
\magnification=\magstep1


$$
{\rm flux\ from\ star\ \ F_{*} } \quad = \quad { { L_{*} } \over {4 \pi \thinspace r_{\rm orb}^2 } }
$$

$$
{\rm Lum\ of\ planet\ \ L_{p} } \quad = \quad  F_{*} \thinspace 
             \alpha \thinspace {\pi R_{\rm P}^2 }
$$

$$
{ { L_{p} \over {L_{*}}  } } \quad = \quad  { {\alpha} \over {4} }
                { \left( {R_{\rm P} } \over { r_{\rm orb} } \right) }^2
$$


$$
{\rm S/N} \quad \sim \quad { { \rm photons\ from\ star } 
          \over 
       { \sqrt{  \rm (photons\ from\ star) + (photons\ from\ sky) } } }
$$


$$
I_{*}(\theta) \quad = \quad {\rm number\ of\ photons\ from\ star\ at\ angular\ distance\ \theta}
$$

$$
\eqalign{
\sigma_{*}(\theta) \quad &= \quad {\rm noise\ of\ photons\ from\ star\ at\ angular\ distance\ \theta} \cr
                         &= \quad \sqrt{I_{*}(\theta) } \cr
 \phantom{=} \cr
P \quad &= \quad {\rm number\ of\ photons\ from\ planet} \cr
}
$$


$$
P \quad > \quad \sqrt{I_{*}(\theta) }
$$


$$
I(\theta) \quad = \quad I_0 \thinspace 
                  e^{-{\theta}^2/(2 s^2)}
    \quad \simeq \quad
             I_0 \thinspace 
                  e^{-{\theta}^2/(0.36 \ {\rm FWHM}^2)}
$$

$$
s \quad \simeq \quad {\rm FWHM} / 2.35
$$

$$
I(\theta) \quad = \quad I_0 \thinspace
                    e^{ -2.76 \thinspace {(\theta/{\rm FWHM})}^2 }
$$

$$
{\rm star\ noise\ at\ \theta,\ in\ mag} 
       \quad = \quad {-2.5 \thinspace \log_{10} { 
            \left( { { \sqrt{I(\theta)} } \over {I(\theta)} } \right)  } }
$$


$$
{ { L_{p} \over {L_{*}}  } } \quad = \quad  
                { \left( {R_{\rm P} } \over { R_{*} } \right) }^2
                { \left( {B_{\lambda}(T_{\rm P}) } \over { B_{\lambda}(T_{*}) } \right) }^4
$$




\bye
