simple MOND question

The Wikipedia page for MOND

https://en.wikipedia.org/wiki/Modifi...onian_dynamics

seems like a good introduction to the subject. In particular, it is
clear how Eq. (1) (which is the main postulate of MOND) leads to Eq.
(2), which is one of the main predictions of MOND (and it is not obvious
how this arises in more standard astrophysics and cosmology).

So far, so good.

But the equation immediately before Eq. (2),

F_N = ma^2/a_0,

can be rewritten as

F_N = ma(a/a_0).

The standard expression, of course, is

F_N = ma.

In the "deep MOND regime", where a is much smaller than a_0, the factor
in parentheses is much less than 1. So, the force in this case is the
standard force multiplied by a factor much less than 1. In other words,
at low acceleration ("deep MOND regime"), the MOND force should be
LESS than the standard force. However, this regime corresponds to the
outskirts of galaxies, where the observed orbital velocity is much MORE
than expected from the standard force law (which is why, if one assumes
the standard force law, one is led to dark matter to explain the
additional force).

What am I missing?

In article , "Phillip Helbig (undress to
reply)" writes:

But the equation immediately before Eq. (2),

F_N = ma^2/a_0,

can be rewritten as

F_N = ma(a/a_0).

The standard expression, of course, is

F_N = ma.

In the "deep MOND regime", where a is much smaller than a_0, the factor
in parentheses is much less than 1. So, the force in this case is the
standard force multiplied by a factor much less than 1. In other words,
at low acceleration ("deep MOND regime"), the MOND force should be
LESS than the standard force. However, this regime corresponds to the
outskirts of galaxies, where the observed orbital velocity is much MORE
than expected from the standard force law (which is why, if one assumes
the standard force law, one is led to dark matter to explain the
additional force).

OK, answering my own question here. F_N is the NEWTONIAN force. What
is confusing is that since the Newtonian force is known (F_N=ma), it is
somewhat confusing to write an expression for it which includes a_0, the
new constant (units of acceleration) introduced by MOND, and where a is
NOT the NEWTONIAN acceleration, but rather the "total" acceleration.
Presumably, the interesting thing is the acceleration predicted by MOND,
which is "a" above. So, in the "deep MOND regime", we have

a = sqrt(a_0*F_N/m).

Since

F_N = GMm/(r^2),

we have

a = sqrt(GMa_0)/r.

Contrasting this with the Newtonian acceleration, it falls off as 1/r
instead of 1/r^2 (in the low-acceleration regime), so the MOND
acceleration is of course larger.

Since the acceleration in a circular orbit is sqrt(GM/r^2), we get

v^4 = GMa_0.

In other words, the circular velocity is (in the low-acceleration
regime) independent of the radius, leading to the famous flat rotation
curves of spiral galaxies.

Mond introduces a new constant with the dimensions of acceleration, but
spiral galaxies don't have a constant ACCELERATION in the
low-acceleration regime, but rather a constant CIRCULAR VELOCITY.

On Sunday, 28 August 2016 20:06:48 UTC+2, Phillip Helbig wrote:

For my own thoughts about MOND read this:
http://users.telenet.be/nicvroom/mond.htm

OK, answering my own question here. F_N is the NEWTONIAN force. What
is confusing is that since the Newtonian force is known (F_N=ma), it is
somewhat confusing to write an expression for it which includes a_0, the
new constant (units of acceleration) introduced by MOND, and where a is
NOT the NEWTONIAN acceleration, but rather the "total" acceleration.

I prefer to call this the MOND acceleration (FN = Newton versus FM = MOND)
When I do a simulation the program looks something like:
for i = 1 to 100
ai = 0
for j = 1 to 100
if ij then
r = abs(ri - rj)
ai = ai + sqrt(G * mj * a0) /r (4)
next j
vi = vi + ai*dt
ri = r1 + vi*dt
next i
This is only a rough impression.
In reality you should use ax, ay, vx, vy, x and y
In the case of Newton equation (4) becomes:
ai = ai + G * mj / r^2 (5)
Equation (4) becomes equal to equation (5) when a0 = G * mj/ r^2

In both eq(4) and eq(5) ai represents a summation.
In Newton's case in eq (5) when you increase the number of objects
the total acceleration stays the same.
When you divide mj by N the right hand part of eq(5) becomes: (summation)
ai = ai + N * G * {mj/N}/r^2 for N objects.
This is the same as eq(5) for 1 object with mass mj

In MOND this is different. When you divide mj by N eq(4) becomes
ai = ai + N * sqrt(G * {mj/N} * a0) /r or
ai = ai + sqrt(N) * sqrt(G * mj * a0) /r
This is not the same as eq(4) for 1 object.
In fact eq(4) becomes larger.
This means that less mass is required the more objects are considered
with MOND.

Presumably, the interesting thing is the acceleration predicted by MOND,
which is "a" above. So, in the "deep MOND regime", we have

a = sqrt(a_0*F_N/m). (1)

Since

F_N = GMm/(r^2), (2)

we have

a = sqrt(GMa_0)/r. (3)

Contrasting this with the Newtonian acceleration, it falls off as 1/r
instead of 1/r^2 (in the low-acceleration regime), so the MOND
acceleration is of course larger.

That is true (also the speed is larger) but the consequence is that
if you want to simulate a specific GRC which MOND much less mass
is required.
In fact in the range of a GRC where v = constant there is no mass.

Since the acceleration in a circular orbit is sqrt(GM/r^2), we get

v^4 = GMa_0.

In other words, the circular velocity is (in the low-acceleration
regime) independent of the radius, leading to the famous flat rotation
curves of spiral galaxies.

IMO not all GRC's are flat.
See https://arxiv.org/pdf/astro-ph/0010594v2.pdf Paragraph 4.3
See http://arxiv.org/ps/astro-ph/0010594v2 specific fig 4

This document is interesting because it discusses both MOND and darkmatter:
https://arxiv.org/abs/1303.7062

About our Milky Way:
http://arxiv.org/abs/1504.01507
The question it what happens after 8kpc

A document with an unexpected conclusion?
https://arxiv.org/abs/1406.2401

Mond introduces a new constant with the dimensions of acceleration, but
spiral galaxies don't have a constant ACCELERATION in the
low-acceleration regime, but rather a constant CIRCULAR VELOCITY.

I agree this sounds strange. This constant acceleration takes care
that there is almost no matter required in the outer region.
As such GRC's with MOND always increase or are flat.

In Wikipedia they use the function mu(a/a0) = a/(a+a0)
When a goes to zero than FM = m * mu(a/a0) * a becomes smaller than
FN = m * a. That means that at large distances masses are not taken
into account (relatif speaking) with MOND, resulting in a flat GRC.

Nicolaas Vroom

In article , "Phillip Helbig (undress to
reply)" writes:

Since the acceleration in a circular orbit is sqrt(GM/r^2), we get

v^4 = GMa_0.

In other words, the circular velocity is (in the low-acceleration
regime) independent of the radius, leading to the famous flat rotation
curves of spiral galaxies.

Mond introduces a new constant with the dimensions of acceleration, but
spiral galaxies don't have a constant ACCELERATION in the
low-acceleration regime, but rather a constant CIRCULAR VELOCITY.

If I pretend that our formulation of GR is incorrect in some manner not yet=
discovered, and that the constant circular velocities observed in galaxies=
, that are independent of R, are caused by spacetime curvature rather than =
a MOND change of Newtonian expectations, how must spacetime curvature be ch=
anging with radius?

Seems to me the circular velocity is constant, but, the radius of curvature=
is getting larger. So it seems like the spacetime curvature would be gett=
ing smaller. But that it would not be getting smaller by as fast as we wou=
ld normally expect using current GR formulation.

Just trying to understand how we would understand the observation **IF** it=
were due to spacetime curvature. Would spacetime curvature be falling off=
by 1/R or some other factor?

Ross