MOND
Modified Newtonian Dynamics
A Humble Introduction
Markus Nielbock
Johannes Kepler
1571 - 1630
Isaac Newton
1643 - 1727
Overview
• Gravitational Law (Newton/Kepler)
• Application: Solar System (Theory/Observation)
• Application: Galaxies (Theory/Observation)
• Modification of Newton‘s Gravitational Law
• Consequences of MOND
(rotation curves, surface density, isothermal spheres)
• Difficulties
• Summary
Newton‘s Gravity
GMm
Gravitation: Fg 
 g m
2
R
m free  weight of m is zero, accelerated with a
m fixed  g assigns a weight to m
/
Centrifugal force:
3. Keplerian Law
Solar System
GM 2
R  2 P
4π
3
Solar System Rotation Curve
GM
v
R
Galaxies
The laws of physics concerning (Newtonian) gravitation seem to be
transferrable from laboratory scales to the solar system.
We are confident, they are valid even on larger scales like galaxies.
Rotation curve:
GM ( R)
v
R
Rotation Curves of Galaxies
Observations contradict theoretical predictions.
1. Orbital velocities are too high.
measured
2. Rotation curves stay flat.
GM ( R)
v
R
gas
„Dark Matter“
MOND
Modified Newtonian Dynamics
Milgrom (1983)
based on Newtonian, non-relativistic gravitational theory
modification of inertia
modification of gravity
if
if
New fundamental constant:
(empirical)
Might be a coincidence.
MOND
Modified Newtonian Dynamics
analytic form of µ unknown, often assumed to be like:
μx  
x
1 x2
MOND
Modified Newtonian Dynamics
if
Gravitational forces in bound systems mostly Newtonian.
Only at large distances from the central mass (e.g. in galaxies), the
acceleration declines below a0 (R = 11.8 kpc for M = 1011 M).
In our solar system, the gravitational acceleration of all planets lies well
above a0.
But: a = a0 for R = 7700 AU  Oort Cloud
Rotation Curves with MOND
What is the rotation velocity with MOND, where
?
GMa0
Gravitational acceleration:
a
Centrifugal force:
v2
a
R
GMa0
R
R
v2

R
v  (GMa0 )
1
4
For a given mass, the rotation velocity converges to a constant value.
This is in accord with observations.
v ~ L
4
Tully-Fisher
Rotation Curves with MOND
The fitting procedure:
• assumption: M/L is constant
• NIR surface photometry preferred (old stars, extinction)
• include neutral hydrogen and correct for helium abundance
• calculate the Newtonian gravitational force for a thin disk
and add a bulge, if necessary
• calculate the MONDian gravitational force with a fixed a0
and use the M/L ratio as the only free parameter
Comparison: MOND vs. Dark Matter
HSB galaxies
Begeman et al. (1991)
Comparison: MOND vs. Dark Matter
LSB galaxies
Begeman et al. (1991)
• MOND fits rotation curves as good as „Dark Matter“ or better
• substantial improvement for LSB galaxies
The Critical Surface Density
Can we find a diagnostic quantity that indicates the validity of MOND?
M
M

 2
A πR
GM
a
a 2

R
πG
A
M
Galaxy
a0
M
Critical surface density:  m 
 228 2
πG
pc
M
Spiral galaxies:   1000 2
rotation curves Keplerian-like
pc
LSB galaxies: rotation curves rising asymptotically
h
NGC 2903
Disk Instabilities
• rotating, gravitating systems unstable
• galactic bar formation
• in MOND:    m (Spirals)
B
• most spiral galaxies should have bars
• corroborated by observations (NIR)
Ks
Isothermal Pressure-Supported Systems
 r
M
radial velocity dispersion: 11  
10 M   100 km


s
4
• Elliptical galaxies
similar to Faber-Jackson relation
Isothermal spheres with r  100  300 km s have galactic mass.
• Molecular clouds
MOND predicts „dark matter“ problem
low-mass extension of Faber-Jackson relation
105 M for typical velocity dispersion ~5 km/s
The Equivalence Principle
Inertia and weight are not equivalent. Mass of weight and mass of inertia
are not the same, but depend on the state of acceleration.
Theory of Relativity?
Difficulties and Problems with MOND
• claims a0 may not be universal
not confirmed: data quality, poor statistics
• The case NGC 2841
poor fit
distance derived from redshift
distance free fitting parameter
excellent fit
Cepheid distance: 14.1 Mpc
Cepheid calib. T-F: 23 Mpc
Supernova (Ia ?): 24 Mpc
Sanders (1996)
Difficulties and Problems with MOND
MOND is derived from classical Newtonian Gravitational Theory, and
therefore is incompatible with General Relativity.
Just like Newtons Gravity, MOND cannot give reliable answers to:
• Cosmology
• Relativistic Phenomena
Summary
Rotation curves of galaxies are not Keplerian/Newtonian.
Apparently contain more matter than is visible (Dark Matter).
Alternative Explanation: Modification of Gravity (MOND)
MOND describes galactic rotation curves very well.
MOND provides predictions verified by observations.
Just like Newton‘s Gravity, MOND cannot explain relativististic effects.
Dark Matter and MOND should be treated equally.