Orbits of Particle Around a Black Hole using Effective Potential

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SUMMARY

The discussion focuses on the analysis of particle orbits around a black hole using the effective potential UGR(r), defined as UGR(r) = -GM/r + l²/2m²r² - Rsl²/2m²r³, where Rs = 2GM/c² is the Schwarzschild radius. Participants are tasked with demonstrating that UGR has minimum and maximum values at specific radii (r- and r+) and describing the characteristics of possible orbits. The energy balance equation K = 1/2 mr² + mUGR = constant is also central to understanding the dynamics of these orbits.

PREREQUISITES
  • Understanding of general relativity concepts, particularly black hole physics.
  • Familiarity with effective potential energy in classical mechanics.
  • Knowledge of calculus for analyzing functions and finding extrema.
  • Basic understanding of orbital mechanics and energy conservation principles.
NEXT STEPS
  • Explore the derivation of the Schwarzschild radius and its implications for black hole physics.
  • Learn how to plot effective potential functions and identify turning points in dynamics.
  • Study the characteristics of stable and unstable orbits in gravitational fields.
  • Investigate numerical methods for simulating particle motion around black holes.
USEFUL FOR

Astrophysicists, students studying general relativity, and anyone interested in the dynamics of particles in strong gravitational fields.

omegas
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Homework Statement


The possible orbit of a particle moving around a black hole can be described using the effective potential UGR(r) (in effect, potential energy per unit mass):

UGR(r) = -GM/r + l2/2m2r2 - Rsl2/2m2r3

where the symbols have their usual meaning and in particular Rs = 2GM / c2 is the Schwarzschild radius for the black hole. We then use the energy balance equation

K = 1/2 mr2 + mUGR = constant

to illustrate the main features of the possible orbits.

(a) Show that the function UGR has a minimum and maximum value at r = r- and r = r+ respectively and determine expressions for these qualities.
(b) Briefly describe the possible orbits.

Homework Equations



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The Attempt at a Solution



Please help.
 
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Begin by plotting the effective potential with respect to r and look for turning points.
 

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