AiSC STI 04 Team Number: 3.0

Team Name: Heavy Thinkers Area of Science: Astronomy, Mathematical

Modeling Project Title: Variable Gravity or How Much

Do You Want to Weigh?  

Team Members• Jeffrey K Raloff • Dale Henderson

• Sponsoring Teacher(s) • Challenge Betsy • Nick Bennett • Project Mentor(s) • Isaac Newton • Albert Einstein • David Kratzer

Abstract:• The current theory of the universe calls for a strange

substance called "dark matter" that provides the mass to hold together the universe and the rate which the galaxies spin.

• This stuff has many strange characteristics much like the mythical ether that was proposed to carry light one hundred years ago.

• No experiments to date have confirmed the existence of this dark matter”.

• Instead what we observe in the universe can be demonstrated if gravity were not held constant .

• We will take current mathematical models of gravity and modify them to see if the variability of gravity would hold together galaxies as they really are.

Background• In 1983 Moti Milgrom proposed a different

solution to the “dark matter” problem.

• Estimates of at least 90% of the matter of the whole universe had to be this “dark matter”.

• Without this matter the galaxies and the universe could not stay together.

MONDThe alternative solution is Modified Newtonian

Dynamics [MOND ]. Traditional Newtonian would have a solar system

with the velocities of the planets decreasing away from the sun.

However, on the macro-galaxy scale, the MOND solution proposes the acceleration of gravity changes at “very large” distances. This is much like the fact that forces on an atomic scale are very different than our scale, requiring a whole new set of laws – quantum dynamics.

When it appliesWhen the accelerations of a star in orbit in a galaxy are much above a

value of 1 angstrom per sec per sec,{a[0]}, which reaches about to the outer edge of our solar system, regular Newtonian dynamics apply. For the larger sizes like the size of a galaxy, the acceleration of gravity g[N] would be: a[0]*mu(x) = g[N], where x = radius.

Two commonly assumed forms which are acceptable to galaxy data are

MOND has been shown to follow the actual shape of velocity versus radius determinations in more than one hundred galaxies so far. In the following slide the blue line represents a typical Newtonian prediction and the pink line the MOND prediction – which follows a real distribution.

21

2 )1()(

xxxmxexm 1)(

xexm 1)(

Simulation of is called V vs R in

excel

MOND Files €

Distance=r NewtVelocity MONDVELocity

1.00 173.205081 177.8279411.05 169.030851 180.0102871.10 165.144565 182.1160291.15 161.514571 184.1511611.20 158.113883 186.1209721.25 154.919334 188.0301551.30 151.910905 189.8828921.35 149.071198 191.6829311.40 146.385011 193.4336421.45 143.838990 195.1380681.50 141.421356 196.7989671.55 139.121669 198.4188481.60 136.930639 200.0000001.65 134.839972 201.5445161.70 132.842233 203.0543181.75 130.930734 204.5311741.80 129.099445 205.9767141.85 127.342908 207.3924451.90 125.656172 208.7797631.95 124.034735 210.1399642.00 122.474487 211.4742532.05 120.971676 212.7837532.10 119.522861 214.0695142.15 118.124885 215.3325162.20 116.774842 216.5736772.25 115.470054 217.7938592.30 114.208048 218.9938702.35 112.986537 220.1744732.40 111.803399 221.3363842.45 110.656667 222.4802792.50 109.544512 223.6067982.55 108.465229 224.7165432.60 107.417231 225.8100862.65 106.399035 226.8879702.70 105.409255 227.9507062.75 104.446594 228.9987832.80 103.509834 230.0326632.85 102.597835 231.0527892.90 101.709526 232.0595792.95 100.843897 233.0534333.00 100.000000 234.0347323.05 99.176941 235.0038413.10 98.373875 235.9611063.15 97.590007 236.9068613.20 96.824584 237.8414233.25 96.076892 238.7650963.30 95.346259 239.6781733.35 94.632045 240.5809313.40 93.933644 241.4736403.45 93.250481 242.3565573.50 92.582010 243.2299283.55 91.927712 244.0939913.60 91.287093 244.9489743.65 90.659683 245.7950973.70 90.045034 246.6325713.75 89.442719 247.4616003.80 88.852332 248.2823803.85 88.273483 249.095099

Newt vs MOND

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0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00

radius (arbitrary units)

velo

city

(ar

bitr

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units

)

NewtVelocityMONDVELocity

N-Body “Galaxy” programsWe would find a program in C++ [StarX] that we

could run and modify to check out MOND.

A main part of a completed project would be Java programs to model the variation in orbital velocity first, and second a small number of stars orbiting a central [black hole] with a Java applet. In these programs we could put the actual parameters from our research to get more realistic models.

Original approach

Excel Formulas

• SQRT(3/A5)*100 • SQRT(SQRT(A5*10))*100 – are the excel

formulas for Newtonian Velocity and MOND velocity respectively. A5 if the radius variable and the constants were chosen to have the graphs start near each other and show the exponential parts.

Started off with…

Then …. Starlogo ModelStarlogo is used to model a number of galaxy

systems with different parameters:1. Basic solar system orbit purely with Newtonian

physics [01]2. Solar system evolution with constant center of

mass [xx]3. Solar system evolution with variable center of

mass [02] [03]4. “big-bang” with constant gravity [04]5. “big-bang” with variable gravity [05vg]

Main hurdles

• Only a crude model of a “galaxy” with a few stars rotating about a center*

• The N-body problem has made any precise modeling of a real galaxy impossible to date, even with super-computers, more extensive models may well be difficult to run in Starlogo or Java.

• However, we still have some success.

lack of “space”lack of “bodies”

Sources – http://• Cosmological models in the relativistic theory of gravitation• Physics Demonstrations on (1L) - Gravity • : Class Model• UA Astronomy - Normal Galaxy Images• Surface of Section• Cosmology JavaLab• [XSTAR] The XStar N-body Solver• Galaxies• The MOND pages• NASA ADS: ADS Home Page• [astro-ph/0107284] How Cold Dark Matter Theory Explains Milgrom's Law• Search results_MOND• 0207469.pdf (application/pdf Object)_equation!• MOND_discussion forum• http://www2.iap.fr/users/alard/mond/• Modified Newtonian Dynamics and the physics aesthetic• PhysicsWeb - homepage• PhysicsWeb - Shadow cast on dark matter

More MOND page background• We usually think first in terms of a modification at some length scale:

galaxies are big, so maybe gravity is different on large scales. This does not work. But there are other scales which are different about galaxies. One of them is the very low centripetal acceleration experienced by stars orbiting within galaxies. This is just as far removed from our laboratories as is the size scale of galaxies. MOND was motivated by two observations: 1) the asymptotic flatness of rotation curves and 2) the slope of the Tully-Fisher relation (M ~ Vflat4). These two things lead to an acceleration scale:

• Newton: GM/R = V2Observed: M = AV4

• where A is a constant which holds irrespective of differences in R. Squaring Newton,

• V4 = (GM/R)2 = M/AA-1 = G2M/R2 = G*(G*surface density)

• and G*surface density has units of acceleration: • A-1 = Ga0

Newtonian Orbital Theory

Equating the force from Newton’s 2nd law of motion and his Law of Gravitation we get the first equation, then deriving v.

2

2

RGM

Rv

RGMv

MOND [material from MOND Pages FAQ]

A[0] = 1.2 x 10-10 m s-2, i.e., about one Angstrom per second per second. This is one part in 1011 of what we feel on the surface of the earth. The precise value depends on the distance scale to galaxies, so perhaps it would be better to say a[0] = 1.2 x 10-10 m s-2 h752, where h = H0/75 is the Hubble Constant (the expansion rate of the universe) in units of H0 = 75 km s-1 Mpc-1. (Currently, most measurements report values in the neighborhood of H0 = 72 km s-1 Mpc-1.)