1. The energy $E$ of a photon is related to its momentum $p$ by \[ E=p c \] where $c$ is the speed of light. (a) Near to a star of mass $M$ that radiates isotropically with luminosity $L$ (note: luminosity means radiation power), there is a small spherical object with mass $m$ (where $m \ll M$ ) and radius $a$. Photons from the star are absorbed upon hitting the object. Prove that the object will be pushed away by the star's radiation pressure if \[ \frac{L}{M}>4 G c \frac{m}{a^{2}} \] (b) Suppose that the condition in part (a) is satisfied for a particular object. Further assume that the object is at rest at an initial distance $R$ from the star. What is the terminal velocity that the object can reach? (c) To get some feelings of this effect of radiation pressure around a celestial body, we consider how the radiation pushes nearby ionized hydrogen (with mass $1.67 \times$ $10^{-27} \mathrm{~kg}$ ). For simplicity, we use the Thomson cross section $\sigma=6.65 \times 10^{-29} \mathrm{~m}^{2}$. What is the maximum luminosity of a celestial body of one solar mass $M=1.99 \times$ $10^{30} \mathrm{~kg}$, such that the celestial body does not eject hydrogen by radiation pressure?

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