HW $\backslash$#$13$ Begin Date: $12/2/202412$:$01$:$00$ AM Due Date: $12/10/202411$:$59$:$00$ PM End Date: $12/21/202411$:$59$:$00$ PM

Problem $9$: $(10\backslash \%$ of Assignment Value$)$

Astronomical observations of our Milky Way galaxy indicate that it has a mass of about $\backslash (8\mathrm{.}0\backslash$cdot $10^\{11\}\backslash )$ solar masses. A star orbiting on the galaxy's periphery is about $\backslash (6\mathrm{.}0\backslash$cdot $\backslash )\backslash (10^\{4\}\backslash )$ light years from its center.

Part $($a$)$

What should the orbital period of that star be in years?

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$-$ Part $($b$)$

If its period is $\backslash (6\mathrm{.}0\backslash$cdot $10^\{7\}\backslash )$ years instead, what is the mass of the galaxy in solar masses? Such calculations are used to imply the existence of "dark matter" in the universe and have indicated, for example, the existence of very massive black holes at the centers of some galaxies.