\leftline{ \underline{\bf Lyman Alpha Clouds: A Consistent Distribution}} \\ \leftline{ \underline{\bf in Column Density}}
Session 78 -- QSOs and Gravitational Lenses
Oral presentation, Thursday, 2:30-4:00, Durham Room

## [78.08] \leftline{ \underline{\bf Lyman Alpha Clouds: A Consistent Distribution}} \\ \leftline{ \underline{\bf in Column Density}}

V.V. Chernomordik$^1$, L.M. Ozernoy$^2$~~~~~~ ($^1$Sci. Council on Cybernetics, Moscow, Russia; $^2$GMU \& USRA/NASA/GSFC)

\def\lax {\ifmmode{_<\atop^{\sim}} \else{${_<\atop^{\sim}}$} \fi} \def\gax {\ifmmode{_>\atop^{\sim}} \else{${_>\atop^{\sim}}$} \fi} \def\kms {\ifmmode{{\rm ~km~s}^{-1}} \else{~km~s$^{-1}$} \fi} \def\kmp {~km~s$^{-1}$~pc$^{-1}$} \def\lo {~$L_{\odot}$} \def\mo {~${\rm M}_{\odot}$} \def\moyr {\hbox{~${\rm M}_{\odot} \,{\rm yr}^{-1}$}} \def\etal {{\sl et~al.~}} \def\eg{e.\thinspace g.} \def\ie{i.\thinspace e.} We propose a convenient analytical approximation for curve-of-growth analysis of the Lyman alpha series, which is valid over a very broad range of parameters of absorbing clouds. The equivalent width of the line as a function of the line center optical depth, $\tau_0=\left ({\sqrt\pi e^2/ m_ec} \right) \left ({\lambda_0fN/ b} \right )$, is given by \begin{displaymath} W = \frac{\sqrt 2 b\lambda_0}{ c} \left [{\rm ln} \left (1+ \frac{\pi}{2} \tau_0 ^2\right )\right ]^{1/2}+\frac{\sqrt N}{N_d}, \end{displaymath} where $N_d=\left ({m_ec^2/ 2\pi e^2} \right )\left (f\lambda_0\right )^{-1} \left ({g_2/ 2g_1} \right )^{{1/ 2}}$. This expression gives the correct asymptotics in the unsaturated, saturated, and damped regimes, as well as provides an accuracy not worse than 10\% in the intermediate cases. With the use of this formula we have obtained a simple analytical approximation to the relationship between a rest-frame equivalent width distribution for Ly-$\alpha$ forest absorption lines, ${\cal N}(W)$, and an HI column density distribution of the observed cloud number, ${\cal N}(N)$. Assuming a simple power-law form for ${\rm d} {\cal N}/{\rm d} N\propto N^{-\beta}$ and that the velocity width $b={\rm const}$, it is shown that $\beta = 1.4$ agrees fairly well with the observed form of ${\cal N}(W)$ in a broad range of column densities $10^{13}~ {\rm cm^{-2}} \lax N \lax 10^{17}{\rm cm^{-2}}$, which corresponds to equivalent width range $0.06~\AA\lax W\lax 0.9~ \AA$. Only this value of $\beta$ makes consistent the procedure of transforming ${\cal N}(W)$ into ${\cal N}(N)$, and vice versa. The above result has a number of important implications, of which worth mentioning is `the proximity effect' (Chernomordik \& Ozernoy 1993, {\sl ApJ} 404 L5).