Power law distributions

Scientists often take a census of items: stars in the solar neighborhood, or insects in the forest, or subatomic particles in a region of space. When one counts items and measures their properties, it is often useful to create a distribution, showing the number of items with some particular property.

For example, suppose that Joe Astro studies the stars in three different clusters. He measures the mass and luminosity of each star in each cluster. Here are his counts:

How can Joe count fractional stars? His eyepieces need to be cleaned, apparently

# Mass     Number of stars with that mass
# (solar)        A        B        C
#------------------------------------------
    0.3000    746.30    50.00     5.46 
    0.8000     67.85    50.00    18.12 
    1.3000     35.54    50.00    25.04 
    1.8000     24.07    50.00    30.42 
    2.3000     18.20    50.00    34.99 
    2.8000     14.63    50.00    39.02 
    3.3000     12.23    50.00    42.67 
    3.8000     10.51    50.00    46.04 
    4.3000      9.21    50.00    49.17 
    4.8000      8.20    50.00    52.12 
    5.3000      7.39    50.00    54.91 
    5.8000      6.72    50.00    57.57 
    6.3000      6.17    50.00    60.10 
    6.8000      5.70    50.00    62.54 
    7.3000      5.29    50.00    64.88 
    7.8000      4.94    50.00    67.14 
    8.3000      4.64    50.00    69.33 
    8.8000      4.36    50.00    71.45 
    9.3000      4.12    50.00    73.51 
    9.8000      3.91    50.00    75.51 
#------------------------------------------

It's clear that the properties of stars in these clusters are quite distinct:

in cluster A, low-mass stars are far more common
in cluster B, stars of all mass occur equally frequently
in cluster C, stars of high mass are slightly more common

Is there a way to put these statements into some sort of mathematical form? In other words, can we come up with some sort of formula which explains how likely it is to find a star of some particular mass in each cluster?



                       Yes!

One type of mathematical form which is very useful is called the power law. It's very useful for three reasons:

it's simple to write down
... and so simple to manipulate, integrate or differentiate
it does a pretty good job of representing actual distributions found in nature

Joe decides to write a power law describing the distribution of stars as a function of mass like so:

Notice the middle section of the right-hand side: it involves one quantity (mass, in this case) raised to some power (γ, in this case). That's why this sort of equation is called a power law. Let's look at each piece of the equation.

N(m): is the number of stars with a mass centered on m and extending a little bit to higher and lower masses
dm: is the width of that range of masses
K: is a constant, which serves to scale the numbers produced by the equation so that they match the number observed
m: is the mass of a star
γ: is the exponent of the power law, which determines how the number of stars changes with mass
dm: is the width of a range of masses again, exactly the same as it was on the left-hand side

It's important to remember that whenever we try to use the equation, we need to specify some range over which we'll count. Look at the start of Joe's table, for example:


# Mass     Number of stars with that mass
# (solar)        A        B        C
#------------------------------------------
    0.3000    746.30    50.00     5.46 
    0.8000     67.85    50.00    18.12 
    1.3000     35.54    50.00    25.04 
    1.8000     24.07    50.00    30.42 
      ...       ...      ...      ...
#------------------------------------------

There are 35 stars in cluster A with a mass of about 1.3 solar masses, right? A more precise statement is that, Joe counted 35 stars in cluster A which had masses in the range between 1.05 and 1.55 stellar masses, We might say, "Joe counted 35 stars in a bin of width 0.50 solar masses, centered on 1.30 solar masses." If we don't include the range (or bin width) in our statement, then our statement is meaningless. After all, Joe counted 127 stars with a mass of about 1.3 solar masses -- if you use a range from 0.55 to 1.95.

Sticklers for clarity might write the power law in a slightly different manner:

There's no ambiguity here: the equation yields the number of stars with masses in the range from m to m + dm.