Dr. Aaron Titus | Department of Physics, High Point University PHY1050      Astronomy of Stars, Galaxies, and the Cosmos home | WebAssign | textbook | course calendar

## luminosity

Luminosity is the energy a star radiates (i.e. gives off) per second.

Energy emitted per second is called power, and the SI unit of power is watts (W). Therefore, the unit of luminosity is watts. Thus, the luminosity of a 100 W light bulb is 100 watts.

Suppose that I have a 60 W bulb and a 100 W bulb. Which bulb has a greater luminosity?
100 W bulb | 60 W bulb

Suppose that I place a 60 W bulb and a 100 W bulb one meter from you. Which bulb will appear brighter?
100 W bulb | 60 W bulb

Now, suppose I take the 100 W bulb and move it the distance of a football field from you and leave the 60 W bulb only 1 meter from you. Which bulb will appear brighter?
100 W bulb | 60 W bulb

Apparent brightness of a star depends on two factors: (1) the luminosity of the star and (2) the distance to the star.

Thus, a star can appear bright either because it radiates a lot of energy per second or because it is very close.

Apparent brightness is proportional to luminosity divided by distance squared. Mathematically, this is expressed as

Because a star's apparent brightness is inversely proportional to the square of the distance to the star, this relationship is called the inverse square law for light.

Suppose that Star A and Star B are identical stars. Thus, they give off the same energy per second. However, Star B is 3 times further from us than Star A. How much brighter will Star A appear than Star B?

Suppose that Star C and Star D have the same apparent brightness. However, Star C is 4 times further than Star D. How much more energy does Star C emit, compared to Star D?

### Absolute Magnitude

Note that the luminosity of a light bulb does not depend on how far away you are from the light bulb or on how bright it appears. It only depends on how the filament of the light bulb is made. Likewise, the luminosity of a star does not depend on the distance to the star. It only depends on "how the star is made." For this reason, astronomers refer to the instrinsic brightness of the star.

The instrinsic brightness of a star is basically the same as luminosity; however, its unit is absolute magnitude.

To distinguish the unit for apparent brightness from the unit for luminosity, astronomers refer to luminosity using absolute magnitude and refer to apparent brightness using apparent magnitude.

To calculate absolute magnitude, we convert the apparent magnitude of a star to the apparent magnitude that a star would have it it were at a distance of 10 parsecs (pc). (The parsec is a unit of distance which you will learn about in the less on parallax.)

For example, suppose that a star has an apparent magnitude of 1 and is at a distance 4 pc from Earth. To calculate the absolute magnitude of this star, we do the following:
"Move" the star to 10 pc.
In this example, we have to move the star from 4 pc to 10 pc, that's 2.5 times the original distance (since 2.5 x 4 = 10).
Square this factor that the distance is changed.
In this example, the distance was increased by a factor of 2.5, and 2.5 squared is 6.25. Thus, the star would appear dimmer by a factor of 6.25.
Add or subtract the magnitude difference correspoding to this increase or decrease in brightness.
In this example, the brightness of the star would decrease by a factor of 6.25. This represents an increase in magnitude of 2. Therefore, the absolute magnitude of the star is the apparent magnitude plus 2. In other words, the absolute magnitude of the star is 1 + 2 = 3.

That's conceptually how we calculate absolute magnitude. However, we can condense it into a single step using a mathematical equation. The absolute magnitude, M, of a star is related to its apparent magnitude, m, and its distance, d, in parsecs, according to

If a star's distance is less than 10 parsecs, then its absolute magnitude is greater than its apparent magnitude because at 10 pc, it would appear dimmer than at its actual distance.

If a star's disstance is greater than 10 parsecs, then its absolute magnitude is less than its apparent magnitude because at 10 pc, it would appear brighter than at its actual distance.

### What makes a star luminous?

There are two factors that affect the luminosity of a star: (1) its radius and (2) its temperature. We can express the proportionality in the following way.

This means that if the radius of a star doubles, its luminosity increases by a factor of 4.

If a star's temperature doubles, then its luminosity increases by a factor of 16.

Likewise, we can compare stars of the same temperature. If Star A has a radius that is two times the radius of Star B, then Star A will have 4 times the luminosity of Star B (assuming they have the same temperature).

We can also compare stars of the same radius. If Star C has a temperature that is two times the radius of Star D, then Star C will have 16 times the luminosity of Star C (assuming they have the same radius).

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