## Magnitude

Apparent magnitude (m): Apparent “brightness” of a star, as seen from Earth. Hipparcus first measured it, using a scale from 1 (brightest) to 6 (dimmest). Note that smaller values correspond to brighter starts! This scale was adopted and extended in the XIX c.:

• The minimum was set to -26.81 (Sun), and the maximum to ~ 29 (faintest object).
• It was set to be a logarithmic scale in terms of brightness, such that a difference of 5 magnitudes is equal to 100. That is: if Δm = 5 , then the ratio of brightnesses is 100. Thus, for Δm = 1 , the same ratio is 1001/5.

With a photometerm of accuracy +/- 0.01, and Δm of accuracy +/- 0.002, can be measured.

Radiant flux (F): Measure of the “brightness” of a star. Therefore related to m by a logarithmic relation. Next formula is the ratio of F for two different stars (1 and 2):

Radiant flux is the light energy of all wavelengths per unit time crossing a 1 cm2 area oriented perpendicular to the direction of light (see also figure at the end of this post).

L is luminosity, the light or energy emitted per unit time.

The formula above is an inverse square law. There is no light adsorbed before the distance r.

L == Luminosity of the Sun == 3.826e33 erg s-1
F
== Solar flux above Earth’s atmosphere == Solar constant == 1.360e6 erg s-1 cm-2

Absolute magnitude (M): Nothing more than the apparent magnitude that a star would have if measured from 10 pc distance.

The relation between M and m for one given star, is:

The distance (d) to a star can be expressed as a function of only m and M:

Distance modulusm – M. It is a measure of the distance to a star.

Note: To avoid confusion, magnitudes and masses of the Sun are expressed as:

• Mass: M
• Magnitude: MSunmSun

Trick: To solve problems about magnitudes and radiant fluxes, it may be better to work with ratios F1/F2, L1/L2, … instead of absolute values. For once, we don’t have to take care of unit conversion, since by using ratios the units automatically cancel.

• m and F are observed properties of a star.
• L and M are intrinsic properties.

In principle, one would measure the observed properties, and then calculate the intrinsic properties by knowing the distance to the star. For pulsating variable stars, however, we can know L and M without any knowledge of the distance. Such stars act as references to determine other distances.

In sum: