A scientist decides to float a small vantablack $($i$.$e$.,$ totally absorbing$)$ sphere at a distance R immediately above an isotropic point source of light, so that its radiation pressure will exert an $10$ Physics $122($P$)$ Final Exam Fall $2023$ upward radiation force to match the downward gravitational force on the sphere. The sphere has density and radius r $$ R$.[$Hint: the area of the sphere$$s shadow is A $=$ r $2]($a$)(4$ pts$)$ What$$s the luminosity L required of the light source? $($b$)(2$ pts$)$ Even if such a source were made, why would the support of the sphere be unstable? $($c$)(2$ pts$)$ If the scientist were to use a reflective coating of aluminum $($i$.$e$.,$ a totally reflecting surface$)$ instead of vantablack, what would be the required luminosity?