狗來福 🐶 大自然 🌲: MOND

https://en.wikipedia.org/wiki/Modified_Newtonian_dynamics

F_{\text{N}}=m\,\mu \left({\frac {a}{a_{0}}}\right)\,a~.

(1)

Here $F$ _N is the Newtonian force, $m$ is the object's (gravitational) mass, $a$ is its acceleration, $μ$ ( $x$ ) is an as-yet unspecified function (called the interpolating function), and $a$ ₀ is a new fundamental constant which marks the transition between the Newtonian and deep-MOND regimes. Agreement with Newtonian mechanics requires

{\begin{aligned}\mu (x)\longrightarrow 1&&{\text{ for }}x\gg 1\end{aligned}}~,

and consistency with astronomical observations requires

{\begin{aligned}\mu (x)\longrightarrow x&&{\text{ for }}x\ll 1\end{aligned}}~.

Beyond these limits, the interpolating function is not specified by the hypothesis, although it is possible to weakly constrain it empirically.^[14]^[15] Two common choices are the "simple interpolating function":

\mu \left({\frac {a}{a_{0}}}\right)={\frac {1}{1+{\frac {a_{0}}{a}}}}~,

and the "standard interpolating function":

\mu \left({\frac {a}{a_{0}}}\right)={\sqrt {{\frac {1}{1+\left({\frac {a_{0}}{a}}\right)^{2}}}~}}~.

Thus, in the deep-MOND regime ( $a$ ≪ $a$ ₀):

F_{\text{N}}=m{\frac {\,a^{2}\,}{\,a_{0}\,}}~.

Applying this to a star or other object of mass $m$ in circular orbit around mass $M$ (the total baryonic mass of the galaxy), produces

{\frac {GMm}{r^{2}}}=m{\frac {\left({\frac {v^{2}}{r}}\right)^{2}}{a_{0}}}\quad \Longrightarrow \quad v^{4}=GMa_{0}~

(2)

By fitting his law to rotation curve data, Milgrom found a₀ ≈ 1.2 × 10⁻¹⁰ m/s² to be optimal.

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 ${\textstyle \mu \left({\frac {a}{a_{0}}}\right)a.}$ In this case, the modified dynamics would apply not only to gravitational phenomena, but also those generated by other forces, for example electromagnetism.^[16]
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

{\textstyle {\frac {GMm}{\mu \left({\frac {a}{a_{0}}}\right)r^{2}}}.}

In this interpretation, Milgrom's modification would apply exclusively to gravitational phenomena.

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