The Cosmological Distance Ladder

Magnitude, Luminosity, and Distance

There are two magnitude scales that we use to classify the brightness of stars.  The first is called the apparent magnitude.  This scale is simply a classification of how bright the stars appear to be.  It is called an "apparent" magnitude scale because when we look into the night sky and see the stars all shining with different brightness, we don't know if the stars are bright or dim because of their distance from us or because of they are inherently bright or dim.

The second scale is called the absolute magnitude scale.  On this scale, all the stars are assumed to be at some standard distance, so differences in brightness on this scale are due entirely to how much light the star emits. 

We classify a star's apparent brightness in the sky by assigning it a magnitude.  The magnitude scale was initially developed by the ancient astronomer Hipparchus, (~150 B.C.E).  He grouped stars according to their apparent brightness.  The brightest stars were "stars of the first magnitude".  The next brightest stars were "stars of the second magnitude," and so on.  Under this naming convention, the higher the magnitude of a star, the dimmer it appears in the sky.

Later studies showed that a difference in magnitude of 5 in Hipparchus' scale corresponded to a difference in brightness of about 100 times.  In the 1800's, this was formalized so that a 5 point difference in magnitude was defined to correspond to a 100 times difference in brightness.  Mathematically, the definition is written as follows:

  Relationship between Flux and Magnitude

In this equation F1 and F2 represent the observed brightness of two stars (specifically total observed flux) with apparent magnitudes of m1 and m2 respectively.  Notice that whenever the difference in magnitude is equal to 5, (i.e. m2-m1 = 5), the ratio of brightness between the two stars is equal to 100.

Equation 1 is generally rewritten as a power of 10 rather than a power of 100 simply because powers of 10 are more commonly used.  This can be done by noting that 100 = 10^2, (a review of logs and exponents can be found here).

Rewriting as a power of 10 rather than a power of 100.

Taking the log of both sides.

 

Now to determine a star's absolute magnitude, we must imagine it is located at some fixed distance away, and determine how much the magnitude would change based on the change in distance from where the star really is, to the reference distance.  We can do this by noting that the total flux observed from a star is proportional to the square of of its distance, as seen in the equation below where F is the observed flux, L is the inherent luminosity of the star, and d is the distance between us and the star.  Note that to compare the amount of flux we would observe at two different distances, we simply take a ratio of the flux at the first distance to the flux at the second distance, which, as can be seen, is simply the square of the ratio of the two distances.

Inverse square relationship between flux and distance.

Ratio of flux received at two different distances.

 

So now we can substitute (d₂/d₁)² for (F₁/F₂) as shown below and we end up with an equation that relates differences in observed magnitude to differences in distance.

Relationship of magnitude to distance.

The standard distance for absolute magnitudes has been agreed upon as 10 parsecs, so d₁ is replaced by 10, (which makes solutions to the equation clean and simple).  We also change the variable names to make clear what they represent:  m₂ is replaced with m, the apparent magnitude of the object being observed; m₁ is replaced with M, the absolute magnitude of the object being observed, (which is the apparent magnitude if the object being observed were at the standard distance of 10 parsecs); and d₂ is replaced by d, the actual distance of the object whose apparent magnitude is m.

Same relationship but with absolute magnitude defined at 10 parsecs.

 

There are two ways in which this equation can be useful.  First, if one knows both the apparent and absolute magnitude of a star, one can immediately calculate its distance. 

THE DISTANCE EQUATION (click on equation to see algebraic derivation).

 

Second, if one knows the apparent magnitude of a star, and its distance, then its absolute magnitude can be determined.

THE ABSOLUTE MAGNITUDE EQUATION (click on equation to see algebraic derivation).

The first equation is very important because it tells us that the distance to an object is easily calculated if we now both its apparent magnitude and its absolute magnitude.  The trick is going to be determining the absolute magnitude since the only way we can know the absolute magnitude is to know the distance of the object being observed which is exactly what we are trying to find.

The second equation tells us that if we know the apparent magnitude and the distance to an object, we can easily calculate its absolute magnitude.  But if we already know the distance to an object, what use really is this second equation?

The importance of the second equation lies in the fact that there are other distance independent properties of stars that can be related to the star's absolute magnitude, but there is no way to know such a relationship exists without making observations of stars whose distance we already know, and whose apparent magnitudes have been converted to absolute magnitudes.

The application then, is to determine the absolute magnitude for as many nearby stars as we can, (using the method of stellar parallax primarily), and then look for other look for other observable properties that would enable us to determine a star's absolute magnitude even without knowing its distance.  If these relationships can be found, then we can use them to determine the absolute magnitude of objects whose distances we don't know, and apply the distance equation to find their distance.

As it turns out there are several observable properties that allow us to determine a star's absolute magnitude.  These are discussed next, and allow us to reach further out into the cosmos.