In recent years, scientists have discovered hundreds of planets orbiting other stars. Some of these planets are in orbits that are similar to that of earth, which orbits the sun $\left(M_{\text {sun }}=1.99 \times 10^{30} \mathrm{kg}\right)$ at a distance of $1.50 \times 10^{11} \mathrm{m},$ called 1 astronomical unit $(1 \mathrm{au}) .$ Others have extreme orbits that are much different from anything in our solar system. Relate to some of these planets that follow circular orbits around other stars. HD $10180 \mathrm{g}$ orbits with a period of 600 days at a distance of 1.4 au from its star. What is the ratio of the star's mass to our sun's mass?
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Step 1: Use the formula for centripetal force in circular motion, where the gravitational force is equal to the centripetal force: \[ M_s \cdot R \cdot \omega^2 = \frac{G \cdot M_s \cdot M_p}{R^2} \] Show more…
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In recent years, scientists have discovered hundreds of planets orbiting other stars. Some of these planets are in orbits that are similar to that of earth, which orbits the sun $\left(M_{\text {sun }}=1.99 \times 10^{30} \mathrm{kg}\right)$ at a distance of $1.50 \times 10^{11} \mathrm{m},$ called 1 astronomical unit $(1$ au $) .$ Others have extreme orbits that are much different from anything in our solar system. Problems $51-53$ relate to some of these planets that follow circular orbits around other stars. HD $10180 \mathrm{g}$ orbits with a period of 600 days at a distance of 1.4 au from its star. What is the ratio of the star's mass to our sun's mass?
In recent years, scientists have discovered hundreds of planets orbiting other stars. Some of these planets are in orbits that are similar to that of earth, which orbits the sun $\left(M_{\text {sun }}=1.99 \times 10^{30} \mathrm{kg}\right)$ at a distance of $1.50 \times 10^{11} \mathrm{m},$ called 1 astronomical unit $(1 \mathrm{au}) .$ Others have extreme orbits that are much different from anything in our solar system. Relate to some of these planets that follow circular orbits around other stars. Kepler-42c orbits at a very close 0.0058 au from a small star with a mass that is 0.13 that of the sun. How long is a "year" on this world?
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 $\mathrm{CM}$ is stationary. Assume $m_{\mathrm{A}}$ and $m_{\mathrm{B}}$ are in circular orbits with radii $r_{\mathrm{A}}$ and $r_{\mathrm{B}}$ about the system's $\mathrm{CM} .(a)$ Show that $r_{\mathrm{A}}=\frac{m_{\mathrm{B}}}{m_{\mathrm{A}}} r_{\mathrm{B}}$ (b) Now consider a Sun-like star and a single planet with the same characteristics as Jupiter. That is, $m_{B}=1.0 \times 10^{-3} m_{A}$ and the planet has an orbital radius of $8.0 \times 10^{11} \mathrm{m} .$ 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 2$r_{A} .$ If astronomers are able to detect angular displacements $\theta$ of about 1 milliarcsec $\left(1$ arcsec $=\frac{1}{3600}$ of a degree), from what \right. distance $d$ (in light-years) can the star's wobble be detected $\left(1 \mathrm{ly}=9.46 \times 10^{15} \mathrm{m}\right) ?(d)$ The star nearest to our Sun is about 4 $\mathrm{ly}$ 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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