Surface X has a temperature T_X and emissivity ε_x. Surface Y has a temperature T_Y and emissivity ε_y. The two surfaces emit radiation at the same rate.

What is the ratio $\frac{{{T_{\rm{X}}}}}{{{T_{\rm{Y}}}}}$?

A. ${\left( {\frac{{{\varepsilon _{\rm{y}}}}}{{{\varepsilon _{\rm{x}}}}}} \right)^{\frac{1}{4}}}$

B. ${\left( {\frac{{{\varepsilon _{\rm{x}}}}}{{{\varepsilon _{\rm{y}}}}}} \right)^{\frac{1}{4}}}$

C. ${\left( {\frac{{{\varepsilon _{\rm{y}}}}}{{{\varepsilon _{\rm{x}}}}}} \right)^4}$

D. ${\left( {\frac{{{\varepsilon _{\rm{x}}}}}{{{\varepsilon _{\rm{y}}}}}} \right)^4}$