Astronomy 162: Professor Barbara Ryden
Tuesday, January 14
HOW BRIGHT IS A STAR?
``He that strives to touch the stars
Oft stumbles at a straw.''
- Edmund Spenser
- 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
(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
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
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).
- 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
(3) Apparent brightness depends on both luminosity
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
Important Equation: The INVERSE-SQUARE LAW relating
apparent brightness and luminosity
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.
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
Updated: 2003 Jan 14
Copyright © 2003, Barbara Ryden