How is absolute magnitude determined

Now it is known that the eye is not quite a logarithmic detector. Your eyes perceive equal ratios of intensity as equal intervals of brightness. For example, first magnitude stars are about 2. Notice that you raise the number 2. Also, many objects go beyond Hipparchus' original bounds of magnitude 1 to 6.

The important thing to remember is that brighter objects have smaller magnitudes than fainter objects. The magnitude system is screwy, but it's tradition!

Song from Fiddler on the Roof could be played here. Apparent Magnitude The apparent brightness of a star observed from the Earth is called the apparent magnitude. The apparent magnitude is a measure of the star's flux received by us.

How do you do that? Which star is brighter and by how many times? Star B is brighter than star A because it has a lower apparent magnitude. Star B is brighter by 5.

In terms of intensity star B is 2. The amount of energy you receive from star B is almost 16 times greater than what you receive from star A.

Absolute Magnitude and Luminosity If the star was at 10 parsecs distance from us, then its apparent magnitude would be equal to its absolute magnitude.

The absolute magnitude is a measure of the star's luminosity the total amount of energy radiated by the star every second. If you measure a star's apparent magnitude and know its absolute magnitude, you can find the star's distance using the inverse square law of light brightness.

If you know a star's apparent magnitude and distance, you can find the star's luminosity see the table below. The luminosity is a quantity that depends on the star itself, not on how far away it is it is an "intrinsic" property.

For this reason a star's luminosity tells you about the internal physics of the star and is a more important quantity than the apparent brightness.

A star can be luminous because it is hot or it is large or both! Because the temperature is raised to the fourth power, it means that the luminosity of a star increases very quickly with even slight increases in the temperature. Because the surface area is also in the luminosity relation, the luminosity of a bigger star is larger than a smaller star at the same temperature. You can use the relation to get another important characteristic of a star. If you measure the apparent brightness, temperature, and distance of a star, you can determine its size.

The figure below illustrates the inter-dependence of measurable quantities with the derived values that have been discussed so far. In the left triangular relationship, the apparent brightness, distance, and luminosity are tied together such that if you know any two of the sides, you can derive the third side.

For example, if you measure a glowing object's apparent brightness how bright it appears from your location and its distance with trigonometric parallax , then you can derive the glowing object's luminosity. Or if you measure a glowing object's apparent brightness and you know the object's luminosity without knowing its distance, you can derive the distance using the inverse square law. For this we recall from the table above that the Sun has an apparent magnitude of It is important to remember that magnitude is simply a number, it does not have any units.

The symbol for apparent magnitude is a lower case m ; you must make this clear in any problem. What does the fact that Sirius has an apparent magnitude of Another way of thinking about this is to ask why is Sirius the brightest star in the night sky? A star may appear bright for two main reasons:. The apparent magnitude of a star therefore depends partly on its distance from us. In fact Sirius appears brighter than Betelgeuse precisely because Sirius is very close to us, only 2.

The realisation that stars do not all have much the same luminosity meant that apparent magnitude alone was not sufficient to compare stars. A new system that would allow astronomers to directly compare stars was developed. This system is called the absolute magnitude, M.

The absolute magnitude, M , of a star is the magnitude that star would have if it were at a distance of 10 parsecs from us. A distance of 10 pc is purely arbitrary but now internationally agreed upon by astronomers. This logarithmic scale is also open-ended and unitless. Again, the lower or more negative the value of M, the brighter the star is. Absolute magnitude is a convenient way of expressing the luminosity of a star. Once the absolute magnitude of a star is known you can also compare it to other stars.

As you may recall from the section on astrometry, most stars are too distant to have their parallax measured directly.

Nonetheless if you know both the apparent and absolute magnitudes for a star you can determine its distance. Let us look again at Sirius and Betelgeuse plus another star called GJ How far away is GJ 75?

It is an unusual star in that its apparent and absolute magnitudes are the same. The reason is that it is actually 10 parsecs distant from us, so by definition its two magnitudes must be the same. What about Sirius? Its apparent magnitude is lower therefore brighter than its absolute magnitude. This means that it is closer than 10 parsecs to us. Betelgeuse's apparent magnitude is higher therefore dimmer than its absolute magnitude so it would appear even brighter in the night sky if it were only 10 parsecs distant.

Is there a quick way of checking whether a star is close or not? This value is negative and Sirius is closer than 10 pc. This value is positive and Betelgeuse is more than 10 pc distance. Astronomers use the difference between apparent and absolute magnitude, the distance modulus, as a way of de terming the distance to a star.

A formal derivation of this equation is given in the next page on luminosity. You should be comfortable in solving this equation given any two of the three variables. Let us know look at how you can solve some examples. Example 3: Given m and d , need to find M. What is its absolute magnitude? Remember in solving magnitude equations log refers to logarithms to base 10 and not natural logarithms or ln.

Example 4: Given m and M , find d. Betelgeuse has an apparent magnitude of 0. How far away is it? This problem requires us to rewrite equation 4.

This is shown below:. Again, this example shows complete working whereas in reality you may not show every step. It is important, however, that you set your working to such problems out clearly so you can check your algebraic manipulation and your substitutions. Working with logs and indices can be tricky so ensure you know how to do these on your calculator. Example 5: Given M and d , find m.

In practice this type of problem is less realistic for actual objects as we can normally measure their apparent magnitudes directly however it may be that we wish to calculate what apparent magnitude a class or type of object may have given the other parameters.

Again, starting with equation 4. In reality Deneb is about pc distant although this value has a large uncertainty. Example 6 : What if d is not given but parallax, p is given?

This is actually very straight forward. Recall from the section on astrometry that there is a direct relationship between distance and parallax. Let us know revisit that photo of Crux and the Pointers from the top of this page. The photo below shows the same region with the prominent stars labelled. They also have their apparent magnitudes shown. Crux is a constellation, one of 88 regions that the celestial sphere has been broken up into and agreed upon internationally by astronomers.

Crux is actually the smallest of the constellations and is easily identified in the southern skies. The prominent nearby stars commonly called the Pointers are actually part of a large constellation called Centaurus.