Measuring the Stars
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Measuring the Stars. 2 Ways to Measure Star Distance. Stellar Parallax. Stellar Brightness (Specroscopic Parallax).

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Measuring the Stars

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Measuring the Stars

2 Ways to Measure

Star Distance

Stellar Parallax

Stellar Brightness

(Specroscopic Parallax)

We already discussed stellar parallax earlier in the course. Remember, the more the star changes its relative position in the sky over the course of a year, the closer the star is to us.

Good! How about we look at Sirius, in the constellation Canis Major and Rigel in the Orion constellation? Sirius is only 2.7 parsecs away, but Rigel, in Orion, is 240 parsecs away. So, as you might expect Sirius looks brighter in the night sky than Rigel.

The magnitude scale used in astronomy ranks the brightness of a star with a number. But the smaller the number the brighter the star!

Can you name any bright stars?

Question: What if we could move these two stars so that they were each the same distance from Earth - then which would be brighter?

A star's absolute magnitude is its apparent magnitude when viewed from a distance of 10 parsecs. This allows astronomer's to compare stars with each other.

A magnitude difference of 5 corresponds to 100 times in brightness.

For example, if the apparent magnitude of a star that is 100 pc away is +6. Because it is 10 times farther away that means the object will appear 100 times dimmer (inverse-square law).

The 100 times dimmer in bright-ness corresponds to 5 magni-tudes. So, the star has an absolute magnitude of +6 – 5 = +1.

Apparent Magnitude

  • 1) Which object appears brighter from Earth: Star C or Star D? Explain.

  • 2) Which object is actually brighter: Star A or Star D? Explain.

  • 3) Rank the objects in order of their distances from Earth. Explain.

  • How would the apparent and absolute magnitudes of Star A change if it were now moved to a distance of 40 parsecs from Earth? Explain.

Star D – smaller apparent magnitude means the star is brighter.

Star A – it has a smaller absolute magnitude.

Farthest to nearest: C, A&D tie, B

Absolute magnitude would not change, but apparent

magnitude would get larger (it would go to about 4).

Spectral Classification

Astronomers categorize stars in spectral classes. You can remember the spectral classes in order of decreasing temperature if you remember the mnemonic:

"Oh, Be AFine Guy/Girl, Kiss Me".

  • Almost stars fit somewhere into this sequence of O, B, A, F, G, K, and M.

    • O stars are hottest and thus appear bluish white;

    • G stars are less hot and thus appear yellowish; and

    • M stars are coolest and thus appear reddish.

NOTE: To understand the relationship between color and temperature, think about the toaster element in your toaster or the heating element on your electric stove. As it heats up, it starts to glow red. As it gets even hotter, it begins to glow orange. As it gets even hotter, the color that it glows will move up the spectrum to yellow (1275 K) and eventually blue-white (1425 K and higher). So hotter objects give off higher frequency light. Keep in mind the colors of the visible spectrum – Red, Orange, Yellow, Green, Blue, Violet.

  • Astronomers then divide each of these letter classes into subclasses. The subclasses are labeled from 0 to 9, going from hottest to coolest.

  • For example, a G3 star is hotter than a G8 star.

  • But a G8 star is hotter than a K9 star.

  • Look at the following charts that summarize the stellar spectral classes, and show some of the key stars in the winter sky.

a Capella is a double star. The temperature, radius, and luminosity are those of the brighter and cooler component.

b The luminosities and radii of Epsilon Orionis, Rigel, and Betelgeuse are only approximate because their distances

are estimated.

radius increases

radius increases

Cooler stars that are very luminous must be very large in radius. Conversely, very hot stars that are not very luminous must be very small in radius.

  • Giants are stars having radii between 10 and 100 times that of the Sun.

  • Supergiantsare larger than giants having radii as large as 1000 times the solar radius.

  • A dwarf refers to any star of radius comparable to or smaller than the radius of the Sun (from 0.01 times the Sun's radius up to about one solar radius)

Few white dwarfs appear because almost no white dwarfs lie close enough to Earth to have been bright enough for the instrument. About 90% of all stars in our solar neighborhood, and pre-sumably a similar percent-age elsewhere in the universe, are main-sequence stars. About 9% of stars are white dwarfs, and 1% are red giants.

Measurement of the apparent brightness of a light source, combined with some knowledge of its luminosity, can yield an estimate of its distance. The procedure is as follows:

  • Determine the star’s spectral type.

  • Assuming the star lies on the main sequence, read the star’s luminosity directly off the HR diagram.

  • Knowing the star’s luminosity, determine its distance by measuring its apparent brightness and using the inverse-square law.

This process of using stellar spectra to infer distances is called spectroscopic parallax. In practice, the width of the main sequence line on the HR diagram translates into a small (10–20 percent) uncertainty in the distance, but the method is still valid.

Over the years, astronomers have developed a system for classifying stars according to the widths of their spectral lines. Because line width depends on pressure in the stellar photosphere, and because this pressure in turn is well correlated with luminosity, this stellar property has come to be known as luminosity class.

  • Binary Stars

  • Most stars are members of multiple-star systems—groups of two or more stars in orbit around one another. The majority are found in binary-star systems, which consist of two stars in orbit about their common center of mass, held together by their mutual gravitational attraction. (The Sun is not part of a multiple-star system; if it has anything at all uncommon about it, it is this lack of stellar companions.)

  • Astronomers classify binary-star systems (or simply binaries) according to their appearance from Earth and the ease with which they can be observed.

    • Visual binaries have widely separated members bright enough to be observed and monitored separately.

    • In the rarer eclipsing binaries, the orbital plane of the pair of stars is almost edge-on to our line of sight. In this situation, we observe a periodic decrease of starlight intensity as one member of the binary passes in front of the other. Media Clip

    • The more common spectroscopic binaries are too distant from us to be resolved into separate stars, but they can be indirectly perceived by monitoring the back-and-forth Doppler shifts of their spectral lines as the stars orbit one another and their line-of-sight velocities vary periodically.

In a double-line spectroscopic binary, two distinct sets of spectral lines—one for each component star—shift back and forth as the stars move. Because we see particular lines alternately approaching and receding, we know that the objects emitting the lines are in orbit. Media Clip

In the more common single-line systems, one star is too faint for its spectrum to be distinguished, so we see only one set of lines shifting back and forth. This shifting means that the detected star must be in orbit around another star, even though the companion cannot be observed directly. If this idea sounds familiar, that's probably because we have discussed it before. All the extrasolar planetary systems discovered to date were found using this single-line method.

  • Note: These binary categories are not mutually exclusive. For example, an eclipsing binary may also be (and, in fact, often is) a spectroscopic binary system.

An Example of How Stellar Mass Can Be Determined

Consider the nearby visual binary system made up of the bright star Sirius A and its faint companion Sirius B. Their orbital period is 50 years, and their orbital semi-major axis is 20 A.U (i.e. 7.5" at a distance of 2.7 pc), implying that the sum of their masses is 3.2 (= 203/502) times the mass of the Sun.

Further study of the orbit shows that Sirius A has roughly twice the mass of its companion. It follows that the masses of Sirius A and Sirius B are roughly 2.1 and 1.1 solar masses, respectively.

  • With few exceptions, main-sequence stars range in mass from about 0.1 to 20 times the mass of the Sun.

  • The hot O- and B-type stars are generally about 10 to 20 times more massive than our Sun.

  • The coolest K- and M-type stars contain only a few tenths of a solar mass.

Important Note: The mass of a star at the time of its formation determines its location on the main sequence.

Expected Stellar Lifetimes

Dividing the amount of fuel available (that is, the star’s mass) by the rate at which the fuel is being consumed (the star’s luminosity), one can estimate the expected lifetime of a star. The expected lifetime of the Sun is about 10 billion years, and is currently about 4.6 billion years old.

Because luminosity increases so rapidly with mass, the most massive stars are by far the shortest lived!

For example, according to the mass–luminosity relationship, the lifetime of a 10-solar-mass O-type star is about 1/1000 (=10 solar mass/104 luminosity) that of the Sun, or 10 million years. So we can be sure that all the O-type and B-type stars we observe are quite young—less than a few tens of millions of years old.

The reason is that their nuclear reactions proceed so rapidly that their fuel is quickly depleted despite their large masses. At the opposite end of the main sequence, the low core density and temperature of an 0.1-solar-mass M-type star mean that its proton–proton reactions churn away much more sluggishly than in the Sun’s core, leading to a very low luminosity and a correspondingly long lifetime. Many of the K-type and M-type stars now visible in the sky could shine on for at least another trillion years.

Slow and steady wins the race!!

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