\leftline{ \underline{\bf Lyman Alpha Clouds: A Consistent Distribution}} \\ \leftline{ \underline{\bf in Column Density}}
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**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).

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