Dark Matter — Question 10

Question 10

A modified-acceleration law can mimic a flat curve but requires wider tests

In a hypothetical low-acceleration model $g=\sqrt{g_Na_0}$, with $g_N=GM_b/r^2$ and $a_0=1.2\times10^{-10}$ m/s$^2$. Let baryonic mass be $10^{41}$ kg and G=$6.67\times10^{-11}$ SI.

1. Derive the asymptotic circular speed and evaluate it.

2. Find the speed ratio if baryonic mass rises by a factor 16, and state why one rotation curve cannot uniquely establish this law.

Question 10 - Solution

A modified-acceleration law can mimic a flat curve but requires wider tests

(a) Method and calculation

$$v^2/r=\sqrt{GM_ba_0}/r\Rightarrow\boxed{v^4=GM_ba_0}.$$$$\boxed{v=(6.67\times10^{-11}\times10^{41}\times1.2\times10^{-10})^{1/4}\simeq168\ {\mathrm{km/s}}}.$$

(b) Interpretation and further calculation

$$\boxed{v^\prime/v=16^{1/4}=2}.$$$$\text{Rotation alone constrains }g(r);\quad\boxed{\text{lensing, clusters and cosmology supply independent tests}}.$$

Physical check

This supplied phenomenological law is compared as a model, not adopted as an established replacement for dark matter.