Question

problem 7: (8% of assignment value)
because dark matter seems far - fetched to some, suppose you read an article
where it was proposed that the milky ways rotation curve could be explained
by assuming that the milky ways supermassive black hole was much more
massive.
does this explanation adequately explain why stars in the milky way orbit faster than expected? choose the correct answer with the best explanation.
yes, a more massive black hole would distort space throughout the milky way more, causing more distant stars to orbit faster as
they fall into the black hole.
no, increasing the mass of the black hole would distort space in the galactic center more but would not affect stars near the sun
yes, a more massive black hole would exert stronger gravitational force on milky way stars, causing the more distant stars to
orbit faster in a uniform way.
no, the supermassive black hole does not gravitationally influence stars in our galaxy.
no, the gravitational force still drops with the distance squared, so increasing the mass of the black hole would still lead to a drop
off in orbital velocities with distance

Explanation:

The gravitational force \(F = \frac{GMm}{r^{2}}\) (where \(G\) is the gravitational constant, \(M\) is the mass of the black - hole, \(m\) is the mass of the star, and \(r\) is the distance between them). The orbital velocity \(v=\sqrt{\frac{GM}{r}}\) (derived from equating centripetal force \(F_{c}=\frac{mv^{2}}{r}\) to gravitational force \(F = \frac{GMm}{r^{2}}\)). If we only increase \(M\) (mass of the black - hole), the orbital velocity \(v\) still depends on \(r\) as \(v\propto\frac{1}{\sqrt{r}}\). The supermassive black - hole is at the center of the galaxy. For stars near the Sun (at a large \(r\) from the black - hole), the gravitational influence of the black - hole is negligible compared to the overall mass distribution of the galaxy. The first option about stars falling into the black - hole is incorrect as stars in stable orbits do not fall into the black - hole. The third option about a uniform increase in orbital velocity is wrong because of the \(v\propto\frac{1}{\sqrt{r}}\) relation. The fourth option is wrong as the supermassive black - hole does have some gravitational influence (but not enough to explain the flat rotation curve).

Answer:

No, the gravitational force still drops with the distance squared, so increasing the mass of the black hole would still lead to a drop off in orbital velocities with distance.