Astronomers use the term apparent magnitude to describe how bright an object appears in the sky from Earth. The idea of a magnitude scale dates back to Hipparchus (around 150 BC) who invented a scale to describe the the brightness of the stars he could see. He assigned an apparent magnitude of 1 to the brightest stars in the sky, and he gave the dimmest stars he could see an apparent magnitude of 6. He did not include the sun, moon, or planets in his system.

The magnitude scale astronomers use today is based on Hipparchus' system, but has been expanded since the invention of the telescope. In this system, the brighter an object appears, the lower its magnitude. Some of the brightest objects (including the sun and planets) visible in the sky have negative values for apparent magnitude. The faintest objects detected with the Hubble Space Telescope have apparent magnitudes of 30.

The following table gives a list of some commonly known objects and their apparent magnitudes

When Hipparchus first invented his magnitude scale, he intended each grade of magnitude to be about twice the brightness of the following grade. In other words, a first magnitude star was twice as bright as a second magnitude star. A star with apparent magnitude +3 was 8 (2x2x2) times brighter than a star with apparent magnitude +6.

In 1856, an astronomer named Pogson formalized the system by defining a typical first magnitude star as a star that is 100 times as bright as a typical sixth magnitude star. In other words, it would take 100 stars of magnitude +6 to provide as much light energy as we receive from a single star of magnitude +1. So in the modern system, a magnitude difference of 1 corresponds to a factor of 2.512 in brightness, because

2.512 x 2.512 x 2.512 x 2.512 x 2.512 = (2.512)⁵ = 100

A fourth magnitude star is 2.512 times as bright as a fifth magnitude star, and a second magnitude star is (2.512)⁴ = 39.82 times brighter than a sixth magnitude star.

The following table shows how the difference in apparent magnitude between two stars (m₂ - m₁) corresponds to the ratio of their apparent brightnesses (b₁/b₂)

This relationship can also be shown by the equation:

(m₂ - m₁) = 2.5log₁₀(b₁/b₂)

Some examples to try:

1. Put these galaxies in order of magnitude from brightest to faintest:

NGC 4085: m = 12.94

M101: m = 8.30

M87: m = 9.60

IC1410: m = 15.94

NGC 5248: m = 10.97

2. How much brighter is a magnitude +2 star than a magnitude +4 star?

3. A variable star periodically triples its light output. By how much does the apparent magnitude change?

Answers:

1. M101, M87, NGC 5248, NGC 4085, IC1410

2. 6.31 times brighter

3. (m₂ - m₁) = 2.5log₁₀(3) ; (m₂ - m₁) = 1.19, so the star's brightness varies by 1.19 magnitudes

Absolute magnitude is a concept that was invented after apparent magnitude when astronomers needed a way to compare the intrinsic, or absolute brightness of celestial objects.

The apparent magnitude of an object only tells us how bright an object appears from Earth. It does not tell us how bright the object is compared to other objects in the universe. For example, from Earth the planet Venus appears brighter than any star in the sky. However, Venus is really much less bright than stars, it is just very close to us. Conversely, an object that appears very faint from Earth, may actually be very bright, but very far away. 

Absolute magnitude is defined to be the apparent magnitude an object would have if it were located at a distance of 10 parsecs. So for example, the apparent magnitude of the Sun is -26.7 and is the brightest celestial object we can see from Earth. However, if the Sun were 10 parsecs away, its apparent magnitude would be +4.7, only about as bright as Ganymede appears to us on Earth.

In practice, the magnitude of a celestial object is measured in certain wavelengths or colors using filters. This is because information about the color of stars is very useful to astronomers and gives them information about the surface temperature of a star.

The surface temperature of a star determines the color of light it emits. Blue stars are hotter than yellow stars, which are hotter than red stars. A hot star like Sirius, with a surface temperature of about 9,400 K emits more blue light than red light, so it looks brighter through a blue filter than through a red filter. The opposite is true of a cooler star such as Betelgeuse, which has a surface temperature of about 3,400 K. Betelgeuse looks brighter when viewed through a red filter than when viewed through a blue filter.

The color index of a star is the difference between the magnitude of the star in one filter and the magnitude of the same star in another filter. Any filters can be used for color indices, but some of the most common are B - V and V - R. B is blue wavelengths, V is green wavelengths and R is red wavelengths. Remember that magnitudes decrease with increasing brightness, so if B - V is small, the star is bluer (and hotter) than if B - V is large.

For example, for a star with B = 6.7 and V = 8.2, the magnitude in the B filter is brighter than the magnitude in the V filter, and B - V = -1.5. For values of B = 6.7 and V = 5.8, B - V = 0.9, and the star emits more green light than blue (this star would appear white).

The diagram below shows the intensity of light at given wavelengths that is emitted by a blackbody (an idealized dense object) at a particular temperature. The B and V color bands show where blue and green wavelengths lie on the curves.

The video below explains how a star's color is related to its temperature, and why we don't see green stars:

Apparent magnitude, absolute magnitude and distance are related by an equation:

m - M = 5 log d - 5

m is the apparent magnitude of the object

M is the absolute magnitude of the object

d is the distance to the object in parsecs

The expression m - M is called the distance modulus and is a measure of distance to the object. An object with a distance modulus of 0 is exactly 10 paresecs away. If the distance modulus is negative, the object is closer than 10 parsecs, and its apparent magnitude is brighter than its absolute magnitude. If the distance modulus is positive, the object is farther than 10 parsecs and its apparent magnitude is less bright than its absolute magnitude.

The following table gives values of d corresponding to different values of m - M.

Use the relationship between an object's apparent magnitude and absolute magnitude m - M = 5 log(d) - 5 to answer the following questions:

1. Suppose you were viewing the Sun from a planet orbiting another star 40 pc away. What would the Sun's apparent magnitude be? (The Sun has absolute magnitude M = +4.8)

2. Star A and star B are both equally bright as seen from Earth, but A is 60 pc away while B is 15 pc away. Which star is intrinsically brighter? By how much?

3. Star C has an absolute magnitude of 0.0, and an apparent magnitude of +14.0. What is the distance to star C?

Answers:

1. The Sun's apparent magnitude would be +7.8.

2. Star A is is 14.8 times brighter than star B.

3. 6309.6 pc (or 10^3.8pc)

Parallax and Stellar Parallax

Astronomers use an effect called parallax to measure distances to nearby stars. Parallax is the apparent displacement of an object because of a change in the observer's point of view. In the diagram below, as the observer moves between the two positions, he would see the same tree, but it would appear to move against the background.

Another way to see how this effect works is to hold your hand out in front of you and look at it with your left eye closed, then your right eye closed. Your hand will appear to move against the background.

This effect can be used to measure the distances to nearby stars. As the Earth orbits the sun, a nearby star will appear to move against the more distant background stars, in the same way the tree apears to move against the more distant mountains in the diagram above. Astronomers can measure a star's position once, and then again 6 months later and calculate the apparent change in position. The star's apparent motion is called stellar parallax.

There is a simple relationship between a star's distance and its parallax angle:

d = 1/p

The distance d is measured in parsecs and the parallax angle p is measured in arcseconds.

This simple relationship is why many astronomers prefer to measure distances in parsecs.

Limitations of Distance Measurement Using Stellar Parallax

Parallax angles of less than 0.01 arcsec are very difficult to measure from Earth because of the effects of the Earth's atmosphere. This limits Earth based telescopes to measuring the distances to stars about 1/0.01 or 100 parsecs away. Space based telescopes can get accuracy to 0.001, which has increased the number of stars whose distance could be measured with this method. However, most stars even in our own galaxy are much further away than 1000 parsecs, since the Milky Way is about 30,000 parsecs across. The next section describes how astronomers measure distances to more distant objects.

An example to try

A star has a parallax angle p of 0.723 arseconds. What is the distance to the star?

Answer

1/0.723 = 1.38 parsecs

While stellar parallax can only be used to measure distances to stars within hundreds of parsecs, Cepheid variable stars and supernovae can be used to measure larger distances such as the distances between galaxies.

Using Cepheid Variables to Measure Distance

Cepheid variable stars are intrinsic variables which pulsate in a predicatable way. In addition, a Cepheid star's period (how often it pulsates) is directly related to its luminosity or brightness.

Cepheid variables are extremely luminous and very distant ones can be observed and measured. Once the period of a distant Cepheid has been measured, its luminosity can be determined from the known behavior of Cepheid variables. Then its absolute magnitude and apparent magnitude can be related by the distance modulus equation, and its distance can be determined.Cepheid variables can be used to measure distances from about 1kpc to 50 Mpc.

For example, if an astronomer observed a Cepheid star with period of 34 days, comparing to previously measured Cepheids, its absolute magnitude is -5.65. If its apparent magnitude was +23.0, the astronomer could use the distance modulus equation:

m - M = 5 log d - 5

rearranged:

d = 10^{(m - M + 5)/5} parsecs

to find the distance to the Cepheid:

d = 10^{(23 - -5.65 + 5)/5} parsecs

d = 10^6.73 parsecs

d = 5.4 × 10⁶ parsecs

Using Type Ia Supernovae to Measure Distance

Type Ia supernovae are all caused by exploding white dwarfs which have companion stars. They can be distinguished from other supernovae because they do not have hydrogen lines in their spectra and have a strong Si II line at 615 nm. The gravitational pull of the white dwarf causes it to take matter from its companion star. Eventually it reaches a high enough mass (about 1.44 solar masses) that it cannot support itself against gravitational collapse and explodes. All type Ia supernovae reach nearly the same brightness at the peak of their outburst with an absolute magnitude of -19.3±0.03. They then follow a distinct curve as they decrease in brightness. So when astronomers observe a type Ia supernova, they can measure its apparent magnitude, knowing what its absolute magnitude is. They can then use the distance modulus to calculate the distance to the supernova, and the galaxy that it is in.Type Ia supernovae can be used to measure distances from about 1 Mpc to over 1000 Mpc.

Now that you have completed this section, you are ready to try the following activities:

• Plotting Supernova Light Curves