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I'm trying to fit a Schechter luminosity function to some data points, but it's not clear from this definition what the domain of support of the PDF should be. I'm familiar with the standard Pareto distribution

$$p(x) = \frac{\alpha-1}{x_0}\left(\frac{x}{x_0}\right)^{-\alpha}$$

which is non-zero over $[x_0,\infty)$.

The Schechter function has a similar form (using slightly different conventions from the Wikipedia article):

$$p(x) \propto e^{-\frac{x}{x_0}}\left(\frac{x}{x_0}\right)^{-\alpha}$$

Can anyone confirm that this distribution has the same support as a standard Pareto distribution? I'm not sure how to normalise it, and the Wikipedia article is light on details.

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It's possible that this may be off topic here, but if so, it can likely be migrated to Cross Validated (or maybe even Mathematics). We'll see what the community thinks. –  David Z Sep 3 '12 at 19:25
    
On topic - the Schechter function only exists for the physical theory of galaxy luminosities - it is not studied in more abstract ways. –  Chris White Nov 23 '12 at 3:05

2 Answers 2

up vote 1 down vote accepted

Understand first that this is dangerously close to numerology in astronomy. Astronomers wanted for whatever reason to have a very "simple" functional form for the distribution of galaxy luminosities, and so of course if you look close enough (or even not that close as the case may be) real data will never be fit terribly well by this form, especially over a large range of luminosities.

As for the support, it is technically $(0,\infty)$ in luminosity $L$ (OP's variable $x$). Using more common notation I'll write the Schechter function as $$ \Phi(L) = \frac{n_0}{L^*} \left(\frac{L}{L^*}\right)^\alpha \mathrm{e}^{-L/L^*}, $$ where $\Phi(L)$ measures the number density per unit volume per unit luminosity of galaxies with luminosity $L$ and my $n_0$ is some authors' $\Phi^*$, but this gives $\Phi$ and $\Phi^*$ different units. You quickly see a problem with normalizability depending on $\alpha$. If $\alpha \leq -1$, then $$ \int_0^{L_\text{max}} \Phi(L)\ \mathrm{d}L $$ diverges for any $L_\text{max}$, indicating infinitely many galaxies (almost all very dim) per unit volume. Fits to real data often do have $\alpha$ hovering near this regime, so the problem is not merely academic.

Of course as pointed out in the original paper by Schechter, $$ \int_0^\infty L \Phi(L)\ \mathrm{d}L = n_0 L^* \Gamma(\alpha+2) $$ for $\alpha > -2$, so the luminosity density is still finite for reasonable values of $\alpha$.

When fitting data, note that the disagreement is usually bad enough within two or three (astronomical) mags of $L^*$ that these datapoints are often not included in the fit. I suppose a smooth weighting function that de-weights these points according to their proximity to $L^*$ could also be used.

In summary: The normalization is tied to the actual number density of galaxies in the sample, which in some cases can be infinite.

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Thanks; this is exactly what I was after. I have little grounding in astropjhysics, so didn't quite grasp the nuances of this formalism. Your answer has cleared things up for me :-) –  Will Vousden Nov 23 '12 at 17:59

The support of the Pareto distribution can be determined by requiring that it be normalized to 1:

$$\int_{x_\text{min}}^\infty p(x)\mathrm{d}x = \int_{x_\text{min}}^\infty \frac{\alpha - 1}{x_0}\biggl(\frac{x}{x_0}\biggr)^{-\alpha}\mathrm{d}x = \biggl(\frac{x_\text{min}}{x_0}\biggr)^{1 - \alpha} = 1$$

which gets you $x_\text{min} = x_0$. So you could apply the same reasoning to the luminosity function:

$$\int_{x_\text{min}}^\infty p(x)\mathrm{d}x = \int_{x_\text{min}}^\infty Ce^{-\frac{x}{x_0}}\biggl(\frac{x}{x_0}\biggr)^{-\alpha}\mathrm{d}x = Cx_0\Gamma\biggl(1 - \alpha, \frac{x_\text{min}}{x_0}\biggr) = 1$$

where $\Gamma$ is the upper incomplete gamma function and $C$ is the normalization constant of your luminosity function. You would have to solve this equation for $x_\text{min}$, which can be done numerically using any of various root-finding algorithms or an implementation of the inverse incomplete gamma function; or you could use a least-squares fitting algorithm to match this functional form to your data and thereby determine both $C$ and $x_\text{min}$.

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