Flat Rotation Curve Astronomy Question

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SUMMARY

The discussion focuses on deriving the orbital period, P, for objects in a spiral galaxy with a flat rotation curve beyond 2 kpc from the galactic center. The flat rotation curve indicates that stars maintain a constant orbital speed of 250 km/s regardless of distance from the center. For a star at 8 kpc, the orbital period is calculated to be approximately 100 million years, while for a hydrogen cloud at 30 kpc, it is about 120 million years. This analysis highlights the relationship between orbital radius and period in the context of galactic dynamics.

PREREQUISITES
  • Understanding of orbital mechanics and Kepler's laws
  • Familiarity with the concept of rotation curves in astronomy
  • Basic knowledge of spiral galaxy structure
  • Ability to perform calculations involving speed, distance, and time
NEXT STEPS
  • Study the derivation of orbital periods in gravitational systems
  • Learn about the implications of flat rotation curves in dark matter research
  • Explore the dynamics of spiral galaxies and their structure
  • Investigate the methods used to measure stellar velocities in galaxies
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Astronomy students, astrophysicists, and anyone interested in the dynamics of galaxies and the implications of rotation curves on dark matter theories.

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You have to show your attempts at answering the question, this is a forum policy. Also, please use the template. The only reason your post is not deleted is that you already got some help.
I've been struggling with part (a) of this question. I'm not quite sure what the relation is. Any help is greatly appreciated! Thanks!

Consider a spiral galaxy with a “flat” rotation curve beyond the central 2 kpc.

a. Derive the general relation giving the orbital period, P, of a star (or other object) in the disc orbiting the centre of the galaxy, as a function of the radius, R (>2kpc), from the galactic centre.

b. If stars in the flat part of the rotation curve are orbiting at 250km/s, calculate the orbital period, P in years, for a star at 8kpc from the galactic centre and a hydrogen cloud at 30kpc.
 
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Well, what does the 'flat' rotation curve mean? How fast are the stars moving? Given how fast they are moving, how long will it take them to complete an orbit of radius R?
 

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