# Chapman's function

1. Feb 20, 2006

### stunner5000pt

According to Chapman theory the photon volume deposition rate for an incoming beam of solar radiation is given by
$$q(h) = I_{t} \sigma n_{0} \exp\left(-\frac{h}{H} - \sigma n_{0} H \sec\chi \exp\frac{-h}{H}\right)$$
where It is the flux of incoming solar radiation, σ is the absorption cross-section, no is absorber density at the surface and H is its scale height. Prove that the deposition curve is displaced upwards by one scale height without changing shape if either σ or no are increased by a factor of e the natural log base. It is not sufficient to show that hmax increases by one scale height.

well i tired multiplying sigma or n by e and i get this
$$q(h) = I_{t} \sigma n_{0} \exp\left(1 -\frac{h}{H} - \sigma n_{0} H \sec\chi \exp(1 - \frac{-h}{H})\right)$$

the thing is i am not quite sure how one would show a displacement of H upward...
are we aiming to get q(h) + H ? Is that the kind of dispacement that this quesiton requires?

as always, help is greatly appreciated!

2. Feb 22, 2006