Magnitude Scale and Distance Modulus in Astronomy

Visual magnitude scale measures relative brightness of objects in the night sky. It has many uses, including calculating distance to stars.

History of the Astronomical Magnitude Scale

The Greek mathematician Hipparchus is widely credited for the origin of the magnitude scale, but Ptolemy popularised it and brought it to the mainstream.

In his original scale, only naked-eye objects were categorised (excluding the Sun), the brightest Planets were classified as magnitude 1, and the faintest objects were magnitude 6, the human eye's limit. Each level of magnitude was considered twice the brightness of the previous; therefore, magnitude two objects are twice as bright as magnitude 3. This is a logarithmic magnitude scale.

With the invention of the telescope and other observational aids, the number of new objects soared, and a modification was needed to the system to categorise so many new objects accurately. In 1856, Norman Robert Pogson formalised the magnitude scale by defining that a first-magnitude object is an object that is 100 times brighter than a sixth-magnitude object. Thus, a first-magnitude star is 2.512 times brighter than a second-magnitude object.

Pogson's magnitude scale was originally fixed by assigning Polaris a magnitude of 2. Astronomers later discovered that Polaris is slightly variable, so they first switched to Vega as the standard reference star and later switched to tabulated zero points for the measured fluxes. This is the system used today.

Difference Between Absolute and Apparent Magnitude

Going back to stars A and B, star A is magnitude two and star B is magnitude 3. According to the magnitude scale, Star A is 2.512 times as luminous as Star B. Here, we are referring to the Star Apparent Magnitude, that is, its brightness as seen from Earth. This is how most magnitudes are presented on TV, in planetarium software and magazines.

But how do we know that Star A is brighter than Star B? Star A and Star B can have the same luminosity, but Star B could be further away than Star A, thus appearing dimmer to us from Earth.

We also have a scale that allows us to compare a star's brightness if it were at a fixed distance from the Earth. This is known as the Absolute Magnitude, and it's set at a universally agreed-upon distance of 10 parsecs. A parsec is a unit of distance that represents the distance from the Earth to an astronomical object with a parallax angle of one arcsecond. We'll delve into the concept of parallax in a future article, but for now, just remember that 1 parsec is equal to 3.26 light-years or 1.92 x 10^13 miles.

Absolute Magnitude is given the symbol M, while Apparent Magnitude is given lowercase m.

Our Sun has an apparent magnitude of -26.73, which easily makes it the brightest object visible in the sky; however, the Sun would not be as bright if it was 10 parsecs away. At this distance, it would only shine at a mere apparent magnitude of 4.6, so it would be faint in the night sky. At 10 parsecs, the Sun's magnitude is called the Absolute Magnitude.

Sirius is the next brightest Star in the sky and has an apparent magnitude of -1.47. However, it only lies 2.64 parsecs away, which is relatively close. If it were moved to a standard 10 parsecs away, it would be absolute magnitude 1.4, 8 times brighter than our Sun at the same distance.

Here's a quick way of remembering the difference between absolute and apparent magnitude:

Apparent magnitude appears to be brightest, Absolute magnitude absolutely is the brightest.

Example Stars on the Magnitude Scale

In this table, we can see examples of various points on the magnitude scale.

Apparent Magnitude, Absolute Magnitude and Distance

There are two main types of magnitude commonly used in astronomy. The first of these, apparent magnitude, is the object's brightness as seen by an observer on the Earth. The apparent magnitude of a star is dependent on two factors:

1. The luminosity of the Star (total energy per second radiated)
2. The distance of the Star from Earth

The second, absolute magnitude, is dependent solely on the Star's luminosity and can be regarded as an intrinsic property of the Star. Absolute magnitude is defined as the apparent magnitude of an object if it were a standard distance from the Earth. The standard distance is 10 parsecs. Since distance is always equal when comparing absolute magnitudes, it can be removed as a factor in the Star's brightness, which is why it can be regarded as an intrinsic property.

Absolute Magnitude and Luminosity

A Star's luminosity, L, is the total amount of energy radiated per unit of time. The absolute magnitude of a star is related to its luminosity in the same way as apparent magnitude is related to flux. Let's compare the ratio of the brightness of two stars, expressed in terms of their luminosities. We obtain a relation for the difference in their absolute magnitudes.

Equation 23 - Absolute Magnitude Relation

Capital letters are used to indicate absolute magnitudes and lowercase letters are used to identify apparent magnitudes.

As we have previously stated, absolute magnitude is the apparent magnitude of an object if it is a distance of 10 parsecs from the Earth.

It is clear from this definition that a star located at 10 parsecs from the Earth will have the same apparent and absolute magnitude. A star further away than 10 parsecs will have a fainter apparent magnitude than the absolute magnitude, and a star closer than 10 parsecs will have a brighter apparent magnitude than the absolute magnitude.

How do we know stars' absolute magnitude? We could travel to every Star and measure the apparent brightness from a distance of 10 parsecs, but that isn't a practical solution at the moment. Luckily for us, however, a very important formula relates to the apparent and absolute magnitudes.

Distance Modulus Equation

Distance Modulus is the difference between the apparent and absolute magnitudes. This can be obtained by combining the definition of absolute magnitude with an expression for the inverse square law and Pogson's relation. Using the distance modulus, it is possible to establish a relationship between the absolute magnitude, M, of a star, its apparent magnitude, m, and its distance, d.

The inverse square law tells us that for a star at a distance d (parsecs), with observed flux F_m, then its flux F_M at 10 parsecs would be given by:

Equation 24 - Inverse Square Law for Flux

We can combine this with equation 23 above to give the distance modulus equation.

Equation 25 - Distance Modulus

Suppose we measure the Star's apparent magnitude, and its distance in parsecs is known. In that case, we can determine the absolute magnitude and, hence, the luminosity of the Star. If we know the Star's absolute and apparent magnitudes, we can use distance modulus to calculate the distance to the Star. This equation is very powerful and will often be used in upcoming tutorials.

The formula for calculating Absolute Magnitude within our galaxy is:

Equation 31 - Absolute Magnitude

Where D is the distance to the Star in parsecs.

Example of Distance Modulus to Calculate Distance

Barnard's Star lays 1.82 parsecs away and has an observed (apparent) magnitude of 9.54.

m - M = 5((log10 D)-1)
M = 9.54 * 5((log10 1.82)-1)
M = 9.54 - (-3.7)
M = 13.24

If Barnard's Star were moved to a distance of 10 parsecs from the Earth, it would have a magnitude of 13.24.

If we already know both Apparent and Absolute magnitudes, it is possible to calculate the distance to the Star:

Equation 63 - Distance Modulus solved for d

d = 100.2(m - M + 5)

Using Barnard's Star again,

d = 100.2(9.54-13.24+5)
d = 100.26
d = 1.82 parsecs

Bolometric Magnitude

Another type of astronomer interest is the bolometric magnitude. So far, the absolute and apparent magnitudes are based on the total visible energy radiated from the Star. We know that not all that energy is received on Earth since it is filtered out by our atmosphere.

Bolometric magnitude is based on the flux throughout the entire electromagnetic spectrum. The term absolute bolometric magnitude is based specifically on the luminosity (or total rate of energy output) of the Star.

The bolometric magnitude M_bol takes into account electromagnetic radiation at all wavelengths. It includes those unobserved due to instrumental pass-band, the Earth's atmospheric absorption, and extinction by interstellar dust. It is defined based on the luminosity of the stars. In the case of stars with few observations, it must be computed assuming an effective temperature.

Equation 39 - Bolometric Magnitude

This can then be reworked to find the ratio of luminosity.

Equation 40 - Luminosity ratio of magnitudes