Chapter 9, Problem 112GP

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 m_B revolves around its star of mass m_A. If no external force acts on this simple two-object system, then its cm is stationary. Assume m_A and m_B are in circular orbits with radii r_A and r_B about the system’s CM. (a) Show that

$r_{\text{A}}=\frac{m_{\text{B}}}{m_{\text{A}}}{r}_{\text{B}}.$

(b) Now consider a Sun-like star and a single planet with the same characteristics as Jupiter. That is, m_B = 1.0 × 10^–3m_a and the planet has an orbital radius of 8.0 × 10¹¹ m. Determine the radius r_A 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 2r_A. If astronomers are able to detect angular displacements q of about 1 milliarcsec $\left(1\text{ arcsec}=\frac{1}{3600}\text{ of a degree}\right)$, from what distance d (in light-years) can the star’s wobble be detected (l ly = 9.46 × 10¹⁵m)? (d) The 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?