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Function: template_area()

0 float template_area(struct Scope *Grid)
This function computes the area of the enclosed region of parameter space shown in Figure [*].

The arguments are:

Grid: Input. This function uses only the minimum mass, maximum mass and the cut-off frequency $f_0$ fields of Grid.
The function returns the numerical value of the area in units of ${\rm sec}^2$. See the example in the following subsection.

The function uses an analytic expression for the area obtained by integration of formulae ([*],[*]) for $\tau_0$ and $\tau_1$ given earlier. For example, to obtain the area of the trapezoidal region bounded above by the maximum-mass curve and below by the $\tau_0$ axis, we integrate

\begin{eqnarray*}
A_{1} &=& \int_{m_{\rm max}}^{m_{\rm min}}
\tau_1 (m_{\rm min...
...2/3} }
\right ]
\biggr\}_{u=m_{\rm min}/m_{\rm max}}^{u=1} \; .
\end{eqnarray*}



Here $a=924/743$ and $A_0$ is a quantity with dimensions ${\rm sec}^2$ given by

\begin{eqnarray*}
A_0={18575 \over 49545216 }{ M_{\odot}^2 \over (\pi M_{\odot} f_0 )^{14/3} }
\left ( {c^3 \over G } \right )^{8/3} \;.
\end{eqnarray*}



The area $A_{2}$ under the minimum-mass curve can be obtained from the formula above by interchanging $m_{\rm min}$ and $m_{\rm max}$. (If you wish to use geomtrized units in which the solar mass is $4.92\times10^{-6}
\> {\rm sec}$ simply set $G=c=1$.) The area under the equal-mass curve $A_{3}$ can be obtained by performing a similar integration along the equal-mass curve

\begin{eqnarray*}
A_{3} &=& \int_{m_{\rm max}}^{m_{\rm min}}
\tau_1 (m,m) {d \t...
...right )^{8/3} \right ]
\left ( {c^3 \over G } \right )^{8/3} \;.
\end{eqnarray*}



These three results can be combined to give the total area enclosed
\begin{displaymath}
A_{total} = A_{1} + A_{2} - A_{3} \;.
\end{displaymath} (9.5.201)

Equation ([*]) is the basis of template_area(); the next example shows an application of this function.
Author: Alan Wiseman, agw@tapir.caltech.edu
Comments: None.


next up previous contents
Next: Example: area program Up: GRASP Routines: Template Bank Previous: Function: m_and_eta()   Contents
Bruce Allen 2000-11-19