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energy hypotheses ?
E Guiot
To cite this version:
E Guiot. A classical alternative to the dark matter and dark energy hypotheses ?. 2015. �hal-01198850�
E. Guiot
guiot.eric_1@yahoo.frAbstract. We report the possibility that an alternative classical force of gravitation could explain several anomalies which has been observed last decades, without ad-hoc hypotheses, as for example « dark matter » or
« dark energy » hypotheses. We present in this paper a simple solution to explain, the « flat » curve of rotation of the galaxies and the inflation, at an increasing rate, of the Universe, with respect for the classical physics.
Moreover, we suggest a test, inside the solar system, in order to valid or invalid our model.
Keywords: dark matter; force of gravitation; galaxies; modified gravitation;
1.Introduction
Several astronomical observations, last decades, seems in contradiction with the classical laws of gravitation. For example, the curve of rotation of galaxies can’t be explain without add, inside these galaxies, a large amount of a mysterious matter, generally called «dark matter » [1]. Moreover, the expansion at an increasing rate [3]of the universe can’t be explain with these laws of gravitation, without consider another hypothetic substance, called this time « dark energy », which should be repulsive [3,4].
However, despite all attempts of the researchers since several decades, to detect these matter and energy, it seems that there is no proof of their existence today [5]. These matter and energy could, consequently, be only a creation of the minds.
A second way, to explain these astronomical observations is to change the laws of gravitation themselves. A first problem is that these laws are well verified inside our solar system. A second problem is that the alternative theories don’t do unanimity, and sometimes seems in contradiction with laws of physics [6].
Consequently, we purposed, in our turn, an alternative theory of gravitation, in two previously paper [7, 8]. This theory is building with respect for classical physics. In this paper, we present a result we obtained recently. This result seems us solve the astronomical contradictions we listed before.
2. Our force of gravitation
We present hear the different assumptions, which lead us to our solution:
a) The classical physics can describe, with a correct approximation, the trajectories of the stars at the periphery of galaxies.
b) The trajectory of a star, which orbits around a center of mass, is a conic.
In our previously papers [7,8] we showed that the assumption b) is respected if the force is given by the relation
2
e
r f r f
r e f r f Af
F R
1 . ' 2 ) 1 1 (
' 2
1 2
2
(1)
Where f is a function of rand f' its derivate. This force is given in a polar system of coordinate, where ris the distance to the foci of the conic. The origin of the system is located at this foci and the angle is measured from the periapsis of the orbit. Ais constant and proportional to the mass of the system.
dt
d
is given by
) ) ( 1
( 2
r r f
e Aa
Where ais the semi-major axis of the conic and eits eccentricity. The radial speed is given by dtr
r d
Where ris given by
eCos e r a
1
) 1
( 2
For example, if we choose
2
) 1 (r r
f we obtain the Newton’s force. If we choose f(r)as a constant we obtain a force of Hooke [8].
We showed that, in all the cases, the force allows well to obtain a conic trajectory. We studied also the two body-problem and the conservation of the kinetic energy.
In order to determine a correct expression of f(r)we list now several other assumptions:
c) When the acceleration is high, this force of gravitation is the Newton’s
d) When the acceleration is low, and in the case of circular motion, this force leads to a “flat” curve of rotation, with respect for the Tully-Fisher law
e) When the acceleration is low, and in the case of parabolic motion, this force can be attractive or repulsive
We purposed several possibilities for f(r), in particular, in a first time, the solution given by )
1 1 ( ) (
0
2 r
r r
r
f
Because this simple solution, with a correct choice of the constant r0 allows to respect the assumption (c) and (d). But it appears this solution doesn’t respect assumption (e). At a second time, we studied the solution given by
) 1 1 ( )
( 2 kr2
r r
f (2)
Where kis a constant of the motion. It appears today that this second solution allows to respect all our assumptions, by a correct choice of k, and we will, consequently, in this paper, study it more detailed.
By using relation (1) we obtain the force, given by
3
e
kAeSin a e
e r r kA a e r
F A R R
. 1
) 1
( 2
2
(3)
The first term of this expression is naturally the Newton’s. When ris small this term is dominant, but when rincreases, progressively, this term disappears. Assumption c) is well respected.
3. Physical interpretation of this force
This force can be decomposed in two forces, by the relations
2
1 F
F F
where the force of Newton F1
is given by
eR
r F A
1 2
And the second part is
e
keSin a e
e r r k a
F R
. 1
) 1
( 2
2
the magnitude of this second force is given by
ar a r e a a
F2 k 2(1 2) 2 2 2
by noticing that the distance OM, where Mis the point-particle and Othe center of the conic, is given by
ar a r e a
OM 2(1 2) 2 22
we obtain
aOM F2 k
We can now determine the center of this force: by using the polar coordinate of the vector OM given by
r e aeSin e aeCos
M
O R
. )
(
And the relation of the conic
er r e Cos a(1 2) We notice that the vector product
0
* 2
F
M O Consequently, OM
and F2
are parallel and the center of the force F2
is located atO. The force F2 is simply a force of Hooke and its constant factor is given by
a K k we can now present our force on the figure 1.
4
We see that our solution is simply an addition on a force of Hooke and a force of Newton.
The physical signification is that, when the acceleration decreases, the force of gravitation, in our model, becomes progressively a force of Hooke. The force of Newton is only valuable for small distances and high accelerations (as inside solar system or centers of galaxies). The force of Hooke can be more attractive than the Newton, but can also be repulsive, in the case of parabolic or hyperbolic motion.
Note that these two well-known forces are central forces, and consequently can be derivative from a potential. Moreover, an important observation is that, when rincreases, the center of the force and the center of mass aren’t located at the same point.
4. Determination of the constant k
We will now give a correct value ofk, in order to respect the other assumptions. We study the two limiting cases, the circular motion and the parabolic motion.
4.1. The motion is circular
In this particular case, the eccentricity of the conic is given by e0and ar. Consequently, the force given by relation (3) becomes
R R kAe r e
F A
2
The curve of rotation is obtained by written the equality of the acceleration and the force kA
r A r
V
2 2
Where Vis the speed of the point-particle. We obtain r kAr
V A (4)
When rincreases this relation becomes
O
M
F F2
F1
Figure 1. Representation of our force of gravitation
5
kAr V And, consequently
kAa V
In order to obtain a flat curve of rotation, we see that khas to be dependant of the semi-major axis, as a
k r
0
1 (5)
Where r0is constant. Moreover, we can determine this constant, because we have to respect the Tully Fisher law [9]. Indeed, this empirical relation suggests a correspondence between the visible mass of galaxies and the velocity as
)
4 ( M V
WhereM is the total mass of the galaxy and Vthe constant speed of the stars. We saw that Ahas to be proportional to the mass thus
14 0
r A V A
And consequently
12
0 A
r
This condition, with respect for physical dimensions, leads to write the relation A
a r0 0
Where a0is an acceleration. This acceleration has to be constant in the entire universe, and can be linked with the Milgrom’s acceleration used in MOND. We see that now Vis constant for circular motion and is in agreement with the Tully-Fisher law. Consequently, our choice of kallows to respect assumption d).
4.2. The motion is parabolic
In this case, the relations of the conic are given by e1and By using relation (3) the force becomes
R
R e
a kA r e r
F A
1
2
And by using (5) we obtain
R
R e
a r ar e A r
F A
1
0 2
Consequently, this force can be, in the case of parabolic motion, alternatively attractive or repulsive.
The sign of the acceleration is depending on the ratio a
r . We can solve the equation to obtain the particular points of this evolution
6
0 1
0
2
a
r ar
A r
A (7)
This equation becomes after simplification
2 0
0 2
3 r ar a r
To solve this equation we use the method of Cardan. We write thus
2 0 3 3
27 2
3 a a r q
p a
The discriminant is given by
5 0 2 4
0 2 3
27 1 4
1 2
3p q r a r a
D
And this discriminant is equal to zero if
4 0
27r a Consequently if 0
4 27r
a we haveD0 : We obtain one real solution but it appears that this solution is always negative and has no physical signification. This point indicates that, in this case, the force of gravitation is always attractive.
But, if 0
4 27r
a we haveD0 : in this case we obtain three real solutions. As previously, it appears that one of them is negative: consequently, only two solutions are possible. Thus, the force of gravitation can be successively attractive and repulsive. We present theses results in the Figure 2.
7
We can distinguish three different cases:
When rrAthe force is always attractive and can be approximately, if ris small (as for example in solar systems), considerate as a force of Newton.
WhenrArrBthe force is repulsive.
When rrBthe force becomes an attractive force of Hooke.
4. Our cosmological model
With the results we obtained, about the circular and parabolic motion, we can now present a simple and schematic cosmological model.
We consider a galaxy. At the periphery of this galaxy, the orbits of the stars are circular. The radius of their orbits are such rr0. Consequently, the curve of rotation is given by
r0
V A
Acceleration (parabolic motion)
r
rA rB
Force of Hooke
Force of Newton
Zone of repulsive force of Gravitation
Figure 2. Evolution of the force of Gravitation with r
0 D
0 D Small distances :
Force of Newton
8
The curve of rotation is well “flat”, without dark matter hypothesis to explain it. Moreover, with the choice
0
0 a
r A
Where a0is a constant acceleration of the universe (comparable or equal to the acceleration used in MOND), the curve of rotation is in agreement with the Tully-Fisher law.
We consider now two clusters of galaxies, which move by describing a parabolic motion. The semi major axis of this motion is aand r0is a constant of the system composed of the two clusters.
We have now two solutions to describe the force of gravitation between these clusters. If 4 0
27r a
Then the force is always attractive. But if
4 0
27r a
The force can be repulsive or attractive. If we consider now the modern results about the inflation of the Universe [2], we can suppose that it is the case today, for the majority of the interactions of the clusters between themselves. But this evolution isn’t definitive: as shown on the figure 1. If our model is correct the force of gravitation could become a day, attractive.
Note that the difference of behavior of the force is only due to the difference of motion: in the first case, the motion is circular (in reality, approximately circular) and in the second case, the motion is parabolic (in reality, approximately parabolic). We show that with the solution given by (5) the same force can easily describe so different behaviors.
5. Discussion
5.1. A test in the solar system
As we suggested it previously [8], we can purpose a test inside the solar system, in order to valid or invalid our model. This test consists “simply” to do experiments about the free-fall motion. If our solution is correct this free fall motion should be describe, with the relation
1
0
2 a
r ar
A r r A
(8)
Where ris the radial acceleration, given by dt r r d2
2
Consequently, we should obtain a modification of the Newton’s acceleration, directed to the center of Mass (in the case of parabolic motion). This modification should be very small but can perhaps be detected. We can note that such modification has recently been suspected inside the solar system, in particular about the motion of several spacecrafts [10]. Therefore, in our solar system, we think that the constant r0can’t be linked to a0with the simple relation
9
2 0
0 r
a A
Because our solar system isn’t an isolated system. But we can expect perhaps that r0is constant in all the solar system, and deduct its value from the experiments. Indeed, naturally, only experiments on the free fall motion, in order to detect a small modification of the Newton’s acceleration, could give a correct result.
Therefore, we don’t expect modification of the trajectories of the planets, because these trajectories are always conics and consequently, not modified.
5.2. Simulations of our model
A second method to valid or invalid our model is to do simulations of it. In particular, we could test it at the periphery of spiral galaxies. We think that results could be correct. Because in fact, our solution given by
) 1
1 ( ) 1 1 ( )
( 0
2 2 0 2
2 A
a a r ar r
r r
r
f
Can be writen
) 1 ( )
( 2 g r
r r
f
With
) 1
( )
( 0
2
A a a r r
g
And, in the case of circular motion
) 1
( ) 1
( )
( 0 0
ac
a A
a a r
g
With the acceleration
a2
ac A
It appears that this function is proximate to the “simple” interpolating function used in MOND and given by
0 1 0
) 1
(
c c
a a a
a
This function has been tested with success [11]. Consequently, in our case, we should obtain correct results. Therefore, note that in our case the function g(r)isn’t empirically.
A second important simulation should be to test our model in the case of the bullet cluster. Because in fact, the astronomical observations inside this particular cluster [12] are generally considerate as an indirect proof of the dark matter existence. Moreover, it appears that the alternative theories of gravitation, as MOND, fail to describe the dynamics of this cluster. Consequently, this test should be important in our case.
We think that this simulation could perhaps lead to correct results. Indeed in our model, the center of mass and the center of force aren’t necessary located at the same point. Consequently, we won’t need,
10
perhaps, to use ad hoc hypotheses, as “dark matter” hypothesis, to explain this difference of localization.
At end, an important test will be to see if our model can be a limiting case of a relativistic theory of gravitation. This point is naturally crucial. We think it is perhaps possible. Indeed, our force of gravitation is only the sum of two central forces, which are consequently the derivative of potentials.
5.3. Is a force of Hooke possible for large distances?
In our model, we suppose that the force of Newton turns progressively into a force of Hooke, when the acceleration decreases. A problem is to know is this evolution is possible. We can give several arguments, in order to defend this idea.
Firstly, the force of Hooke exists already in the classical theory of the gravitation: it is simply the force which describes the motion inside a homogeneous sphere. Moreover, in another physical domain, this force was used, for example, to build the Thomson’s model.
Secondly, we showed that the addition of these two forces leads well to conics trajectories, and allows respecting all our assumptions. In particular, we don’t need to introduce a new and hypothetic force to describe the reality. All our assumptions are simply respected, and we can build a simple cosmological model, without “dark matter”, and above all, without “dark energy”. Our force can be repulsive, and this point is a cogent argument for it. For example, MOND, or Dark matter hypothesis, or classical theories of gravitation, can’t explain the expansion of the universe at an increasing rate, without hypothetic hypotheses, as “dark energy”.
6. Conclusion
In this paper, we present a simple classical theory of gravitation, in order to explain several anomalies which have been detected since several decades. This theory is building with respect for classical physics and is predictive. The fundamental idea is that the force of gravitation is modified when the accelerations decreases, and becomes gradually a force of Hooke. We show that this
modification can easily explain the curve of rotation of the galaxies, and the inflation of the Universe.
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