= Monotonicity of blackbody planet-star contrast
{title2=$dr_\lambda/d\lambda>0\quad(T_p<T_s)$}
For <blackbodies> with $T_p<T_s$, the contrast $r_\lambda=(R_p/R_s)^2(e^{x_s}-1)/(e^{x_p}-1)$ increases with wavelength, where $x_i=hc/(\lambda k_BT_i)$. Indeed $d\log r/d\log\lambda=G(x_p)-G(x_s)>0$, with $G(x)=x/(1-e^{-x})$ increasing for $x>0$ because $1-(1+x)e^{-x}>0$. The contrast approaches $(R_p/R_s)^2T_p/T_s$ in the <Rayleigh-Jeans law> limit. This behavior differs from the peaked absolute <blackbody> flux.
Back to article page
