Astronomy 162: Professor Barbara Ryden

HOW BRIGHT IS A STAR?

``He that strives to touch the stars
Oft stumbles at a straw.''
- Edmund Spenser

Key Concepts

• Luminosity is the rate at which a star radiates energy into space.
• Apparent brightness is the rate at which a star's radiated energy reaches an observer on Earth.
• Apparent brightness depends on both luminosity and distance.

(1) Luminosity is the rate at which a star radiates energy into space.

We know that stars are constantly emitting photons in all directions. The photons carry energy with them. The rate at which photons carry away energy from the star is called the star's luminosity. Luminosity is frequently measured in watts (that is, joules per second). However, since stars are so very luminous, it is more convenient to measure their luminosities in units of the Sun's luminosity, 3.9 x 1026 watts.

How can we determine the luminosity of a star? Unlike lightbulbs, stars are not stamped with a label proclaiming their wattage. Suppose you point your telescope at a star. You can determine the rate at which the photons from a star deposit energy within your telescope, but your telescope is very small and very far from the star, and thus collects only a minuscule fraction of all the photons which the star emits.

(2) Apparent brightness is the rate at which a star's radiated energy reaches an observer on Earth.

What you actually measure with a telescope (or with your eyes) is not luminosity, but a different quantity, called apparent brightness. The apparent brightness of a star is the rate at which energy (in the form of light) reaches your telescope, divided by the area of your telescope's mirror or lens. (It is important to normalize the result by dividing by the area of the mirror - all other things being equal, a mirror twice as big will collect twice as much energy.) Apparent brightness is thus measured in watts per square meter.

For instance, the apparent brightness of the Sun is b = 1370 watts/meter2. That is, if you had a perfectly efficient solar panel one meter on a side, if you held it perpendicular to the Sun's rays, it would generate 1370 watts of electricity. (In practice, of course, the Earth's atmosphere absorbs some of the sunlight, and solar panels are not perfectly efficient, but the Sun is still a potent source of energy, even at a distance of 150 million kilometers.) The star with the next highest apparent brightness is Sirius (in the constellation Canis Major). The apparent brightness of Sirius is b = 10-7 watts/meter2. (To light up a 10 watt bulb with the energy of Sirius, you'd need a solar panel ten kilometers on a side.)

Another method of describing apparent brightness, which you may encounter if you read popular astronomy books, is the apparent magnitude scale. The `apparent magnitude' system goes back to the time of the ancient Greeks. The Greek astronomers noted that stars have different apparent brightness. The very brightest stars they could see were called `stars of the first magnitude'. The very faintest stars they could see were `stars of the sixth magnitude'. Stars of intermediate apparent brightness were of the second, third, fourth, and fifth magnitude. With the invention of the telescope, the magnitude system was extended to stars of lower apparent brightness -- seventh magnitude, eighth magnitude, and so forth. The magnitude system can also be extended to objects of higher apparent brightness.

Examples:

• The Sun has an apparent magnitude m = -26.7
• Sirius has an apparent magnitude m = -1.4
• Proxima Centauri has an apparent magnitude m = 11.0
Note that the system of apparent magnitudes is bass-ackwards, with bright objects having small apparent magnitudes. During ancient times, magnitudes were seat-of-the-pants estimates. More recently, the magnitude system has been systematized so that a difference of 5 magnitudes corresponds to a factor of 100 in apparent brightness. (Thus, a star with m=6, just bright enough to be seen with the naked eye, has an apparent brightness 100 times greater than Proxima Centauri, which has m=11).

(3) Apparent brightness depends on both luminosity and distance

How DO you find a star's luminosity?
• (1) Measure the apparent brightness of the star.
• (2) Determine the distance of the star (from parallax).
• (3) Compute the luminosity (from the inverse-square law)

b = L / ( 4 pi d2 )

• b = apparent brightness of the star (in watts/meter2)
• L = luminosity of the star (in watts)
• d = distance to the star (in meters)
• pi = approximately 3.14159265 (but you knew that already)

If you rewrite the inverse-square law in the form L = 4 pi d2 b, you can compute the luminosity from the distance and apparent brightness.

Example:

Sun: b = 1370 watts / meter2

d = 1 AU = 1.5 x 1011 meters (150 billion meters)

L = 4 pi d2 b = 3.9 x 1026 watts (390 trillion trillion watts)

By measuring apparent brightness and distance for stars in our immediate neighborhood, we see that stars have a wide range of luminosities. Most stars in our neighborhood are less luminous than the Sun. (Some, such as the luminosity-challenged star Proxima Centauri, are less than one ten-thousandth the luminosity of the Sun.) A few stars, however, have very high luminosity. The most luminous stars in our galaxy are a million times as luminous as the Sun.
Prof. Barbara Ryden (ryden@astronomy.ohio-state.edu)

Updated: 2003 Jan 14