**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 Δ*m*= 5 , then the ratio of brightnesses is 100. Thus, for Δ*m*= 1 , the same ratio is 100^{1/5}.

With a **photometer**, *m* of accuracy +/- 0.01, and Δ*m* 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 cm^{2} 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 modulus**: *m* – *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:
*M*_{Sun},*m*_{Sun}

*Trick*: To solve problems about magnitudes and radiant fluxes, it may be better to work with ratios *F*_{1}/*F*_{2}, *L*_{1}/*L*_{2}, … 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:

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