Name: Partner(s): Day/Time: Version: intro

# Brightness and Surface Brightness

A tapestry jewelled hangs over the night;
Have you looked up to see where it gleams?
There are rubies and sapphires and diamonds white
Interwoven with mists of lost dreams.

--Cordella Lackey, Sky Tapestry

## Overview

• Understand Luminosity, Brightness, and Surface Brightness
• Understand the Inverse Square Law for Light

### Luminosity and Brightness

An object's luminosity is its intrinsic rate of emitting energy in the form of light. The standard, metric unit of energy is the joule, so luminosity is measured in joules/second, which are known as watts. You know that the luminosity of light bulbs is given in watts, so this describes the intrinsic amount of energy per second emitted by the bulb, which is independent of the distance to any observer. Brightness is the amount of light that we detect from an object. As described below, the detection of this energy does depend on the distance to the light source. Our light detector may be our eye, a digital detector inside a camera, a photographic emulsion, or any other device that records the reception of light. Note that every light detector receives light on a given area, over which it collects and detects light.

As light is emitted from a source, it usually spreads out uniformly over a given spherical area (see Figure 1). This spherical area = 4πD2, and so it increases with D2, the square of the distance D from the source. The source appears equally bright at all positions on the surface of the sphere around it. This brightness B is the source's luminosity L distributed equally on the area of this sphere:

Equation 1

Thus we measure brightness in watts/cm2.

Now consider viewing the source at a distance 3 times farther away. The same amount of light must be diluted to illuminate the sphere at the new distance D2, and the new sphere has an area of 4πD22. How does the new brightness B2 compare to the original, B1?

Equation 2

We see that only the distance between the source and the observer matters. Since D2 = 3D1, we see that

Equation 3

This means that since your detector has a fixed light-collecting area (your eye, camera, etc.) you receive only 1/9th the energy on your detector at a distance that is 3 times farther from the source (see Figure 1). In otherwords, brightness decreases as the distance squared from the source. This is known as the Inverse Square Law for light.

Figure 1: Inverse Square Law. Each individual square in the figure represents the area of the detector. The farther the source, the larger the area over which the light spreads, proportional to the square of the distance to the source.

### Magnitudes

In addition to watts/cm2, brightness of astronomical objects is often measured in magnitudes. This system in use today was basically invented by Hipparchus, a Greek astronomer from the 2nd century BC.  He divided all stars observable to the naked eye into six classes, with 1st magnitude stars being the brightest, and 6th magnitude being the faintest that he could see. In the 19th century, photographic technology enabled astronomers to measure brightness more accurately, and the magnitude scale was revised to a precise mathematical definition. The eye's response to light is not a simple linear relation, and so to preserve Hipparchus's magnitude scale, astronomers define 5 magnitudes to be a ratio of 100 in brightness, according to the formula:

Equation 4

Where B is the brightness and m is the magnitude. Equation 4 can be re-arranged as:

Equation 5

Equation 5 shows us that 1 mag is roughly a factor of 2.5 in brightness.  For example:

• If m1 - m2 = 1, Star 1 is 2.5 times fainter than Star 2. Recall that larger numbers are fainter!
• If m1 - m2 = 5, Star 1 is 100 times fainter than Star 2.
• If m1 - m2 = -3, Star 1 is 16 times brighter than Star 2.

Notice that the magnitude scale is relative; it only compares the brightness between objects. So a fixed reference is needed, and the star Vega was chosen as this reference. It is defined to have a magnitude of 0.0. There are around a dozen stars with brightnesses < 1 mag. Excluding the Sun, the star Sirius has the brightest apparent magnitude, at -1.46 mag. On this scale, the faintest stars visible to the naked eye are still around magnitude 6, although under perfect conditions experienced observers can see down to 7th magnitude. In most areas, light pollution from poorly designed outdoor lighting makes the limiting magnitude, the faintest visible magnitude, much smaller.

The eye’s response to light actually depends in part on the color of light. For example, red lights don't shut down the eye's receptors as much as blue, so we can use a relatively bright red light and still maintain our ability to see when the lights are turned off. This may also affect our perception of an object's brightness.

### Surface Brightness

In addition to point sources like stars, there are many other light-emitting objects like comets, nebulae, the Milky Way, and even the sky itself. Light from these objects appears to come from an extended surface rather than just from a point, so they are called extended objects.  Since the light from these objects is spread out over a small area of the sky, astronomers measure their surface brightness

The surface brightness is a measure of brightness per area on the sky. Since distance on the sky is measured as angular distance, in degrees, minutes, and seconds of arc (see the Coordinate Systems activity), an area on the sky is measured as an angular area, for example, in square degrees or square arcseconds. Imagine taking one of the Pointer Stars in the Big Dipper and spreading its light over a 2 arcsecond by 2 arcsecond square of the sky. This region has an area of 4 arcsec2. Thus, the surface brightness is given in mag/arcsec2. Since the light is spread out, the average brightness from any point on this surface is much fainter than the star as a point source. Therefore, the surface brightness is fainter than the magnitude of the star.**

Glowing clouds of gas called nebulae often have a surface brightness around 16 – 22 mag/arcsec2. The Milky Way has a surface brightness around 21 mag/arcsec2. While the naked-eye detection threshold for point sources is around 6th mag, the detection threshold for extended sources depends more on the relative surface brightness of the object compared to the background sky. You will explore this in the planetarium.

The sky surface brightness varies depending on altitude, humidity, the amount of light pollution, color, and more (details are available under Additional Resources below.) We can estimate the sky glow based on the magnitude of the faintest star visible to the naked eye.

**Important: we cannot divide the star's magnitude by the angular area to obtain the surface brightness. This must be obtained from the brightness in watts/cm2 instead, and is beyond the scope of this activity.

Table 1
NELM SB (mag/arcsec2)
0 13
1 15
2 16
3 17
4 18
5 19
6 21
7 23

Table 1 gives the approximate surface brightness of the sky for several values of the Naked Eye Limiting Magnitude (NELM). To estimate the NELM, we will use charts designed by the GLOBE at Night program. With these charts, match the visible stars in the constellation Orion to one of the pictures. The picture number corresponds roughly to the NELM. You can practice doing this at the website: http://www.globeatnight.org/observe_practice.html.

An extended object must have a brighter surface brightness than the sky for it to be visible to the naked eye.  A sky of 18 mag/arcsec2 will completely wash out a galaxy whose surface brightness is 20 mag/arcsec2. However, this is partly dependent on the color.  Looking at Orion in a dark sky, most observers can see that Betelgeuse is a red star and Rigel is a blue star, and that they are similar in brightness.  In a city where streets are lit with yellow sodium vapor lights, the sky tends to have an orange color.  In that case, Betelgeuse shows less contrast with the orange sky than Rigel, so Rigel will appear brighter.  Similarly, a reddish nebula with a surface brightness of 18 mag/arcsec2 may be completely invisible in a 20 mag/arcsec2 sky because its

Figure 2: The field of view covers an area on the object that is 9 times larger when the same object is at a

Whereas the brightness of an object depends on its distance, the surface brightness is independent of distance. At a greater distance from the observer, the same field of view will include a larger area on that object, as shown in Figure 2. However, recall that the brightness of the light received from that object decreases with the square of the distance, as discussed above. Therefore, although light from a larger area on the object is received in the same field of view, the brightness of that light is reduced by the same factor, so that the surface brightness stays the same. For example, Figure 2 shows that if the same object is three times farther, the same field of view includes 9 times as much area on the object, but the brightness of the object is 9 times less. Thus, the surface brightness, which is the brightness per area, of the object remains the same, regardless of its distance.