How and why is the spectrum of a star related to its surface temperature?

Answer 1

Good question!

The spectrum of a star is related to its surface temperature in at least two ways, (i) through its overall colour, changing from "red" to "blue," etc. as it gets hotter, and (ii) the particular pattern of "spectral lines" observed in its own spectrum. The presence of particular lines, and their strength, reflect both the abundance of the element making such lines in the star's surface regions, and the temperature there.

That's a very short answer; a much longer answer (not for the faint-hearted!) follows

To go further, and delve into many more details, we first need a few preliminaries.

Preliminaries:

1. The surfaces of stars have an "effective temperature," T_eff, related to how big the star is and how much energy is coming from below to be radiated away. Ignoring the technical definition of that, let's simply say that the physical temperature T at the surface layer where the average photon has a 50-50 chance of making it out into space is close to this idealized T_eff. (This 50-50 place is what we generally mean by a star's "surface." Its technical name is the "photosphere," for "the sphere where the observed photons come from.")

2. Light, temperature, and colour. For an idealized, hot "Black Body" cavity (another technical definition left aside), the intensity of light as a function of frequency or wavelength has a characteristic shape for its distribution, peaking somewhere in the middle of the range for either of these two variables.

Towards the end of the 19th century, physicists became very interested in (i) just what this characteristic shape was, (ii) how it changed with temperature, and (iii) why it existed. They studied everything from how a homely iron poker heats up (going from initial grey-black through colours like dull brown, red, orange, yellow ...) and even such things as bright as the intense light coming out of small doors opened in the hottest possible steel furnaces. (Those furnaces reached the closest we could then get to stellar surface temperatures; they were also the nearest thing available to the idealized "Black Body cavity." )

What they noticed was that, by and large, for things most like ideal Black Bodies, the characteristic shape of the distribution got stretched out linearly with temperature, T in the frequency direction (that is, the stretching was proportional to T), while it was increased in the intensity direction by a factor of T^3. (Frequency and intensity are respectively the x- and y-axes of graphs plotting these variables. I may be simplifying historically here; if you plot wavelength and intensity, as might first have been done, the dependencies are different. Sometimes scientific understanding only becomes clearer when the "best" or "right" variables are employed; choosing those can be an art in itself.)

Because of the "stretching out" of the overall distribution in frequency with T, the peak itself shifted over in that same definite way. That is, the peak's position in frequency was simply proportional to the temperature (or in wavelength, inversely proportional to temperature). This peak position and the shape as it crosses wavelength regions to which we're sensitive helps determine the "colour" we'd associate with a Black Body of a particular temperature.

(Just how the position of the peak and/or the distribution of the intensity on either side of it and/or many other factors ultimately get interpreted as "colours" is far beyond my expertise; can we take that as read?)

Max Planck was the first person to understand this characteristic shape and its general properties, around 1900. (He was in a good position to do this, acting as what we would today call a consultant for Siemens, the big German company. Among other things, Planck had learned that experienced steel workers would just throw open the small door in their furnace walls, to judge the temperature by the colour and intensity of the radiation! The right temperauture is needed to make good steel.)

Planck's solution to the problem of light's distribution under these circumstances involved the introduction of the idea that light was "quantized." He imagined that the cavity's walls had liitle light absorbing or emitting "oscillators" in them. Energy exchanges between light in the cavity and these imagined "oscillators" occurred in bits (or "quanta") with Energy proportional to Frequency (later immortalized as E = h nu, where nu is a Greek letter physicists use for frequency; h is Planck's Constant, named after him, of course.) This was an utterly bizarre concept to classical physicists, but Planck had concluded that something bizarre was needed, partly for the following reasons.

Since the mid-19th century, it had been known that the TOTAL amount of radiation batting back and forth in unit volume of any idealized hot cavity at a given temperature, T, was proportional to T^4, that is, it was "aT^4" where 'a' was some new constant with an initially unknown value. (This was a remarkable, but now considered fairly straightforward consequence of Classical Thermodynamics!) Indeed, the relationship of this "internal radiation energy" to the amount that would come out of a given small area opened up in the wall, had also been established theoretically. Because of this, by Planck's time they already knew fairly well the experimental size of the constant multiplying T^4, though they still DIDN'T know why it had that particular value. (I'll return to this point later. But note how the two stretching factors I previously mentioned --- T and T^3 --- when multiplied together DO give T^4 for the area under the intensity curve, or total radiation! That alone strongly supported the theoretical deivation for the total energy content already reached by purely classical arguments.)

I'm now close to the fundamental reason why the distribution of light in an idealized Black Body spectrum goes DOWN for high frequencies, having always been increasing for frequencies up to that point. (A microscopic classical approach said it should always go UP as frequency increased, indeed without limit --- the bizarre and so-called "ultraviolet catastrophe"!) The "ultraviolet catastrophe" tells us that something was seriously wrong with the detailed, microscopic classical argument.

First, let's state something fairly obvious: There really IS a finite amount of radiation per unit volume for an idealized cavity, for any given value of T. So: something was wrong with the microscopic classical argument, which gave an infinite result. What should replace it?

Well, if you think about distributing that fixed amount of energy over all possible frequencies, Planck's idea was that it would cost you so much to put any of that energy into very high frequencies (with E prop. to frequency), that the PROBABILITY of actually having radiation with such a high frequency would be very low, indeed EXPONENTIALLY LOW as frequencies increased without limit! And THAT'S why there's a "characteristic" exponentially decaying tail to light's distribution at high frequencies, no matter how high the temperature, a tail completely unexpected in classical physics.

Most physics students and all professional physicists know this much. What is less well known is that, because of his connection with Siemens, Planck had played around with mathematically representing the observed shape of light's spectrum, and had noticed that the data COULD be fitted by a certain blend of two limiting functional forms --- it was "curve fitting" of the highest order! After a few trial attempts, his empirically GUESSED form finally fitted SO well, that he just KNEW he had to come up with some very new, and probably very original hypothesis to explain it. At the least, he had to show how this strange shape was somehow a natural consequence of some new hypothesis.

The QUANTUM was that hypothesis, and his use of the mathematics of thermodynamics and statistical mechanics with individual energy packets proportional to frequency PROVED that this hypothesis fitted the data almost perfectly. What's more, from his work he could in fact do two more things: (i) calculate the explicit value of 'h' needed to fit the observed data, and (ii) calculate the value of the 'a' in "aT^4," the "radiation energy density." (It ALSO fitted the previously known empirical value for 'a' well.) His later Nobel Prize was in the bag!

One more thing with these preliminaries:

COLOUR only exists for Black Body or near Black Body radiation in our world because of the existence of the quantum of light! Leaving aside the quandary of the ultra-violet catastrophe, classical physics predicted the same RELATIVE amounts of energy at any given frequency or wavelength, completely INDEPENDENT of temperature. Since our physiological responses are LOGARITHMIC, every Black Body would have the same "colour" (a kind of "washed-out white") and the same would generally hold for the stars.

Enough of that. NOW we can FINALLY address the stars and their spectra:

3. The overall shape of the spectrum: As T_eff or other measures of temperature in the surface regions of a star go up, the radiation has a distribution quite close to that of an ideal radiator of a similar temperature, T say. (The reason that it's only close is because stars are "leaky" rather than "ideal radiators" --- the energy at any layer is leaking out towards the surface and ultimately into space. That makes the outermost visible layers in particular less and less like an ideal radiator --- a greater fraction of their own internal energy content is being radiated away per second.)

However, overall, the general shape of the distribution mimics that of the comparably hot, equivalent "Black Body" radiator. When T_eff is cool, the peak is at low frequencies (large wavelengths) which give dull red colours; when T_eff is hot, the peak is at high frequencies (small wavelengths), giving bright blue, ultraviolet or "white-hot" colours. So much for the general T_eff, colour characteristics.

4. Spectral lines: The electron clouds around different kinds of atom have their own characteristic, distinct internal energy levels --- another, but later explained consequence of the quantum theory. Both absorption or emission of energy by the electron clouds can only involve transitions between these discrete energy levels, that is specific energy differences. By Planck's Law, those specific energies correspond to specific frequencies of light. Absorption or emission of energy by the electrons produces less energy in the radiation field, or more, respectively. Thus, the observed consequence is either "dark lines" in the spectrum (where some radiation is missing) or "bright lines" (where there's extra radiation). Which kind dominate depends upon circumstances.

For radiation coming from stars at the most probable frequencies or wavelengths, absorption tends to dominate. Speaking very loosely, the cooler material further out absorbs some of the hotter radiation trying to pass through it. (Experts will know that this is a crude oversimplification.)

However, how large this effect will be is also a function of having the relevant atoms present, in whatever abundances will actually produce the particular lines. The actual lines seen for given T_eff's, and the way that a particular line would itself generally be (say) weak at low T_eff's, get stronger (i.e. darker) as T_eff increased, and then fade away for higher T_eff's, was all itself a BIG mystery.

That mystery was brilliantly solved, out of the blue, by an Indian physicist/astrophysicist named M.N. Saha, in1920. Again, by using statistical mechanics and quantum ideas, he showed that:

(a) As T increased, electron clouds initially became "more excited", producing a greater population of atoms primed to absorb light of the visible frequencies. (If there aren't such "properly primed" atoms around, there's nothing there to absorb the light and thus darken the spectrum in that frequency or wavelength region.)

(b) However, as T_eff (or the material's T) increases, another consequence comes into play. Atoms ionize, and ionize increasingly, stripping off their successive electrons! So, even though a greater fraction of atoms in any given stage of ionization are becoming "primed to absorb," there come to be fewer and fewer of them in total as T goes up; they're being ionized to the next state (with one less electron) --- whose characteristic lines will be quite different. The limit comes when that kind of atom has been completely ionized, leaving no electrons to take light out of the radiation field.

The combination of these two effects --- broad changes in colour with temperauture and broad changes in the kinds of spectral lines present --- explains in overall terms everything seen in typical stellar spectra.

The "Saha Equation" and its consequences describe all this. These methods are still used to this day.

When Saha burst upon the scene with his explanation in four major papers in 1920, all the world's leading astrophysicists recognized that it was a truly major contribution to our understanding. His work was certainly "of Nobel Prize class," but, there being no Nobel prize for astronomy or astrophysics, he did not receive it. (In recent years astrophysicists have occasionally received the Nobel Prize, but only under the Physics heading, and probably against the spirit of Alfred Nobel's will. But that's another story.)

With Saha's work, a quantitative attack could now begin upon interpreting stellar spectra in terms not only what kinds of atom were present in stellar surfaces, but also in what proportion.

A footnote: Early in the 19th century, a famous philosopher declared that "What the stars are made of will remain forever the essence of the unknowable." Within twelve years the German physicist Fraunhofer began his study and first interpretation of spectral lines in the Sun, the lines now named after him. He identified the elements responsible for them. (So much for confident philosophical pronouncements!) One could say that Fraunhofer's work initiated the field of stellar astrophysics. This early heroic phase culminated in Saha's work, which made all later quantitative interpretation possible.

This is probably much more than you wanted to know; but at least you now know both the long, as well as the short of it!

Live long and prosper.

Answer 2

Pretty much for the same reason why a raging hot fire is brighter than the red embers of a dying one. Cooler outer layers probably start to form absorption lines, masking out enough of the high energy higher spectrum light waves to give the star a darker appearance.

Answer 3

From what I remember in astronomy class, the hotter the surface of a star is, the more ionized the hydrogen on the surface becomes. Ionized hydrogen does not give off spectral lines like normal hydrogen does, so the fainter you see spectral lines from hydrogen, the hotter the surface of the star must be.

Answer 4

As any object heats up, the amount of light it emits of each color changes. Cool objects give off mainly infra-red, and almost no blue at all. Medium objects will give off mainly yellow, with some blue. So the coolest stars are red (M stars), next are orange (K stars), then yellow (G, like the Sun), white ( F and A stars), and blue (B and O stars). The extremely rare W stars have their strongest emissions in ultraviolet.

Answer 5

stars are fire, fire has temperatures, green fire is heater than blue, blue heater than red...
that's the reason...