The Basic Mathematics Of The Stellar Magnitude System
Generally, stellar magnitude refers to the apparent brightness of an astronomical object. The brightness magnitude system applied to the stars is also applied to other astronomical objects such as planets, comets, nebulae, galaxies and even the sun and moon.
The Basic Mathematics Of The Stellar Magnitude Ranking System
Stellar brightness is generally measured in units referred to as magnitudes. It is like a system of class or rank where the higher the number the lesser the rank. For example, 1st class outranks 2nd class which in turn outranks 3rd class, etc. Thus, a 1st magnitude star is brighter or outranks a 3rd magnitude star in a sense analogous to the way a 1st class actor may be said to be a brighter movie star than a third class actor. As a result, the general rule is, the higher the stellar magnitude value, the fainter the object, so an object of magnitude 1 appears relatively brighter than an object of magnitude 6. This explains why magnitude 6 is not brighter than magnitude 1, which may seem a more natural way to do it at first glance.
To mathematically examine how this magnitude system works,
let there be two stars of different apparent magnitudes:
m1 = Magnitude of the brighter star
m2 = Magnitude of the fainter star
b = Apparent brightness ratio between m1 and m2
with the brighter star appearing b times brighter than the fainter star.
Here, the distances to the stars can be ignored. We do not need to think about the distances when simply ranking stars by their relative apparent brightness in the sky.
In the equations below:
The stellar magnitude system is designed mathematically so that a difference of exactly 5 magnitudes equates to an apparent brightness ratio (b) of exactly 100-fold. It is a base-10 logarithmic system of astronomical brightness scaling analogous to the Richter or decibel scales used to measure energy intensity.
The general relationship between apparent difference in magnitude vs apparent ratio in brightness may be expressed by the simple equation
Eq. 1
This equation answers the general question:
How much brighter (b) does m1 appear to be than m2 ?
According to the established definition:
When m2 minus m1 = 5, then b = 100.0
That is, a 5-magnitude difference equates to a brightness difference of 100-fold.
In pseudocode, this equation may be expressed as:
NOTE:
The symbol ** means to the power of
It was stolen from the FORTRAN programming language.
In the algorithm, the 5th root of 100 is expressed as 100 to the power of 1/5 or 0.2, which can be treated as a constant and simply replaced by its numerical value instead:
Combining the Brightness of Two Stars Into a Singular Magnitude
To the naked eye what may appear to be a single star may in fact be two or more stars very close together or apparently so due to the line of sight. Several double and multiple star systems appear as a single star to the unaided eye. What we see is a combination of their individual magnitudes.
Let
m = Singular magnitude of two stars combined
where
m1 = Magnitude of brighter star
and
m2 = Magnitude of fainter star
The mathematical relationship between the magnitudes is
Eq. 2
Combining the Brightness of Multiple Stars Into a Singular Magnitude
If there are three or more stars, their singular combined magnitude can be found from the following equation.
Let
m = Singular magnitude of N combined stars
mn = Magnitude of nth star of a group of N stars in any order
The general formula for N combined magnitudes is
Eq. 3
Apparent Magnitude With Respect to Distance
Having covered the general relationship between apparent magnitude and brightness, we can now proceed to the next step and add distance to the problem. Using the following equations, we can mathematically move a star around in space and compute its apparent brightness at any given distance from any given known starting values. This also allows us to mathematically compare the relative brightness of any two stars side-by-side at any common distance. For example, how bright a star would our sun appear to be if viewed from the same distance as Alpha Centauri ? (see Eq. 5) Or how bright would Alpha Centauri appear to be if it replaced our sun ?
Let
m1 = Apparent magnitude of a star at distance d1
m2 = Apparent magnitude of the same star at distance d2
Distances can be in any convenient, consistent units, such as AUs, light years, etc.
The relationship between apparent magnitude and distance may be expressed in terms of any of the four variables as in the following equations.
Eq. 4
Eq. 5
Eq. 6
Eq. 7
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The Basic Mathematics Of The Stellar Magnitude Ranking System
Stellar brightness is generally measured in units referred to as magnitudes. It is like a system of class or rank where the higher the number the lesser the rank. For example, 1st class outranks 2nd class which in turn outranks 3rd class, etc. Thus, a 1st magnitude star is brighter or outranks a 3rd magnitude star in a sense analogous to the way a 1st class actor may be said to be a brighter movie star than a third class actor. As a result, the general rule is, the higher the stellar magnitude value, the fainter the object, so an object of magnitude 1 appears relatively brighter than an object of magnitude 6. This explains why magnitude 6 is not brighter than magnitude 1, which may seem a more natural way to do it at first glance.
To mathematically examine how this magnitude system works,
let there be two stars of different apparent magnitudes:
m1 = Magnitude of the brighter star
m2 = Magnitude of the fainter star
b = Apparent brightness ratio between m1 and m2
with the brighter star appearing b times brighter than the fainter star.
Here, the distances to the stars can be ignored. We do not need to think about the distances when simply ranking stars by their relative apparent brightness in the sky.
In the equations below:
- log(x) = Common base 10 logarithm of (x)
- antilog(x) = 10 to the power of (x)
The stellar magnitude system is designed mathematically so that a difference of exactly 5 magnitudes equates to an apparent brightness ratio (b) of exactly 100-fold. It is a base-10 logarithmic system of astronomical brightness scaling analogous to the Richter or decibel scales used to measure energy intensity.
The general relationship between apparent difference in magnitude vs apparent ratio in brightness may be expressed by the simple equation
Eq. 1
How much brighter (b) does m1 appear to be than m2 ?
According to the established definition:
When m2 minus m1 = 5, then b = 100.0
That is, a 5-magnitude difference equates to a brightness difference of 100-fold.
In pseudocode, this equation may be expressed as:
Apparent Brightness Ratio (b) Between Brighter m1 and Fainter m2
NOTE:
The symbol ** means to the power of
It was stolen from the FORTRAN programming language.
In the algorithm, the 5th root of 100 is expressed as 100 to the power of 1/5 or 0.2, which can be treated as a constant and simply replaced by its numerical value instead:
Apparent Brightness Ratio (b) Between Brighter m1 and Fainter m2
Combining the Brightness of Two Stars Into a Singular Magnitude
To the naked eye what may appear to be a single star may in fact be two or more stars very close together or apparently so due to the line of sight. Several double and multiple star systems appear as a single star to the unaided eye. What we see is a combination of their individual magnitudes.
Let
m = Singular magnitude of two stars combined
where
m1 = Magnitude of brighter star
and
m2 = Magnitude of fainter star
The mathematical relationship between the magnitudes is
Eq. 2
Combining the Brightness of Multiple Stars Into a Singular Magnitude
If there are three or more stars, their singular combined magnitude can be found from the following equation.
Let
m = Singular magnitude of N combined stars
mn = Magnitude of nth star of a group of N stars in any order
The general formula for N combined magnitudes is
Eq. 3
Apparent Magnitude With Respect to Distance
Having covered the general relationship between apparent magnitude and brightness, we can now proceed to the next step and add distance to the problem. Using the following equations, we can mathematically move a star around in space and compute its apparent brightness at any given distance from any given known starting values. This also allows us to mathematically compare the relative brightness of any two stars side-by-side at any common distance. For example, how bright a star would our sun appear to be if viewed from the same distance as Alpha Centauri ? (see Eq. 5) Or how bright would Alpha Centauri appear to be if it replaced our sun ?
Let
m1 = Apparent magnitude of a star at distance d1
m2 = Apparent magnitude of the same star at distance d2
Distances can be in any convenient, consistent units, such as AUs, light years, etc.
The relationship between apparent magnitude and distance may be expressed in terms of any of the four variables as in the following equations.
Eq. 4
Eq. 5
Eq. 6
Eq. 7
© Jay Tanner - PHP Science Labs - 2012