FrontierScience / dbe975d5-c431-4d73-8f7c-f8bba77ea458 / For the sake of convenience in calculation and estimation, consider a…

For the sake of convenience in calculation and estimation, consider a "cylindrical" celestial body (although this assumption is not realistic, it allows for a simple order-of-magnitude estimate). The radius of this celestial body is `R`, and its height is also `R`, with its rotational axis perpendicular to the plane of its orbit around the Sun. This body rotates around its axis with an angular speed `ω`. The period of revolution around the star is much larger than the period of rotation and its effects can be ignored.\
Assume this celestial body is composed of uniform material with density `ρ`, specific heat capacity `C`, and thermal conductivity `K`. For a specific point on its lateral surface, the perpendicular component of incident radiation (from the Sun and other background radiation) at time `t` can be approximated as `ε(t)=ε₀+ε₁cosωt`.  `ε`  and `ε₁` are known. Also `ε₁>0 .` \
We can imagine this celestial body as having only a thin surface layer where temperatures oscillate with time, while temperatures remain nearly constant at depths beyond a certain distance from the surface.

The average temperature of this celestial body is `T₀`, its emissivity is `ϵ`, and its albedo is `A.` The temperature on the lateral surface at a given location oscillates with time (due to the differential heating) with an amplitude of `T₁` ( `T₁>0`) and lags behind the incident flux by a temporal phase difference of `φ`

Consider the body to be a Lambertian source.

Considering the radiation flux emitted by the lateral surface of a cylinder, calculate the magnitude of the force `\(\vec{F}^\mathrm{}_Y\)` acting on the body due to thermal radiation. (The force due to the radiation from the top and bottom surfaces cancel anyway) acting on this object due to the emitted radiation. You may use the approximation that `T₁≪T₀`.

Answer in terms of `ϵ` , the Stefan-Boltzmann constant `σ` , `T₀,T₁,R` and speed of light `c` .

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.

For the sake of convenience in calculation and estimation, consider a "cylindrical" celestial body (although this assumption is not realistic, it allows for a simple order-of-magnitude estimate). The radius of this celestial body is `\(R\)`, and its height is also `\(R\)`, with its rotational axis perpendicular to the plane of its orbit around the Sun. This body rotates around its axis with an angular speed `\( \omega \)`. The period of revolution around the star is much larger than the period of rotation and its effects can be ignored.\
Assume this celestial body is composed of uniform material with density `\(\rho\)`, specific heat capacity `\(C\)`, and thermal conductivity `\(K\)`. For a specific point on its lateral surface, the perpendicular component of incident radiation (from the Sun and other background radiation) at time `\(t\)` can be approximated as `\(\varepsilon (t)=\varepsilon _0+\varepsilon _1\cos \omega t\)`.  `\( \varepsilon \)`  and `\(\varepsilon _1\)` are known. Also `\( \varepsilon_1 > 0  \) .` \
We can imagine this celestial body as having only a thin surface layer where temperatures oscillate with time, while temperatures remain nearly constant at depths beyond a certain distance from the surface.

The average temperature of this celestial body is `\(T_0\)`, its emissivity is `\(\epsilon\)`, and its albedo is `\(A\).` The temperature on the lateral surface at a given location oscillates with time (due to the differential heating) with an amplitude of `\( T_1 \)` ( `\( T_1 >0 \)`) and lags behind the incident flux by a temporal phase difference of `\( \varphi \)`

Consider the body to be a Lambertian source.

Considering the radiation flux emitted by the lateral surface of a cylinder, calculate the magnitude of the force `\(\vec{F}^\mathrm{}_Y\)` acting on the body due to thermal radiation. (The force due to the radiation from the top and bottom surfaces cancel anyway) acting on this object due to the emitted radiation. You may use the approximation that `\( T_1 \ll T_0 \)`.

Answer in terms of `\(\epsilon\)` , the Stefan-Boltzmann constant `\( \sigma \)` , `\( T_0, T_1 , R \)` and speed of light `\( c \)` .

Think step by step and solve the problem below. At the end of your response, write your final answer on a new line starting with “FINAL ANSWER”. It should be an answer to the question such as providing a number, mathematical expression, formula, or entity name, without any extra commentary or providing multiple answer attempts.