"Modern Research"-projekt

Is the change of Newtons law a viable option instead of Dark matter?

By p.kamphuis

1.1 Introduction

When we look at the rotation curves of galaxies we see that at the outer regions these curves do not drop like we expect them to do (When the mass we deduced from the light emmited by the galaxy is all there is)but become a flat line instead.Most common explanation for this is:"We cannot see all the mass in the outer radius of the galaxies".I'm looking at an other explanation for this phenomena.

1.2 Modified theory of non-relativistic dynamics(M.O.N.D)

One form of explanation of the unexpected flatness of the rotation curves are the so called Modified theory of Non-relativistic dynamics (MOND).This theory is based on the following assumptions.
a)A breakdown of Newtonian dynamics(Second law and/or gravity)occurs in the limit of small accelerations.
b)In this limit the acceleration, a,Of a test particle in a gravitating system is given by a(a/a₀)=g_N where g_N is the conventional gravitational field and a₀ is a constant with the dimension of acceleration.
c)The transition from the Newtonian regime to the small acceleration asymptotic region occurs within a range of order a₀ about a₀.
So based on these assumptions we are led to
m*µ(a/a₀)a=F
µ(x» 1)=1, µ(x« 1)=x
replacing newtons second law.

1.3 How is a₀determined?

a₀ is determined to be a constant with a value of about 2 x 10^-8 (H₀/50 km/s/Mpc)² cm/s²
a₀ is determined in four different ways
a)First we can determine a₀ from the V⁴to M relation. this gives a value of about a₀=2x10^-8 h²₅₀(P/P₀)^-1cm s^-2where P is the M/L ratio for standard matter in galaxies in solar units and P₀ is a model value.
b)Second from the V⁴to M relation aplied to our own galaxy the Milky Way:a₀=6x10^-8(V_G/220 km s ^-1)⁴(M_G/3x10¹⁰M_&odot)^-1 cm s^-1 where M_G is the Galaxy`s mass and V_G its asymptotic velocity
c)From the requirement that the preferred surface brightness,observed in galaxies ,equals the one found neccesary to obtain rotation curves that stay flat to very small radii:a₀=3x10^-8(P/2)cm s^-2
d)Assuming that the discrepancy factor between the Oort density and the observed one is µ(V²_&odot/r_&odota₀=0.7 where V_&odot is the Sun's orbital velocity and r_&odot is the Sun's orbital radius.From this follows:
a₀=2x10^-8(V_&odot/220 km s ^-1)²(r_&odot/8.5 kpc)^-1 cm s^-2
Method a) requires exact knowledge of M/L and H₀,b)does not depend on these constants but is susceptible to uncertanties in the galaxy`s mass and asymptotic velocity,c)requires a detailed analysis of induvidual galaxies and may require accurate knowledge ofµ(x),d)requires also information on µ(x) and involves uncertanties in the velocity and radius of the sun.
In all these determinations a₀ follows from the theorie it was never actually measured in an experiment so it gives no hard evidence for the existence of MOND.

1.4 Can MOND be measured experimentally?

Can we measure what the function µ should be ?
Actually it doesn't matter a lot what µ is.It could be of a number of forms as long as it obeys the assumptions of MOND stated above.
µ could this way be of the form
µ(x)=x(1+x²)^-0.5 or
µ(x)=1-e^-x
What µ exactly should be can only be found by comparison of observed rotation curves.
Can we detect this force described by Mond? I believe that this force is not yet detected.The only indication it exist that i could find was a acceleration of the pioneer 10 and 11.
MOND could be an explanation for this acceleration but so could a number of explanations.But maybe oneday MOND will be only theorie that survives I just don't know.

So how does this all remove the need for dark matter?

To explain this I plotted three different simulations of rotation curves in one plot(fig1).

fig1
In this plot the first half isn't interesting because there we are sure how to determine I simulated this part for all three simulations with V_R,U_R,K_R=w*R.
In the second half I chose for V_R and U_R a constant mass like it would be at a time when there was only luminous mass in a galaxy.
And for K_R I chose a mass that was still increasing beyond the half of the plot this way simulating Dark mass.
So the second part of V_R simulates the MOND theorie and is calculated by:
V_R=G*M*a₀ making the rotationcurves asymptotically flat as soon as the mass stops increasing.This asymptotically flatness is somthing we observe in the observed rotation curves.By fitting the M/L ratio and/or the value of a₀ we can plot a theoratical curve over an observed one thus eliminating Dark matter.
How to derive V_R from MOND
The second part of K_R is simulated by the normal dynamics:
K_R=(G*M/r)^0.5
With the right mass distibution this could fit any observed rotation curve.This is one of the weaknesses of the dark matter theorie.It hasn't much restrictions.
The second part of U_R is simulated by the same dynamics as K_R but this time with a constant mass.
In this rotation curve we clearly recognize the rotation curves we get when we only plot the luminous mass.
In most observed rotation curves we do not see the asymptotical flatness direct after the diversion from v=w*r but we notice a little hump like the rotation curve of only luminous mass.This is to be explained by the fact that the MOND part simulated here only works with little accelerations with greater accelerations we have the normal newtonian dynamics so it would look like the luminous mass curve until these slow accelerations were reached.

Conclusions

I found MOND to be viable option instead of Dark matter.It surely is more specific then Dark Matter.And where I can see no explanation for the asymptotical flatnesses in rotation curves with the Dark Matter problem(Why should the distribution of DM be almost always so that we get this flatness)MOND does give an explanation for this.Also MOND makes use of only two parameters when fitting to the rotation curves DM uses three.So overall I find MOND a better theorie then Dark Matter except that there is no evidence for it and there is for Dark matter. For this evidence look at our main page Dark matter?

More information

More information about how the rotation curves are actually determined can be found at olof
More information about how the luminous mass of the systems are determined can be found at Douwe

Literature

Authors: MILGROM, M.
Title: A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis
Bibliographic Code: 1983ApJ...270..365M

Authors: MILGROM,M
Title: A modification of the newtonian dynamics:implications for galaxy systems
Bibliographic Code: 1983ApJ...270..384M

Authors: MILGROM,M
Title: A modification of the Newtonian dynamics -Implications for galaxies
Bibliographic Code:1983ApJ...270..371M

Authors: BEKENSTEIN, J.; MILGROM, M.
Title: Does the missing mass problem signal the breakdown of Newtonian gravity?
Bibliographic Code:1984ApJ...286....7B

Authors: BEKENSTEIN, JACOB D. SANDERS, ROBERT H.
Title: Gravitational lenses and unconventional gravity theories
Bibliographic Code:1994ApJ...429..480B

Authors: BEGEMAN, K. G.;BROEILS, A. H.;SANDERS, R. H.
Title: Extended rotation curves of spiral galaxies - Dark haloes and modified dynamics
Bibliographic Code: 1991MNRAS.249..523B

Authors: SANDERS, R. H.
Title: The Published Extended Rotation Curves of Spiral Galaxies:Confrontation with Modified Dynamics
Bibliographic Code:1996ApJ...473..117S

Authors: FELTEN, J.E.
Title: Milgrom's revision of Newton's laws - Dynamical and cosmological consequences
Bibliographic Code:1984ApJ...286....3F