All detectors have noise . Thus each pixel has a noise associated with it. This noise comes from the electronics in the digital camera itself as well as from the large scale background against which the image is taken. If the signal from a faint star is below the noise level of that pixel or a region of pixels, it will not be detected! A detection means that the signal from the source has risen about the noise of the detector. Things that compromise the detection of faint stars are:

• Poor image quality where the image of the star is diffused over too many pixels. This is why observations from the Hubble Space Telescope are better than from the ground, because the atmosphere no longer smears the image of the star over the detector.

• Too bright of background. Observations taken in the full moon have a higher background and therefore a higher level of background noise which effectively swamps out the signal from fainter stars.

• Insufficient exposure time: If you don't expose long enough, the intrinsic noise of the digital camera will compromise the ability to detect faint sources.

Stars can also appear faint on detectors and/or escape detection if they are too distant. The light from a star decrease as the inverse square of the distance. So, if two stars have identical intrinsic brightness, but one of them is 3 times farther away, then that star will appear to be 9 times fainter on the detector. We simulated this with the inverse square law applet and learned that there are limiting distances beyond which stars can not be detected. We will return to this issue below via some transmitter questions as we have not yet really covered this.

Remember, the light from stars is superposed on the background diffuse light of the night sky (which has multiple sources). Thus to measure the brightness of a star on your detector, you must subtract off the background.

Atoms emit and absorb photons because of various energy levels that the electron can occupy. Each transition between energy levels in an atom produces a photon of energy exactly equal to the energy difference between those levels. Since each chemical element has a unique set of energy levels, then each chemical element has a unique spectral fingerprint.

These electronic transitions give rise to spectral lines with exact wavelengths. We used the elements applet of measure wavelengths for certain atoms in the periodic table of elements. Later we were able to sleuth out what kinds of elements where in a particular stellar atmosphere by matching the wavelengths recorded in the absorption line spectra to the wavelengths of the elements in the laboratory spectra. In this way, we can actually determine what kinds of elements are in stellar atmospheres.

The energy distribution of stars is well approximated by a blackbody law. Thus the peak emission is inversely proportional to wavelength . This means that the color of a star is an indicator of its temperature. In astronomy, we measure "color" by using filters and measuring the relative amounts of light in say a blue filter compared to a red filter. This is what the second homework assignment is concerned with. Measuring these index colors as a function of temperature.

The strength of certain absorption lines of certain elements of stars depends upon the temperature of the stellar atmosphere. We have experimentally determined this in class using several applets. This has lead to the OBAFGKM spectral classification system which, as we have determined is a Temperature Sequence in the sense that O stars are the hottest and M stars are the coolest.

• We thus have two methods of determining the atmospheric temperature of a star: 1) Spectral classification or 2) B-V or V-R color.

Back now to the inverse square law. When we determine the absolute brightnesses of stars we will be working in units of Solar Luminosities. 1 Solar Luminosity is the energy output of the Sun. A star that is 10 times as bright as the Sun would then have 10 Solar luminosities. This is the intrinsic energy output of the star.

What we of course measure on our detectors (those pesky digital things) is the energy deposited by the star light. This is the flux.

Flux is measured in energy deposited per square centimeter on the detector every second.

The flux received by our detector is governed by two primary things:

• the intrinsic energy output of the star

• the distance to a star.

Here we have two stars on our detector.

Star 1 as measured through a 3x3 box has a mean flux of approximately 6600 - 1100 = 5500 units (you should verify this!).

Star 2 has a mean flux of approximately 1500 - 1100 = 400.

So star 1 has a flux that is approximately 15 times higher than star 2. Does that means that star 1 is intrinsically brighter than star 2? there is no way of knowing without measuring distances. Maybe star 2 is really much brighter (intrinsically) than star 1 but its very very far away and so appears to be fainter on our detector.

But we do know that the flux received by a star of fixed intrinsic brightness falls off as the inverse square of the distance to that star. This is shown in the following table:

1. If the star in the above table had a distance of 5 light years, the flux received by the detector would be:
   1. 1600 units
   2. 2400 units
   3. 3200 units
   4. 4800 units
   5. 6400 units
   Now consider this scenario
   
   

   Star	Distance	flux received by detector	Relative Brightness to Star A
   A	10 Light years	1600 units	1
   B	20 Light years	1600 units	4
   C	40 Light years	1600 units	16
   D	80 Light years	1600 units	64

2. Star E has a distance of 40 light years and a measured flux of 3200 units. Relative to Star A, the brightness of star E is
   1. the same
   2. 2 times greater
   3. 16 times greater
   4. 32 times greater
   5. 64 times greater

   You observe 4 stars in the sky. Their properties are summarized in the table below:

   Star	Flux on Detector (Arbitrary Units)	Distance (Light Years)	Observed Color
   A	2	2	Blue
   B	4	4	Red
   C	5	4	Very Blue/White
   D	16	1	Yellow

3. Which of these stars has the coolest surface temperature:
   1. A
   2. B
   3. C
   4. D

4. Which of these stars has the highest luminosity (i.e. which is producing the most energy)?
   1. A
   2. B
   3. C
   4. D

5. Which of these stars has the lowest luminosity?
   1. A
   2. B
   3. C
   4. D

6. What is is the ratio of luminosities between Star D and Star A
   1. they are the same
   2. 2 to 1
   3. 4 to 1
   4. 8 to 1

   Measuring Stellar Distances

   

   Figure 1.3 Schematic Representation of stellar Parallax. Distant stars act as a fixed reference coordinate system. Nearby stars, when observed 6 months apart, will show a small movement with respect to the background of fixed stars. At position 1, the nearby star would be viewed against a background that contained star B while 6 months later, at position 2, the nearby star would be viewed against a background that contained star A.

   In class demo with real students here.

   Some notes about historical use of stellar parallax that illustrates an important principle of science:

   Tycho Brahe (1546-1601) realized that it was possible to actually prove that the Earth revolved around the Sun if the stars are a distant fixed background. The method is known as stellar parallax which is the reflection of the annual motion of the Earth around the Sun as measured against the fixed background of stars. The principle of stellar parallax is shown in Figure 1.3. When observations of a nearby star are made over a timescale of 6 months (over which the Earth has completed 1/2 of its orbit about the Sun), the position of that star appears to shift with respect to the fixed background of more distant stars. It was this shift that Tycho was determined to measure but he did not have a telescope. Instead, he made very accurate naked eye measurements. However, he failed to detect any positional shifts in the stars and, in his mind, was therefore unable to confirm the concept of stellar parallax.

   Based on his failure to detect stellar parallax, Tycho concluded that the Heliocentric model must be incorrect. In so doing, Tycho practiced very bad science as the alternative explanation,that the stars are too far away to allow for a naked eye determination of stellar parallax, was apparently never considered. This is an important lesson in cosmological model making. Many predicted effects are often below the sensitivity of the observing equipment. Just because the effects aren't detected doesn't mean they aren't there. Remember the examples of the CCD detector and noise that we did earlier in the class. If the signal to noise is too weak, then the object is not detected.

   
   
   

   

   Parallax measurements are difficult from the earth because

   the angle is very small because the stars are very distant (first parallax measurement was done in 1839).

   position measurments of stars are "blurred" out by our atmosphere. Therefore it takes years of measurment to beat down this error.

   Parallax with no error

   Parallax with realistic error