When you observe a star, whether with your naked eye or a modern digital detector, there are really only two attributes that you can measure for that star:

Its apparent brightness. This is basically the amount of energy that the star deposits on a detector, whether its your eyeball or some pixel in a CCD detector. The reason you shouldn't look directly at the sun is not because the sun is "too bright" but rather the amount of energy received by your eye is very harmful to it.

Its apparent color. Even with your naked eye you can see that some stars are blue while others are red. Some of this color diversity is seen in the image below:

We will later learn that blue stars are hot (and usually young), with relatively short lifetimes and red stars are cool (usually old) and have long lifetimes. For now, however, all we care about is that stars do come in different colors.

Apparent brightness:

The apparent brightness of a star imaged with some detector (i.e. the amount of energy which it registers on the detector) is due to 3 things:

• The intrinsic energy output of the star (we will call this the Luminosity)

• The distance of the star from the detector (e.g. the distance between the earth and the star)

• The quality of the detector and/or the observing conditions.

While the first two items are the most important, it nevertheless is important to remember that different detectors can yield different information.

In class excercise Let's now do the following to learn about the relationship between exposure time and detection of a stellar image. Use the the provided worksheet for this one and the exercise at the end. Hand in the worksheet at the end of class.

Procedure:

Make measurements in a 10 x 10 box:
What is the minimum exposure time required to believe that you can even see the star with your detector.

What is the minimum exposure time so that the difference between the numbers in the green box labelled mean and red box labelled mean is approximately 100. Remember, the star is centered in the green box while the detector background (e.g. the brightness of the virtual night sky in this applet) is registered in the red box.

Case 1

Case 2

Case 3

Case 4

From this exercise, as well as those done previously, it should be apparent to you that for any given exposure of the sky with any given telescope plus detector there will be many stars that are simply too faint to register on the detector.

But the major thing to understand today, is the relationship between apparent brightness and the distance to the star.

The intensity of light observed from a source of constant intrinsic luminosity falls off as the square of the distance from the object. This is known as the inverse square law for light intensity.

The inverse square law for intensity

Thus, if you double the distance to a light source the observed intensity is decreased to (1/2)² = 1/4 of its original value. Generally, the ratio of intensities at distances d₁ and d₂ are

Because of this fall of in light intensity with distance, the intrinsic energy output of stars (also called their luminosity can only be known if their distances are known. Later on this term we will be measuring distances to nearby stars using the Parallax applet.

Thus, if you have a known distance to a star and are able to measure the flux at that distance, the total energy output of the star can easily be ascertained (this will be shown in class).

A simple way to think about this, is the following:

• Suppose you are in a large, darkened room and in the center of that room is a 100 watt light bulb. When you turn on the light bulb that is the only source of light there is. The light fills the room so the entire room contains all of the energy.

• Now let's suppose that you stand 1 meter away from the light bulb. Let's say you have a 1 square centimeter detector and you hold it up and you measure the energy incident on it. For the sake of this argument, let's assume that energy is equivelent to 4 million photons per second per square centimeter.

• Now you move twice as far away or 2 meters away. Since the surface area of a sphere (or the area of a 2D circle) goes as the square of the radius, the same amount of energy (or number of photons per second from the lightbulb) now has to go through an area 4 times larger. As a result, the flux of the light at 2 meters is down by a factor of 4, so you measure 1 million photons per second. If you were to move out to 4 meters, you would see 250,000 photons per sq. cm per second.

• The instrinsic energy output of the lightbulb (its total number of emitted photons per second) can be found by taking your measured flux and multiplying by the total number of square centimeters corresponding to the surface area of a sphere at your distance. The surface area of a sphere is:
  4pR²

  This means the total energy output is your measured flux multiplied by the surface area:

  So, at R = 1 meter f = 4 million photons per square cm per second ; the total surface area is 4p square meters (or 40,000 square centimeters) so the total Energy emitted is 40,000 x 4 million photons per second.

  Now notice, at R = 2 meters f is down to 1 million photons per square cm per second but the total surface are has increased to 16p square meters or 160,000 square centimeters. So the total Energy is 160,000 x 1 million photons per second, which is the same number as before.

  Thus if you can measure f at a known distance, then you know the intinsic energy output. The ability to measure f, however, depends upon the DETECTOR!

  Example of Inverse Square Law Applet (Point Mode)

  In the following, make sure the Point mode is selected (the applet starts up by default in area mode).

  First Set of Exercises: (Flux at specific Distance; distance = 36)

  Case 1

  Case 2

  Case 3

  Case 4

  Case 5

  Second Set of Exercises: (Limiting Distance)

  Case 1

  Case 2

  Case 3

  Case 4

  Case 5