Solution 6: Mass Ratio (Galaxy vs Sun)
1. Central Mass Formula:
From Kepler’s law: $M = \frac{4\pi^2 r_{orb}^3}{G T_{orb}^2}$.
From Kepler’s law: $M = \frac{4\pi^2 r_{orb}^3}{G T_{orb}^2}$.
2. Applying to Both Systems:
Galaxy Mass: $M_G = \frac{4\pi^2 R^3}{G T^2}$
Sun Mass: $M_S = \frac{4\pi^2 r^3}{G t^2}$
Galaxy Mass: $M_G = \frac{4\pi^2 R^3}{G T^2}$
Sun Mass: $M_S = \frac{4\pi^2 r^3}{G t^2}$
3. Ratio:
\[ \frac{M_G}{M_S} = \frac{R^3 / T^2}{r^3 / t^2} = \left(\frac{R}{r}\right)^3 \left(\frac{t}{T}\right)^2 \]
\[ \frac{M_G}{M_S} = \frac{R^3 / T^2}{r^3 / t^2} = \left(\frac{R}{r}\right)^3 \left(\frac{t}{T}\right)^2 \]
Answer: (a)