Measuring the Stars

Measuring the Stars

The Sun’s Neighborhood A plot of the 30 closest stars to the Sun, projected so as to reveal their three-dimensional relationships. Notice that many are members of multiple-star systems. All lie within 4 pc (about 13 light-years) of Earth. The gridlines represent distances in the Galactic plane. The next nearest neighbor to the Sun beyond the Alpha Centauri system is called Barnard’s Star.

2 Ways to MeasureStar 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.

Inverse-square Law As light moves away from a source such as a star, it is steadily diluted while spreading over progressively larger surface areas (depicted here as sections of spherical shells). Thus, the amount of radiation received by a detector (the source’s apparent brightness) varies inversely as the square of its distance from the source.

Inverse Square Light.htm.lnk

Can you name any bright stars?

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!

DistanceApparent

Magnitude

Sirius 2.7 pc -1.46

Rigel 240 pc +0.14

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

DistanceApparent Magnitude

Absolute Magnitude

Sirius 2.7 pc -1.46 +1.4

Rigel 240 pc +0.14 -6.8

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.

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Apparent Magnitude

apparent magnitude

absolute magnitude

Star A 1 1Star B 1 2Star C 5 4Star D 4 4

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.

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

"Oh, Be A Fine 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.

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.

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.

SPECTRALCLASS

COLOR SURFACE TEM-PERATURE (K)

PRINCIPAL FEATURES EXAMPLES

O Bluish-white

30,000 Relatively few absorption lines. Lines of highly ionized atoms. Hydrogen lines appear only weakly.

Naos

B Bluish-white

11,000 - 30,000 Lines of neutral helium. Hydrogen lines more pronounced than in O-type stars.

Rigel, Spica

A Bluish-white

7,500 - 11,000 Strong lines of hydrogen. Also lines of singly ionized magnesium silicon, iron, titanium, calcium, and others. Lines of some neutral metals show weakly.

Sirius, Vega

F Bluish-white to

white

6,000 - 7,500 Hydrogen lines are weaker than in A-type star but still conspicuous. Lines of singly ionized metals are present, as are lines of other neutral metals.

Canopus,Procyon

G White toyellowish-

white

5,000 – 6000 Lines of ionized calcium are the most conspicuous spectral features. Many lines of ionized and neutral metals are present. Hydrogen lines are weaker even than in F-type stars.

Sun, Capella

K Yellowish-orange

3,5000 - 5,000 Lines of neutral metals predominate. Arcturus,Aldebaran

M Reddish 3,500 Strong lines of neutral metals and molecules.

Betelgeuse,Antares

STAR SPECTRALCLASS

SURFACE TEMPERATURE

(K)

RADIUS(sun = 1.0)

DISTANCE(light years)

LUMINOSITY(sun = 1.0)b

Epsilon Orionis B0 24,800 37 1600? 470,000?

Rigel B8 11,550 74 880? 90,000?

Regulus B7 12,210 3.6 69 270

Sirius A A1 9,970 1.7 8.7 23

Procyon A F5 6,510 2.1 11 7

Sun G2 5,780 1.00 1.6 x 10-5 1.00

Capellaa G5 5,200 14 41 130

Epsilon Eridani K2 5,000 0.7 11 0.28

Aldebaran K5 3,780 61 60 700

Betelgeuse M2 3,600 1200 1400? 21,000?

Sirius B White dwarf 30,000 0.0073 8.7 0.003

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.

Stellar Spectra Comparison of spectra observed for seven stars having a range of surface temperatures. These are not actual spectra, which are messy and complex, but simplified artists’ renderings illustrating a few spectral features. The spectra of the hottest stars, at the top, show lines of helium and multiply ionized heavy elements. In the coolest stars, at the bottom, there are no lines for helium, but lines of neutral atoms and molecules are plentiful. At intermediate temperatures, hydrogen lines are strongest. All seven stars have about the same chemical composition.

The H-R Diagram A plot of luminosity against surface temperature (or spectral class), known as a Hertzsprung-Russell diagram, is a useful way to compare stars. Plotted here are the data for some stars we've discussed. The Sun, of course, has a luminosity of 1 solar unit. Its temperature, read off the bottom scale, is 5800 K—a G-type star.

radi

us in

crea

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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.

Stellar Sizes Star sizes vary greatly. Shown here are the estimated sizes of several well-known stars. Only part of the red-giant star Antares can be shown on this scale; (The symbol R   means “solar radius.”) Giants are stars having

radii between 10 and 100 times that of the Sun.

Supergiants are 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)

H-R Diagram of Nearby Stars Most stars have properties within the shaded region of the H–R diagram known as the main sequence. The points plotted here are for stars lying within about 5 pc of the Sun. Each dashed diagonal line corresponds to a constant stellar radius, so that stellar size can be indicated on the same diagram as stellar lum-inosity and temperature.

H-R Diagram of Bright Stars An H-R diagram for the 100 brightest stars in the sky is biased in favor of the most luminous stars — which appear toward the upper left — because we can see them more easily than we can the faintest stars.

Hipparcos H-R Diagram This is a simplified version of the most complete H-R diagram ever compiled. It represents more than 20,000 data points, as measured by the Euro-pean Hipparcos spacecraft for stars within a few hundred parsecs of the Sun.

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.

Stellar Distances Knowledge of a star’s luminosity and apparent brightness can yield an estimate of its distance. Astronomers use this third rung on our distance ladder, called spectroscopic parallax, to measure distances as far out as individual stars can be clearly discerned—several thousand parsecs.

1) Determine the star’s spectral type.

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

3) 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.

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:

Stellar Luminosities Stellar luminosity classes in the H–R diagram. A star’s location in the diagram could be specified by its spectral type and luminosity class instead of its temperature and luminosity.

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.

Stellar Masses More than any other stellar property, mass determines a star’s position on the main sequence. Low-mass stars are cool and faint; they lie at the bottom of the main sequence. Very massive stars are hot and bright; they lie at the top of the main sequence.

Chart of Stellar Mass Distribution The distribution of masses of main-sequence stars, as determined from careful measurement of stars in the solar neighborhood.

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.

Stellar Radii and Luminosities (a) Dependence of stellar radius on mass for main-sequence stars; actual measurements are plotted here. The radius increases roughly in proportion to the mass over much of the range. (b) Dependence of luminosity on mass. The luminosity increases roughly as the fourth power of the mass (indicated by the straight line).

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!!