A Summing Up — Question 9

Question 9

Changing the gravitational law and adding mass are degenerate in one orbit

A circular orbit has measured speed v at radius r. Model A uses Newtonian G and inferred total mass $M_A=v^2r/G$. Model B has only baryonic mass $M_b=.40M_A$ but uses an effective constant $G_{\mathrm{eff}}$.

1. Find the value of Geff needed to reproduce that orbit.

2. If the same Geff applies to a second radius where measured baryonic mass fraction is .70, find the predicted-to-observed squared-speed ratio and speed ratio.

Question 9 - Solution

Changing the gravitational law and adding mass are degenerate in one orbit

(a) Method and calculation

$$v^2r=GM_A=G_{\mathrm{eff}}M_b.$$$$\boxed{G_{\mathrm{eff}}/G=M_A/M_b=1/.40=2.5}.$$

(b) Interpretation and further calculation

$$\boxed{v_{\mathrm{pred}}^2/v_{\mathrm{obs}}^2=(G_{\mathrm{eff}}/G)(M_{b,2}/M_{A,2})=2.5(.70)=1.75}.$$$$\boxed{v_{\mathrm{pred}}/v_{\mathrm{obs}}=\sqrt{1.75}=1.32288}.$$

Physical check

A parameter adjustment fitting one radius need not fit another. This tests a constant rescaling model, not every possible modified-gravity theory.