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In the standard deep MOND regime,

$$g= \sqrt{(GMa_{0}/r^2)}.$$

Why is the deep MOND regime not simply the sum of Newtonian acceleration and the acceleration scale constant?

$$g= GM/r^2 + a_{0}.$$

Galactic rotation curves yield very similar values to the standard deep MOND formulation. Does it violate the Tully-Fisher law?

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  • $\begingroup$ Is this of any help? physics.stackexchange.com/a/605772/226902 $\endgroup$
    – Quillo
    Commented Sep 16 at 22:42
  • $\begingroup$ No, it is just the standard deep MOND regime explained. $\endgroup$ Commented Sep 17 at 19:42
  • $\begingroup$ Why would you expect it to follow the form you proposed? The point of MOND was to develop a formula that would fit galaxy rotation curves. Since that curve fitting exercise worked, if you modify that formula in a "big" way (not just adding a small correction to the fit), then your modified formula won't fit the data. $\endgroup$
    – Andrew
    Commented Sep 20 at 1:29
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    $\begingroup$ The OP may want to include "Bullet Cluster" in other search terminology, as it seems to have posed problems for MOND. $\endgroup$
    – Edouard
    Commented Sep 20 at 4:24
  • $\begingroup$ @Andrew I think the sum of Newtonian acceleration and the acceleration scale constant also fits galaxy rotation curves. It is not a "big" modification with respect to deep MOND.. $\endgroup$ Commented Sep 20 at 13:15

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Why the Deep MOND Regime is Not Simply Newtonian Acceleration Plus a Constant

The deep Modified Newtonian Dynamics (MOND) regime fundamentally differs from simply adding a constant acceleration to Newtonian dynamics because MOND aims to address the discrepancies in galactic rotation curves and gravitational dynamics without invoking dark matter. In the deep MOND regime, the characteristic behavior arises at extremely low accelerations, below a critical threshold $$a_0 \approx 1.2 \times 10^{-10} \, \text{m/s}^2.$$ Unlike the Newtonian model, which predicts gravitational acceleration $$a = \frac{GM}{r^2},$$ the MOND framework introduces a modification such that at accelerations $$a \ll a_0,$$ the effective acceleration $a$ scales as $$a = \sqrt{a_0 a_N},$$ where $a_N$ is the Newtonian acceleration and $a_0$ is the critical acceleration parameter introduced by MOND (Milgrom, 1983). This implies that the relationship between acceleration and distance is not linear and does not simply involve adding a constant term to the Newtonian force.

If the deep MOND regime were simply Newtonian acceleration plus a constant, one would expect the force law to follow $$a_\mathrm{eff} = a_N + c,$$ where $c$ is a constant acceleration. This would not lead to the observed asymptotic flatness of rotation curves in galaxies, where the orbital velocity $v$ becomes constant at large radii. In contrast, MOND predicts that at large distances, where $$a \ll a_0,$$ the acceleration $$a \approx \sqrt{GMa_0}/r,$$ which yields a constant rotational velocity $$v^4 = GMa_0.$$ This behavior matches observations, unlike the unrealistic scenario that would arise from merely adding a constant acceleration: the resulting dynamics would suggest a linear increase in force at large distances, inconsistent with the gravitational behavior in galactic outskirts (Sanders & McGaugh, 2002).

Therefore, the deep MOND regime embodies a fundamentally different modification of gravity where the acceleration transitions smoothly from the Newtonian $a_N$ to the MONDian $\sqrt{a_0 a_N}$. This transition respects the symmetries and invariances that a simple additive constant would violate. Additionally, MOND's behavior is derived from an action principle and is consistent with scale invariance at low accelerations, a property absent in a Newtonian-plus-constant framework (Famaey & McGaugh, 2012). Thus, the deep MOND regime’s acceleration cannot be reduced to Newtonian acceleration plus a constant but is a unique, non-linear theory that correctly captures galactic-scale gravitational phenomena.

Kindly find below my citations-

# Paper Key Insight Citations
1 A Modification of the Newtonian Dynamics as a Possible Alternative to the Hidden Mass Hypothesis (Milgrom, 1983) Introduces the fundamental MOND equation, establishing the acceleration relation (a = \sqrt{a_0 a_N}), explaining why a simple additive constant is insufficient. 1
2 Modified Newtonian Dynamics as an Alternative to Dark Matter (Sanders & McGaugh, 2002) Demonstrates how MOND successfully explains the flat rotation curves of galaxies, a phenomenon inconsistent with adding a constant acceleration. 1
3 Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions (Famaey & McGaugh, 2012) Discusses the theoretical underpinnings of MOND, emphasizing its consistency with scale invariance at low accelerations. 1
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  • $\begingroup$ Thank you for your answer. But there is evidence that for MOND to correctly explain galaxy cluster dynamics, should increase its force at large distances. Moreover, the "Newtonian acceleration plus a constant of acceleration" yields more or less the same results in galatic rotation curves (not exactly flat, but similar values). $\endgroup$ Commented Sep 21 at 19:29

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