Lecture 14: Stellar Spectroscopy and the HR Diagram,Stellar Masses

This subject is one of the keys to modern astronomy!

A Little History

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Annie Jump Cannon

Grayspec.tif (17446 bytes) Spectral Sequence Defined by Cannon

A set of letters was used to denote the various categories:

A -- lines of H are prominent

B -- lines of He are prominent

G -- line of ionized Ca are prominent

M -- strong lines from neutral metals and molecules apparent

Numbers were used to denote sub-categories so for example, Vega was indicated as an A0 star while stars similar to the Sun were denoted as G2.

Seven major categories were distinguished along with a number of categories which we know now represent rare types of stars.

Stellar Photospheres

Spectral lines are formed in the photosphere of a star so we must understand the structure of a photosphere:

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Formation of Continuous Spectra in Stars
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Temperature Dependence of Spectra Look at hydrogen whose absorption lines are strongest in A stars whose temperatures are ~7500ºK-11,000ºK;
Lines in the visible spectrum are H(3->2), H(4->2), H(5->2) and so on with all lines involving level 2. For atoms to exist with electrons above level 2, the temperature must be high enough to have most H atoms excited at least to level 2:
So a star must have T at least this high to produce lines involving level 2. If the star is too hot, the electrons will jump to higher levels. At high enough T, the H will be completely ionized:

By computing the temperature ranges for various atoms and ions, it became obvious that the correct ordering of the letter spectral types is OBAFGKM:
 
Spectral Class Color T°K Spectral Lines
O blue-white 30,000-50,000 Few lines, HeII, little or no H
B blue-white 11,000-30,000 HeI, some H
A blue-white 7,500-11,000 Strong H, some singly ionized metals
F white 6,000-7,500 H becoming weaker, ionized metals stronger
G yellow 5,000-6,000 CaII strongest line, more neutral metals
K orange 3,500-5,000 Neutral metal lines dominant
M red <3,500 Molecules and neutral metals strong

 

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The HR Diagram: A Summary of Knowledge about Stellar Properties

eh.jpg (67943 bytes)Ejnar Hertzsprung

HR = Hertzsprung- Russell after the two men who first made such diagrams, Ejnar Hertzsprung and Henry Norris Russell, in the early part of the 20th century.

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Notice that stars do not appear just any where in this diagram which plots luminosity (or absolute magnitude) on the y-axis and temperature (or color or spectral type or B-V) along the x-axis.

The bulk of the data points lie in a band stretching from luminous and hot to dim and cool. This is the main sequence.

Stars in other regions of the diagram are no longer burning H in their cores, and have left the main sequence. Note that stars too young to have reached the main sequence will also appear in a different location; these objects are often obscured by dust and require special study to understand their temperatures and luminosities.

The giants, supergiants, and white dwarfs can all be distinguished spectroscopically even if their distances and hence luminosities are not known. The line profiles change shape depending on the surface gravity of the star.

Photospheres are comprised of gas so

The pressure and gravitational force must balance:

For stars of comparable temperatures, those with higher surface gravities will have higher pressures and vice-versa.

If one compares spectral lines from low pressure gas and high pressure gas, one finds that the high pressure gas produces broader spectral lines.

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These spectra come from stars with the same T but P increasing downwards in the plot.

Why? Because the collision rate between atoms is higher for denser gas. When an atom collides with another particle, its energy levels are temporarily shifted making it possible for the atom to absorb slightly redder slightly bluer wavelengths thus making the spectral line appear broader.

Main sequence stars -- have relatively high surface gravities (but not as compared to white dwarfs!)

Giant stars -- have expanded over the size they had while on the main sequence so their surface gravity is lower and their spectral lines are narrower

Supergiant stars -- have expanded the most and so have the thinnest atmospheres and narrowest lines

Luminosity classes are added to denote whether stars are on the main sequence (V), giants (III) , or supergiants (I, Ia, II). Giants and supergiants are luminous than main sequence stars of the same temperature because their surface areas are much larger.

This system of using roman numerals to denote the luminosity class is referred to in the text as the MK (for Morgan and Keenan) or as the Yerkes classification.

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Spectroscopic Parallaxes

By taking a high quality spectrum of a star, the temperature and luminosity class can be determined. If the star's apparent magnitude has also been measured, then by using an HR diagram the star's luminosity can be derived.

Stellar Masses

If a star is isolated in space we have no direct method of measuring its mass (even the nearest stars are far too distant to exert a gravitational force that we could measure).

If a star is a member of a binary pair, we can measure the masses of the two stars using the general form of Kepler's 3rd Law:

so if we can measure r and P, we have the sum of the masses. We can solve for the masses separately if we measure their distances from the center of mass:

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Observationally, a pair of stars orbiting each other and moving through space describe a kind of corkscrew pattern:

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One last issue remains - we only observe the projection of the orbits onto the plane of the sky. We need to know the inclination (or tilt relative to the plane of the sky) of the orbits. Any ellipse will project into another ellipse regardless of the inclination angle. However, both stars' orbits must be ellipses where the center of mass is a focus. Only one projection will give this arrangement.

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A spectroscopic binary is a case of two stars which do not appear as separate objects, but whose spectra can be distinguished as coming from two stars. The spectral lines will exhibit a pattern of changing redshifts and blueshifts that result from two stars orbiting each other:

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Note that

where i=inclination of the orbit with respect to the plane of the sky. Because the Doppler shift gives a velocity in km/sec directly, the distance to the stars doesn't need to be known to derive the sum of the masses, but unless the inclination is known, only a lower limit on the masses can be computed.

Mass-Luminosity Relation

Therefore, it is not surprising that a star's luminosity depends on its mass: A fit to all the data in this figure yields

A more sophisticated fit takes into account a difference in slope between the low and high mass ends: