1. Levitation in Stars and Dusty Winds. We have seen that photons carry not only energy but also momentum, with each photon of wavelength carrying momentum p, = E,/c = h/1. This means that photon interactions can transfer momentum, and in turn that a star's electromagnetic flux is capable of exerting a force; we investigate that force here. (a) [5 points) Consider a flux F of photons all moving in the same direction. If the light lands perpendicular (i.e., directly) on a surface of area A, find an expression for the rate of energy flow onto the surface. From this find an expression for the rate of momentum deposit onto the surface. (b) [5 points] Use Newton's laws to trivially explain why the momentum flow rate you found represents the radiation force on the surface. Then go on to show that the radiation pressure on the surface the force per unit area - is P = F/c. (c) (5 points) You hold you hand a distance of 0.1 meters from a 100 Watt lightbulb. Estimate the size of your palm and calculate the radiation force on it in Newtons. What mass object would have a weight equal to this force? Comment on the strength the radiation force in everyday circumstances. (d) [5 points) Now consider a particle in or above the atmosphere of a star. The star has mass M and luminosity L. The particle absorbs or scatters photons from the star, and thus receives momentum from the starlight. i. Imagine the particle acts as if it cast a shadow of area o: this of course is the cross section for photons interact with the particle. Find an expression for the upward force on the particle due to the radiation pressure it absorbs, if the particle's distance from the star is r. ii. Trivially find an expression gravitational force, or weight, of the particle, also at distance r. iii. Finally, show that the ratio of these two forces is independent of r. Briefly explain why. [5 points) Show that the particle will be pushed away from the star (or "levitate") if the star's luminosity is larger than a critical value 47GMmc Lcrit = (1) o [5 points] Consider a typical asymptotic giant branch star of mass M = 2.5M. and luminosity L = 5000L. We will model the atmosphere of the star as 99% hydrogen by mass, and the remaining 1% as microscopic solid particles: dust grains. Consider a spherical grain of size a and density Pgr. Let the grain's cross section for absorbing photons be o = naFind an expression for the ratio of radiation to gravitational force on the grain. For a = 0.1um = 10-5 cm and per = 3 g/cm, show that the grain is strongly levitated, i.e., that the upward radiation force on the grain greatly exceeds its weight. For a wind to be launched from radiation pressure, the grain must carry with it the surrounding gas, which represents 100 times more mass. Using this, show that a wind will indeed be launched. This leads to strong mass loss in AGB stars and the formation of planetary nebulae. 1. Levitation in Stars and Dusty Winds. We have seen that photons carry not only energy but also momentum, with each photon of wavelength carrying momentum p, = E,/c = h/1. This means that photon interactions can transfer momentum, and in turn that a star's electromagnetic flux is capable of exerting a force; we investigate that force here. (a) [5 points) Consider a flux F of photons all moving in the same direction. If the light lands perpendicular (i.e., directly) on a surface of area A, find an expression for the rate of energy flow onto the surface. From this find an expression for the rate of momentum deposit onto the surface. (b) [5 points] Use Newton's laws to trivially explain why the momentum flow rate you found represents the radiation force on the surface. Then go on to show that the radiation pressure on the surface the force per unit area - is P = F/c. (c) (5 points) You hold you hand a distance of 0.1 meters from a 100 Watt lightbulb. Estimate the size of your palm and calculate the radiation force on it in Newtons. What mass object would have a weight equal to this force? Comment on the strength the radiation force in everyday circumstances. (d) [5 points) Now consider a particle in or above the atmosphere of a star. The star has mass M and luminosity L. The particle absorbs or scatters photons from the star, and thus receives momentum from the starlight. i. Imagine the particle acts as if it cast a shadow of area o: this of course is the cross section for photons interact with the particle. Find an expression for the upward force on the particle due to the radiation pressure it absorbs, if the particle's distance from the star is r. ii. Trivially find an expression gravitational force, or weight, of the particle, also at distance r. iii. Finally, show that the ratio of these two forces is independent of r. Briefly explain why. [5 points) Show that the particle will be pushed away from the star (or "levitate") if the star's luminosity is larger than a critical value 47GMmc Lcrit = (1) o [5 points] Consider a typical asymptotic giant branch star of mass M = 2.5M. and luminosity L = 5000L. We will model the atmosphere of the star as 99% hydrogen by mass, and the remaining 1% as microscopic solid particles: dust grains. Consider a spherical grain of size a and density Pgr. Let the grain's cross section for absorbing photons be o = naFind an expression for the ratio of radiation to gravitational force on the grain. For a = 0.1um = 10-5 cm and per = 3 g/cm, show that the grain is strongly levitated, i.e., that the upward radiation force on the grain greatly exceeds its weight. For a wind to be launched from radiation pressure, the grain must carry with it the surrounding gas, which represents 100 times more mass. Using this, show that a wind will indeed be launched. This leads to strong mass loss in AGB stars and the formation of planetary nebulae