Astronomy/Gauging the distant to stars
I am aware of stellar parallax method. But my problem is that I don't understand how astronomers measure the distance of earth to stars that are farther than 500 light years. Can you explain how they decode the star light to understand the size and the distance of the Earth to those stars? I have read about this on Wikipedia and elsewhere and I still do not get the idea of how the spectrum of light is related to the distance.
As you obviously know, the stellar parallax method becomes unreliable for stars at great distances, because the uncertainty of the measurements is at least 0.002 arcsec, and even for stars at a distance of 500 light years that is about a third of their parallax, meaning that their distances have an average uncertainty of 30 to 40%. So if a star is really 500 light years away its parallax measurement might yield values anywhere between 300 and 700 light years -- not terribly satisfactory, if you want accurate values. Of course if the distance is less the parallax is larger, so for a star only 250 light years away, parallax measurements should yield values between 200 and 300 light years, which is a much better value, though still not "precise".
As a result, for stars at larger distances we cannot, in general, directly measure their distance. Instead, we have to rely on special circumstances. If the star is in a cluster, we can measure the apparent brightness and color index of the stars in the cluster, and use those to construct a Hertzsprung-Russell Diagram. (See the diagrams at the end of Stellar Properties on my website, at http://cseligman.com/text/stars/stellarproperties.htm
for the examples discussed in the next paragraph.)
For stars of known distances, as shown in the first diagram referenced above by blue dots, we can see that the majority lie on what we call the Main Sequence, a diagonal line in which hotter stars are brighter. For stars in a cluster, as shown in the second diagram referenced above, we find a similar situation. For stars of known distance, their apparent brightness can be compared to their distance, and adjusted to account for how far away they are, to calculate the "absolute magnitude", the brightness they would appear to have if they were 10 parsecs (32.6 light years) away. For stars with a color index (a measure of how blue or red they are, and since hotter stars are bluer and cooler stars redder, how hot they are). The two diagrams on the page referenced show that for stars with a color index or temperature similar to that of the Sun, the absolute magnitude is about 5. That is, if the Sun or a similar star were 10 parsecs away, it/they would appear to be about 5th magnitude (too faint to see in bright skies, but easily visible even to the naked eye in dark skies, and easy to see with binoculars or telescopes even in brighter ekies).
Now, in the diagram for M55, the stars similar to the Sun are shown as also having an absolute magnitude of about 5. That is, it is assumed that since similar to the Sun in terms of their temperature (determined by studies of their spectral features or "spectral class", as discussed in the top half of the page in question) or color index (as shown in the Hertzsprung-Russell Diagrams), they must also be similar in their brightness. There is quite a spread of measured brightness and color index in the "HR" diagrams, but the average position of the diagonal is relatively accurate -- certainly, far better for distant stars than the very uncertain parallax measurements.
What is done for such "cluster" stars is to measure their apparent brightnesses (called m, the "apparent" magnitude), construct an HR Diagram (as shown in the diagram for M55), and compare it to the diagram for nearby stars, changing the brightness meausrements until they match the brightnesses that nearby stars would have at a distance of 10 parsecs (the "absolute" magnitude, called M). The difference between the two values, M - m, is called the "distance modulus". For stars in an imaginary cluster at a distance of 10 parsecs, the two numbers would be the same; but if the cluster was 100 parsecs away the stars would look 100 times fainter, and m would be 5 less than M, so if we find that a cluster has a distance modulus of 5, we can say that the stars in the cluster must be about 100 parsecs (about 325 light years) away. There is some uncertainty in this result because of the vertical spread of brightness measurements, but it is still at least as good if not far more accurate than parallax measurements for stars at such distance. In the case of a distant cluster the distance modulus might be as much as 15 or 20, meaning that stars like the Sun would have apparent magnitudes of 20 or 25, indicating that the cluster was about 10 thousand light years away (for M - m = 15), or as much as 100 thousand light years away (for M - m = 20).
There are obviously two problems with this method: first, at very large distance the stars may be too faint to accurately observe the brightness and color index, and it only works for stars in clusters. For so-called "field" stars we can measure their color index and brightness and by assuming that they are on the Main Sequence, guesstimate their distance; but some stars are much brighter than Main Sequence stars, and unless there are clues in their spectra that suggest that, their estimated distances will be well off the mark.
So you are correct in wondering how we can measure how we can measure the distance of distant stars, when parallaxes start to fail at distances of 500 light years or less. The answer is, we can't be at all certain of their distances unless there are special circumstances that allow us to use their apparent brightness to estimate their absolute magnitude, and from their distance modulus, estimate how far they must actually be.
Fortunately, there are a few stars for which special circumstances allow us to make such estimates. In particular, Cepheid variables (discussed on my website at http://cseligman.com/text/stars/variables.htm
) happen to have a strong relationship between their period of variability and their brightness. The circumstances that cause them to get brighter and fainter are in some ways like a pendulum, swinging back and forth from one unstable position to another, but with a period related to its length (longer pendulums take longer to swing back and forth). Similarly, it turns out that larger Cepheids take longer to vary in brightness than smaller ones, and as it happens the cause of their variability depends on their temperature, so all Cepheids have about the same temperature, and as a result about the same brightness per square foot. So bigger ones, which are brighter, take longer to vary in brightness, while smaller ones, which are fainter, take less time to vary in brightness. This is the method that was used by Hubble to measure the distance of the Andromeda Nebula and other nebulae that were what we now call galaxies. There are problems with this method, as it turns out there are two types of Cepheids, a more common fainter kind and a less common brighter kind, and in our galaxy we mostly see the more common fainter ones, and in the Andromeda Galaxy we mostly see the less common brighter ones, so Hubble thought the Cepheids he observed were fainter than they really are, which gave him an estimated distance several times too small. But over time we have developed ways of recognizing this difference, and Cepheid variables can now be used to measure distances of galaxies up to about 50 million light years away. Note that this does not mean that we were able to measure the distance of the other stars in the galaxies directly. We just assume, given the apparently small size of such galaxies, that all their stars must be at about the same distance from us, so if we can infer the distance of some stars in those galaxies (either from Cepheids, HR Diagrams or other methods), then we can infer the distance of all the sars in those galaxies.
There is undoubtedly a discussion of the galactic (or perhaps intergalactic) distance scale in one of the references you have previously consulted, so I will close for now, as this answer is a bit rushed due to my having to go out fairly soon, and is not as thorough and perhaps not as clear as I would hope it might be if I had more time to go over the topic. But you can see that although for stars much more than a few hundred light years away you are correct in feeling that there is no way to be sure for any given star that a stellar parallax measurement will give us a correct measurement of its distance; and in fact for the vast majority of stars even in our own galaxy, unless there is some special circumstance, their estimated distances are just that -- estimates, and not actual measurements. But there are enough clusters, Cepheid variables, stars that be reasonably stated to be Main Sequence stars even though not in a cluster and other special circumstances, that although for a given star at a large distance its stated distance may be wrong, many of the more distance stars have estimated distances that are more accurate than not.