’1 ). ( e - E1’. T / ħ – 1 ). T > (-1)κ * [ ħ3 / E’1 ] * T
Thus - ( e - E1’. T / ħ – 1 ) > (-1)κ * [ ħ2 ]
But e - E1’. T / ħ < 1 . Thus [ 1 – (e - E1’. T / ħ) ] > 0.
Thus [ 1 – (e - E1’. T / ħ) ] > (-1)κ * [ ħ2 ] ……………………………..(19)
But the result (19) is valid for all κ natural number. Thus [ 1 – (e - E1’. T / ħ) ] > ħ2
Thus e - E1’. T / ħ < 1 – ħ2 . But ( 1 – ħ2 ) < e- 2 . Thus e - E1’. T / ħ < e-2 . Thus T > ( 2 ħ / E’1 )
Thus f < ( E’1 / 2.ħ ) …………………………….(20)
Where f is the frequency of oscillation of the particle that has been released from the solar wind.
Here T is the Orbital Time period for one complete oscillation, while moving towards the Black
hole.
Here f is the frequency of electromagnetic waves emits by the solar wind particle (that is
accelerating towards the black hole).
But here, [nħ2π / 2.M1 rs ] - GM1.M / R’1 = E’1 .
By (20): f < [ nħ.π / 4.M1 rs ] – [ GM1.M / 2.ħ R’1 ] ……………………..(21)
Here f is the frequency of electromagnetic waves emits by the solar wind particle (that is
accelerating towards the supermassive black hole).
Here R’1 = r1 . rs / (r1 - rs) But 1 / [ 2.ħ. R’1 ] = (r1 - rs) / (2.ħ. r1. rs). But rs = 2GM / c2 . Where M
is the mass of the super massive black hole. c is the speed of light in vacuum. G is the Newton’s
universal gravitational constant.
15
Then 1 / [ 2.ħ. R’1 ] = (r1 - rs) / (2.ħ. r1. rs) = ( r1 - 2GM / c2 ) / (2.ħ. r1 . 2GM / c2 )
= [ r1 .c2 - 2GM ] / [ 4. ħ. G. M. r1 ] .
Then [ GM1.M / 2.ħ R’1 ] =
( [G.M1. M ]* [ r1 .c2 - 2GM ] ) / [ 4. ħ. G. M. r1 ] = M1 * ( r1 .c
2 - 2GM ) / 4. ħ. r1
But ( r1 - rs ) > 0. Thus r1 – 2GM / c2 = r1. c2 – 2GM / c2 > 0. Thus = ( r1. c
2 – 2GM ) > 0.
Thus M1 * ( r1 .c2 - 2GM ) / 4. ħ. r1 > - 2.G.M.M1 / 4.ħ. r1 = - G.M.M1 / 2.ħ. r1
Thus by (21):
f < [ nħ.π / 4.M1 rs ] – [ GM1.M / 2.ħ R’1 ] < [ nħ.π / 4.M1 rs ] – [G.M.M1 / 2.ħ. r1 ]
Thus f < [ nħ.π / 4.M1 rs ] – [ G.M.M1 / 2.ħ. r1 ] ……………………………(22)
; where n is a natural number
But the usual mass of a Supermassive black hole is
M = (1.989 × 1030 * 16 * 109 ) = 31.824 * 1039 kg
But rs = 2GM / c2 = 2 * 6.754 × 10−11 * 31.824 * 1039 / (2.998 * 108 )2 = 47.82 * 1012 meters
Then by (22) , ( f ) < [ nħ.π / 4.M1 rs ] - [ G.M.M1 / 2.ħ. r1 ] < [ nħ.π / 4.M1 rs ]
But we know that the solar wind mostly consists of mostly electrons, protons and alpha particles.
Case 1 (Consider P1 as an electron)
Then M1 = 9.11 * 10-31 kg . Then by (22):
f < n * 6.63 * 10-34 * 3.142 / ( 4* 9.11 * 10-31 * 1.89 * 1022 ) for all n natural number. Then
[ n * 6.63 * 10-34 * 3.142 / ( 4* 9.11 * 10-31 * 1.89 * 1022 ) ]min =
6.63 * 10-34 * 3.142 / ( 4* 9.11 * 10-31 * 1.89 * 1022 )
= 0.302 * 10-25 . Thus f < 0.302 * 10-25 Hz . But the minimum frequency of a photon that is
possible is greater than 0.302 * 10-25 Hz .
16
Case 2 (Consider P1 as a proton)
Then M1 = 1.67 * 10-27 kg . Then by (22):
f < n * 6.63 * 10-34 * 3.142 / ( 4* 1.67 * 10-27 * 1.89 * 1022 ) for all n natural number. Then
[ n * 6.63 * 10-34 * 3.142 / ( 4* 1.67 * 10-27 * 1.89 * 1022 ) ]min =
6.63 * 10-34 * 3.142 / ( 4* 1.67 * 10-27 * 1.89 * 1022 )
= 1.65 * 10-29 . Thus f < 1.65 * 10-29 Hz . But the minimum frequency of a photon that is
possible is greater than 1.65 * 10-29 Hz .
Case 3 (Consider P1 as an alpha particle)
Then M1 = 1.65 * 10-27 kg . Then by (22):
f < n * 6.63 * 10-34 * 3.142 / ( 4* 1.65 * 10-27 * 1.89 * 1022 ) for all n natural number. Then
[ n * 6.63 * 10-34 * 3.142 / ( 4* 1.65 * 10-27 * 1.89 * 1022 ) ]min =
6.63 * 10-34 * 3.142 / ( 4* 1.65 * 10-27 * 1.89 * 1022 )
= 1.67 * 10-29 . Thus f < 1.67 * 10-29 Hz . But the minimum frequency of a photon that is possible
is greater than 1.67 * 10-29 Hz .
Thus for all cases 1 , 2 , 3 ; P1 particle cannot emit EM radiations anyway under the previously
conditions.
Therefore the particles of solar wind, those move towards the supermassive black hole do not emit
any electromagnetic radiations to the outer space………………………..(23)
17
Section 2 : How the supermassive black hole controls the orbiting velocity of nearby
stars/planets
According to the well-known knowledge, GMM’ / r2 = M’ .V2 / r
M = mass of the supermassive black hole according to planet’s reference frame
M’ = mass of the considering planet according to planet’s reference frame
r (t) = the distance between supermassive black hole and the planet at time ‘t’ according to the
planet’s reference frame
V = the orbiting linear velocity of the planet around the central object
Then V = √ (GM / r) ……………………………………………(24)
But the equation (24) is valid only according to classical physics theories. We know that there is a
magnetic field that is spreading from the supermassive black hole’s magnetic axis, towards the
space. But usually there is another magnetic field that is due to the magnetic field of the planet
(which orbits around the black hole).Also there are some other magnetic fields that is existing in
that system. Those are the magnetic fields due to the magnetic fields of other planets/stars near to
the considering main planet.
Then there is a magnetic attraction/repulsion among those magnetic fields. Depending on the
direction of the considering main planet’s magnetic axis and the total equivalent magnetic axis of
rest of the matters nearby to the main planet, the velocity of the main planet (A1) changes.
Then let’s try to find the total equivalent magnetic force (F1) that acts on A1 (due to the total
magnetic field because of the black hole and the magnetic fields of rest of other planets/stars nearby
to A1) as below.
F1 = ( πμ0 / 4 )* (M1. M2. R1 2 . R2
2 ) * [ 1
𝑥2 +1
(𝑥+𝐿1+𝐿2)2 −1
(𝑥+𝐿1)2 −1
(𝑥+𝐿2)2 ]……………(25)
μ0 is the permeability of space, which equals to 4π*10−7 T·m/A
M1 , M2 identified as the magnetization of the magnet in A1 and the magnetization of the total
equivalent magnet of the host planets/black hole respectively (the virtual magnets which produced
by the magnetic field of the black hole/ host planets of A1)
18
x is the distance between the two virtual magnets in meters (in the black hole/nearby planets/
nearby stars) - identified as the distance between magnet in A1 and the equivalent magnet of the
host planets/ black hole.
R1 , R2 are the radiuses of two magnets in meters - identified as the radius of the magnet in A1 and
the radius of the total equivalent magnet of the host planets/black hole respectively (the virtual
magnets which produced by the magnetic field of the black hole/ host planets of A1)
L1 , L2 are the lengths of two magnets in meters- identified as the lengths of the magnet in A1 and
the length of the total equivalent magnet of the host planets/black hole respectively (the virtual
magnets which produced by the magnetic field of the black hole/host planets of A1).
But we know that the magnetization of A1 ( = M1 ) < magnetization of the supermassive black
hole/host planets (= M2 ).
And radius of the magnet in A1 (virtual magnet that causes A1 ‘s magnetic field) (= R1) <
radius of the total equivalent magnet of the host planets/supermassive black hole (= R2)
Thus by (25): F1 < ( M1. M2. R1 2 . R2
2 ) < ( M2 . R2 )2 . Thus F1 < ( M2 . R2 )
2 ………….(26)
Here R2 is also a vector.
But the gravitational force acting on A1 , due to other host planets and host star of it (F2) at time t
= GmM’ / L2 (t)_
Here L (t) is the distance between A1 and the gravitational center of the host planets and the host
star of A1 at time t. m is the total mass of the of the host planets and the host star of A1.
The gravitational force acting on A1 due to the supermassive black hole (F3) (supermassive black
hole has located at the center of the galaxy which the A1 has located) = GMM’ / r2 (t)_
Here r(t) is the distance between A1 and the supermassive black hole at time t.
Here L2 (t)_ means that L2 (t) is a vector. And r2 (t)_ means r2 (t) is a vector.
Then the projection of F2 on the direction of F1 = GmM’ / L2 (t). cos θ
And the projection of F3 on the direction of F1 = GMM’ / r2 (t). cos ϕ
19
Here θ is the angle between the direction of F1 and F2 and ϕ is the angle between the direction of
F3 and F1 .
But A1 is orbiting around its host star. As well as A1 and its solar system is revolving around the
supermassive black hole (Supermassive black hole is at the center of the galaxy which A1 has
located). Then let F4 is the total centrifugal force acting on A1 . Then F4 = M’. V0 / D2
The projection of F4 on the direction of F1 = [ M’. ( V0 / D2 ) .cos δ ] for some V0 and D.
Here δ is the angle between the direction of F1 and F4 .
Here V0 is the total equivalent linear velocity of A1 (along the direction which is perpendicular to
the total centrifugal force acting direction at time t). And D is the distance between A1 and total
centrifugal force acting center.
Thus total external force acting on A1 (due to the supermassive black hole/host objects) at time t =
F = F1 + F2 + F3 - F4 (Because the total centrifugal force acting in the opposite direction of rest of
the forces acting on A1).
= F1 + GmM’ / L2 (t). cos θ + GMM’ / r2(t). cos ϕ - [ ( M’. V0 / D2 ).cos δ ]
But A1 is stable in the system at time t. Thus F = 0.
Thus F1 + GmM’ / L2 (t). cos θ + GMM’ / r2(t). cos ϕ - [ ( M’. V0 / D2 ).cos δ ] = 0.
Thus F1 + GmM’ / L2 (t). cos θ + GMM’ / r2(t). cos ϕ = [ ( M’. V0 / D2 ).cos δ ] ……………..(27)
By (25): F1 = ( πμ0 / 4 )* (M1. M2. R1 2 . R2
2 ) * [ 1
𝑥2 +1
(𝑥+𝐿1+𝐿2)2 −1
(𝑥+𝐿1)2 −1
(𝑥+𝐿2)2 ]
Consider 1
𝑥2+
1
( 𝑥+𝐿1+𝐿2)2= [ 2𝑥2 + (𝐿1)2 + (𝐿2)2 + 2. 𝐿1. 𝐿2 + 2 𝑥. 𝐿1 + 2. 𝑥. 𝐿2 ]
1
𝑥2(𝑥+𝐿1+𝐿2)2
1
(𝑥 + 𝐿1)2 +
1
(𝑥 + 𝐿2)2= [ 2𝑥2 + (𝐿1)2 + (𝐿2)2 + 2𝐿1. 𝑥 + 2. 𝐿2. 𝑥 ]
1
(𝑥 + 𝐿1)2. (𝑥 + 𝐿2)2
But 𝑥2(𝑥 + 𝐿1 + 𝐿2)2 = x2 [ x2 + L21 + L2 2 + 2.L1 . L2 + 2x. L1 + 2x.L2 ]
= x4 + x2 L21 + x2 L2 2 + 2.x2 L1.L2 + 2x3. L1 + 2x3 L2
And (𝑥 + 𝐿1)2. (𝑥 + 𝐿2)2 = (x2 + L21 + 2.x. L1 )* (x2 + L2
2 + 2.x. L2)
20
Thus obviously, 𝑥2(𝑥 + 𝐿1 + 𝐿2)2 < (𝑥 + 𝐿1)2. (𝑥 + 𝐿2)2
Thus obviously 1
𝑥2+
1
( 𝑥+𝐿1+𝐿2)2 >
1
(𝑥+𝐿1)2 +
1
(𝑥+𝐿2)2
Thus by (25), F1 > 0.
By (27): GMM’ / r2(t). cos ϕ < [ ( M’. V0 / D2 ).cos δ ] < ( M’. V0 / D
2 )
Thus GM * [ D / r ]2 * [cos ϕ] < V0
GM * [ cos ϕ / r2 ] < GM * [ D / r ]2 * [cos ϕ] < V0
Thus GM * [ cos ϕ / r2 ] < V0 ………………………………….(28)
By (27): F1 + GmM’ / L2 (t). cos θ + GMM’ / r2(t). cos ϕ = [ ( M’. V0 / D2 ).cos δ ]
F1 + GmM’ / L2 (t). cos θ + GMM’ / r2(t). cos ϕ < F1 + GmM’/ L2 (t) + GMM’
Thus ( M’. V0 / D2 ).cos δ < F1 + GmM’/ L2 (t) + GMM’
V0 < { (D2 / M’ ). F1 + Gm* [ D2 / L(t) ] + G.M. D2 } / cos δ
< { ( D2. F1 ) + Gm* [ D2 / L(t) ] + G.M. D2 } / cos δ = D2 [ F1 + (Gm / L (t)) + GM ] / cos δ
Thus V0 < D2 [ F1 + (Gm / L (t)) + GM ] / cos δ ………………………(29)
By (28) and (29): GM * [ cos ϕ / r2 ] < V0 < D2 [ F1 + (Gm / L (t)) + GM ] / cos δ ……..(30)
Here V0 is the total equivalent linear velocity of A1 along its path of orbiting, M is the mass of the
supermassive black hole, D is the distance between A1 and total centrifugal force acting center,
r(t) is the distance between A1 and the supermassive black hole at time t, F1 is the total equivalent
magnetic force that acts on A1 , m is the total mass of the of the host planets and the host star of
A1 , L (t) is the distance between A1 and the gravitational center of the host planets and the host
star of A1 at time t, ϕ is the angle between the direction of F3 and F1 , δ is the angle between the
direction of F1 and F4 .
21
Result Evaluation
According to (23): the particles of solar wind, those move towards the supermassive black hole do
not emit any electromagnetic radiations to the outer space. Therefore the nearby area of a
supermassive black hole is also dark rather than the region of the supermassive black hole.
According to (30): GM * [ cos ϕ / r2 ] < V0 < D2 [ F1 + (Gm / L (t)) + GM ] / cos δ
That implies that there is a range for the total equivalent linear velocity of a planet that along its
path of orbiting. And that limit of the linear velocity depends on the mass of the host supermassive
black hole as well.
Discussion
There are particles those eject by the solar surface (as the particles of the solar wind) and when
those particles accelerate towards the supermassive black hole, those solar wind particles cannot
emit any electromagnetic radiations to the outer space. Therefore no one can see those solar wind
particles in the sky those are accelerating towards the supermassive black hole.
And there is a range for the orbital velocity of a planet if that planet is considerably influenced by
a supermassive black hole.
Conclusion
Charged particles those are accelerating towards a supermassive black hole cannot emit any
electromagnetic wave. Therefore the nearby area of a supermassive black hole is dark and cannot
observe directly. Moreover, there is a range for the orbital velocity of a planet that is orbiting
around a star under the influence of a supermassive black hole.
Therefore we can apply this mathematical method to explain more observations in
astrophysics/cosmology and in physics as well.
22
References
Wikipedia Magnetic Field, Last modified on 31 January 2018
Retrieved from: https://en.wikipedia.org/wiki/Magnetic_field
Wikipedia Solar Wind, Last modified on 13 January 2018
Retrieved from: https://en.wikipedia.org/wiki/Solar_wind
The production of EM waves
Retrieved from:
http://labman.phys.utk.edu/phys222core/modules/m6/production_of_em_waves.html
Wikipedia Super massive black hole, Last modified on 03 February 2018
Retrieved from: https://en.wikipedia.org/wiki/Supermassive_black_hole
Wikipedia Force between magnets, Last modified on 04 February 2018
Retrieved from: https://en.wikipedia.org/wiki/Force_between_magnets
Wikipedia Magnetization, Last modified on 14 December 2017
Retrieved from: https://en.wikipedia.org/wiki/Magnetization
Appendix
The magnetic force (F1) that acts on a planet can be written as below (When there is an own
magnetic field with the planet and by considering the magnetic fields of the host black hole and
with the magnetic field of the host planets):
F1 = ( πμ0 / 4 )* (M1. M2. R1 2 . R2
2 ) * [ 1
𝑥2 +
1
(𝑥+𝐿1+𝐿2)2 −
1
(𝑥+𝐿1)2 −
1
(𝑥+𝐿2)2 ]……………(25)
μ0 is the permeability of space, which equals to 4π*10−7 T·m/A
23
M1 , M2 identified as the magnetization of the magnet in A1 and the magnetization of the total
equivalent magnet of the host planets/black hole respectively (the virtual magnets which produced
by the magnetic field of the black hole/ host planets of A1)
x is the distance between the two virtual magnets in meters (in the black hole/nearby planets/
nearby stars) - identified as the distance between magnet in A1 and the equivalent magnet of the
host planets/ black hole.
R1 , R2 are the radiuses of two magnets in meters - identified as the radius of the magnet in A1 and
the radius of the total equivalent magnet of the host planets/black hole respectively (the virtual
magnets which produced by the magnetic field of the black hole/ host planets of A1)
L1 , L2 are the lengths of two magnets in meters- identified as the lengths of the magnet in A1 and
the length of the total equivalent magnet of the host planets/black hole respectively (the virtual
magnets which produced by the magnetic field of the black hole/host planets of A1).
