Solving Extrasolar Planet Orbital Radius & Angular Displacement Problems

In summary, observing the wobble of a star caused by an extrasolar planet can help detect the planet's presence. The center of mass of a system with a star of mass ma and planet of mass mb remains stationary if there is no external force acting on it. The orbital radius of the planet is related to that of the star by ra = (mb/ma)rb. For a Sun-like star and a Jupiter-like planet, with mb = (1.0 x 10-3)ma and an orbital radius of 8.0 x 1011 meters, the radius ra of the star's orbit about the system's CM can be determined. The star's wobble, when viewed from Earth, can be detected at
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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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Related to Solving Extrasolar Planet Orbital Radius & Angular Displacement Problems

1. What is an extrasolar planet?

An extrasolar planet, also known as an exoplanet, is a planet that orbits a star other than our sun. These planets are located outside of our solar system and are typically discovered through various observation methods, such as radial velocity or transit techniques.

2. How is the orbital radius of an extrasolar planet calculated?

The orbital radius of an extrasolar planet can be calculated using Kepler's third law, which states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. This can also be calculated using the planet's distance from its host star and the star's mass.

3. What is angular displacement in relation to extrasolar planets?

Angular displacement refers to the change in position of an object from a reference point, measured in degrees or radians. In the context of extrasolar planets, angular displacement is used to calculate the change in position of a planet as it orbits its host star.

4. What factors can affect the orbital radius of an extrasolar planet?

The orbital radius of an extrasolar planet can be affected by several factors, including the mass and size of the planet, the mass and luminosity of its host star, and any gravitational interactions with other nearby objects. Additionally, the presence of other planets in the same system can also impact the orbital radius of a planet.

5. How are extrasolar planet orbital radius and angular displacement problems solved?

Solving extrasolar planet orbital radius and angular displacement problems involves using mathematical equations and data from observations to determine the various parameters of a planet's orbit. This can include calculating the planet's semi-major axis, orbital period, and angular displacement at different points in its orbit. Advanced computer simulations and modeling techniques are also used to aid in solving these problems.

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