Solving Extrasolar Planet Orbital Radius & Angular Displacement Problems

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SUMMARY

This discussion focuses on the detection of extrasolar planets through the analysis of their gravitational effects on their host stars. It establishes the relationship between the orbital radii of the star and planet, specifically showing that ra = (mb/ma)rb. Using a Sun-like star and a Jupiter-like planet, the orbital radius ra is calculated to be 8.0 x 10^11 meters. The discussion also addresses the detection of angular displacements of 1 milliarcsecond, leading to the conclusion that stars wobbling at this scale can be detected from a distance of approximately 1,000 light years, allowing astronomers to estimate the number of stars in the Milky Way suitable for this detection method.

PREREQUISITES
  • Understanding of gravitational dynamics in two-body systems
  • Familiarity with circular orbital motion and center of mass calculations
  • Knowledge of angular measurement units, specifically arcseconds and milliarcseconds
  • Basic principles of astronomy related to extrasolar planet detection
NEXT STEPS
  • Study gravitational dynamics in two-body systems using Newton's laws
  • Learn about the methods of detecting extrasolar planets, including the radial velocity method
  • Explore the concept of angular displacement and its applications in astronomy
  • Research the distribution of stars in the Milky Way Galaxy and implications for extrasolar planet searches
USEFUL FOR

Astronomers, astrophysicists, and students studying celestial mechanics or extrasolar planet detection techniques will benefit from this discussion.

Fahad Jan
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An extrasolar planet can be detected by observing the wobble it produces on the star around which it revolves. Suppose an extrasolar planet of mass mb revolves around its star of mass ma . If no external force acts on this simple two-object system, then its CM is stationary. Assume ma and mb are in circular orbits with radii ra and rb about the system's CM.

(a) Show that ra =(mb/ma)rb

(b) Now consider a Sun-like star and a single planet with the same characteristics as Jupiter. That is, mb = (1.0 x 10-3)ma and the planet has an orbital radius of 8.0 x 1011 meters. Determine the radius ra of the star's orbit about the system's CM.

(c) When viewed from Earth, the distant system appears to wobble over a distance of 2ra . If astronomers are able to detect angular displacements ? of about 1 milliarcsec (1 arcsec = 1/3600 of a degree), from what distance d (in light years) can the star's wobble be detected (1 ly = 9.46 x 1015 m)?

(d) That star nearest to our Sun is about 4 ly away. Assuming stars are uniformly distributed throughout our region of the Milky Way Galaxy, about how many stars can this technique be applied to in the search for extrasolar planetary systems?
 
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