a-cosmology-group/acg#304

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#304 MOND phenomenology
RedshiftDriftRedshiftDrift, edited OpenGravitation Models/Derivations

MOND experts: what’s the best determined value for the MOND acceleration $a_0$?

The best I found is $a_0 = 1.20 \pm 0.02 \mbox{ (rand)} \pm 0.24 \mbox{ (syst)} \times 10^{−10} h_{75}^2 \mbox{ m/s}^2$, from the 2016 paper doi: 10.1103/PhysRevLett.117.201101, with $h_{75} \equiv H_0/(75\mbox{km/s/Mpc})$.

Does anybody know of a value of $a_0$ now measured with a smaller systematic uncertainty?

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Comments (13)
HanDeBruijnHanDeBruijn

Key formula for the MOND acceleration in my MOND = UAC section with the Variable Mass theory is

$$ a_0 = \frac{G M}{R^2} $$

And the following values have been plugged in for making the calculation.

# Gravity constant (G)
G := 6.67408*10^(-11);
# Speed of light
light := 299792458;
# Number of seconds in a year
year := 31556926;
# Radius of milky way (R)
R := 52850*light*year;
# Mass of our sun
sun := 1.989*10^(30);
# Mass of milky way (M)
M := 2.06*10^(11)*sun;
# MOND acceleration
a_0 := G*M/R^2;

Resulting in $a_0 := 1.09 \times 10^{-10}\mbox{ }m/s^2$ .

Notes.

  • With Wikipedia as a reference and searching for “2023” we find: In September 2023, astronomers reported that the virial mass of the Milky Way Galaxy is only $2.06\times 10^{11}$ solar masses, only a 10th of the mass of previous studies.
  • Data may be easily corrupted by the assumed presence of e.g. Dark Matter. So I would appreciate a second opinion on the above numbers, especially those for radius $R$ and mass $M$ .
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RedshiftDriftRedshiftDrift, edited

MOND acceleration applies to the rotation curves of all disk galaxies, why use $R$ of the Milky Way?

In any case, I am asking for a measured value, not a theoretical one.

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HanDeBruijnHanDeBruijn, edited

@RedshiftDrift . >MOND acceleration applies to the rotation curves of all disk galaxies, why use $R$ of the Milky Way?

Does a systematic error $\pm 0.24$ suggest that $a_0$ is a true “constant” for all galaxies ? And the Milky Way is just an example, of course. Would you rather prefer the Andromeda?

# Gravity constant (G)
G := 6.67408*10^(-11);
# Speed of light
light := 299792458;
# Number of seconds in a year
year := 31556926;
# Solar mass
sun := 1.989*10^(30);
# Radius of Andromeda (R)
R := 152000*light*year;
# Mass of Andromeda (M)
M := 1.1*10^(12)*sun;
# MOND acceleration
a_0 := G*M/R^2;

Giving $a_0 = 0.7\times 10^{-10}\mbox{ } ms^{-2}$ . >In any case, I am asking for a measured value, not a theoretical one.

Calculating $a_0$ from the definition $F= m (a^2/a_0)$ , how does that count as a “measured value” ?

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RedshiftDriftRedshiftDrift, edited

@HanDeBruijn The systematic error on $a_0$ results from our inaccurate knowledge of the luminosity-to-mass ratio $\Upsilon_\star$ for all galaxies (hence it’s called ‘systematic’). Milgrom and other astronomers have shown that $a_0$ is dependent on $\Upsilon_\star$, but $a_0$ is the same for all galaxies.

Since your equation gives a result that varies with the mass and size of the specific galaxy, you are not calculating a fundamental constant in UAC theory. In any case, the measured values of $m_{galaxy}$ you use all depend on $\Upsilon_\star$ (that’s how they know the mass of galaxies). So your numbers also have the systematic uncertainty of $\Upsilon_\star$, and therefore cannot be more accurate than the $a_0$ given above.

Calculating $a_0$ from the definition $F = m ( a^2 / a_ 0 )$, how does that count as a “measured value” ?

Every measured value depends on a model or a theory. In MOND, it turns out that for whatever $F$, $m$ and $a$ observed in galaxies, the same $a_0$ comes out. Think of $F = Gmm’/r^2$ : for any measured $F$, $m$, $m’$ and $r$, the same constant $G$ comes out. $G$ is the measured value of a fundamental constant in Newtonian gravity theory.

${\rm }$ What I am asking above can be rephrased as: what’s the best known value for $\Upsilon_\star$ ? Or is there a measurement of $a_0$ that does not require the mass of galaxies?

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HanDeBruijnHanDeBruijn, edited

The only “theoretical” mathematics of the (PDF) paper seems to be contained in this equation:

$$ \large g_{obs} = \frac{g_{bar}}{1-e^{-\sqrt{g_{bar}/g_\dagger}}} \qquad (4) $$

where $g=$ acceleration, obs = observed, bar = baryonic, $\dagger =$ their one fit parameter is the acceleration scale. However, I find (4) much too complicated. With a series expansion for the exponential function it may be simplified to

$$ \large g_{obs} \approx \frac{g_{bar}}{\sqrt{g_{bar}/g_\dagger}} = \sqrt{g_{bar}.g_\dagger} $$

So we have the geometric mean of two accelerations. And with MOND = UAC as before, tacitly assuming that $a_0=g_\dagger$ :

$$ g_{obs} \approx \sqrt{\frac{GM}{r^2}.\frac{GM}{R^2}} = \frac{GM}{Rr} = \frac{V_f^2}{r} $$

Resulting in a flat Velocity profile $V_f$ indeed:

$$ V_f = \sqrt{\frac{GM}{R}} $$

Remaining symbols have been defined in a previous posting and the references. But now we actually have two different theories for MOdified Newtonian Dynamics, namely

$$ F \approx m\frac{a^2}{a_0} \quad \mbox{(common)} \qquad \mbox{and} \qquad F \approx m\sqrt{a.a_0} \quad \mbox{(this thread)} $$

where $F=$ force, $m=$ mass and the acceleration $a_0$ on the left - in principle - is not the same as the $a_0$ on the right.

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RedshiftDriftRedshiftDrift, edited

@HanDeBruijn there are two versions of MOND, one as a modification of gravity (with $g_\dagger$), and one as a modification of inertia (with $a_0$). $a_0$ and $g_\dagger$ have the same value. Milgrom doesn’t know which one to use. There are no experimental result that tells us which one to choose.

Milgrom’s law can be interpreted in two ways:

One possibility is to treat it as a modification to Newton’s second law, so that the force on an object is not proportional to the particle’s acceleration a but rather to $\mu \left( \frac {a}{a_0} \right) a$. In this case, the modified dynamics would apply not only to gravitational phenomena, but also those generated by other forces, for example electromagnetism.

Alternatively, Milgrom’s law can be viewed as leaving Newton’s Second Law intact and instead modifying the inverse-square law of gravity, so that the true gravitational force on an object of mass m due to another of mass M is roughly of the form $G \frac{Mm}{\mu(a/a_0) r^2}$. In this interpretation, Milgrom’s modification would apply exclusively to gravitational phenomena.

In the latter case $a_0 \equiv g_\dagger$. These equations do not describe the same force:

$$ F_i \approx m_i \frac{a_i^2}{a_0} \quad \mbox{ and} \quad F_g \approx m_g \sqrt{a_g \ g_\dagger}. $$

with $i$ for Newtonian ‘inertia’, and $g$ for Newtonian ‘gravity’.

The interpolation function $\mu \left( \frac {a}{a_0} \right)$ is not described by MOND. Two examples of an interpolation function are given on the Wiki page. My understanding is that McGaugh chooses

$$ \mu \left( \frac {g_{bar}}{g_\dagger} \right) g_{bar} = \large g_{obs} = \frac{g_{bar}}{1-e^{-\sqrt{g_{bar}/g_\dagger}}} $$

because it better fits the central part of galaxy RAR measurements.

“Milgrom’s law is not a complete and self-contained physical theory, but rather an ad hoc empirically motivated variant of one of the several equations that constitute classical mechanics.” - wikipedia

When you write MOND = UAC, which version of MOND is it?

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HanDeBruijnHanDeBruijn, edited

When you write MOND = UAC, which version of MOND is it?

Honestly, I didn’t know that there is also a gravitational version and that the latter is employed in the paper. Thanks! Whatever. Once established that $a_0=GM/R^2$, I have shown that UAC explains for both the two versions of MOND. It’s a bit fake to say this, however, because (MOND = UAC) is a completely different (i.e. Variable Mass) theory.

Oh yes and thank you too for the ( \dagger $\to \dagger$ ).

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HanDeBruijnHanDeBruijn, edited

Two or three other quantities are related to Milgrom’s “constant” $a_0$. Calculation sequence:

Mpc := 3.08567758*10^22;
H := 74.03*1000/Mpc;
c := 299792458;
a_P := H*c;
a_0 := evalf(a_P/(2*Pi));
  • Lightspeed times Hubble parameter: $H.c\approx 7.2\times 10^{-10} \mbox{ }m/s^2$ . Which happens to be close to the Pioneer anomaly.
  • The Pioneer anomaly $a_P\approx H.c$ divided by $(2\pi)$ is (close to) Milgrom’s “constant” $a_0 \approx 1.15\times 10^{-10}\mbox{ }m/s^2$. As is calculated in the Rotation section of The Big Circus model by Sahil Gupta. @sahil5d . May I ask you where the miraculous factor $(2\pi)$ comes from?
  • Last but not least the Standard gravitational parameter in a galaxy divided by radial distance squared $(G.M/R^2)$ is related to $(H.c)$ , as can be seen in the picture below (and in replies above).

verlinde

A more precise account of the relationships, together with better physical underpinning, would be quite welcome !

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RedshiftDriftRedshiftDrift

I’m working on it!

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