Stellar objects

By the end of this topic you should be able to:

-describe the method of parallax, the method of spectroscopic parallax and the Cepheid method for determining distances in astronomy;-define the parsec;-state the definitions of absolute and apparent magnitude.

Parallax methodThe parallax method takes advantage of the fact that, when an object is viewed from two different positions, it appears displaced, relative to a fixed background of stars, because of the fact that in the intervening time the Earth has moved in its orbit around the Sun.

Since the parallax angle is very small,tan p ≈ p

Parallax methodThe parallax angle is the angle at the position of the star that subtends a distance equal to the radius of the Earth’s orbit around the Sun, a distance known as one astronomical unit, 1 AU = 1.5 x 1011 m.

Parallaxes are measured quite accurately provided they are not too small. For example, parallaxes down to 1 arcsecond (1/3600 of a degree) are easily measured.If the star is too far away, the parallax is too small and the method fails.Typically, measurements from observatories on Earth allow distances up to 300 ly to be determined with the parallax method. Using measurements from satellites above the Earth’s atmosphere, larger distances can be determined.

ParsecThe parallax method can be used to define a common unit of distance in astronomy, the parsec. One parsec is the distance to a star whose parallax is 1 arcsecond.

1 pc = 3.09 x 1016 m = 3.26 ly

This means that if the parallax of a star is known to be p arcsecond, the distance is 1/p parsecs

Absolute and apparent magnitudesThe ancient astronomers devised a relative system of classifying stars according to how bright the appeared to an observer on Earth. Each star was given a number called the apparent magnitude m-the higher the apparent magnitude, the dimmer the star. In this system six classes of brightness were defined, and assigned numbers from 1 to 6. A magnitude 6 star was supposed to be 100 times dimmer than a magnitude 1 star.

The modern magnitude scale is defined as follows:

Given a star of apparent brightness b, we assign to that star an apparent magnitude m defined by

b/b0 = 100-m/5 = 2.512-m

(The value b0 = 2.52 x 10-8 W/m2 is taken as the reference value for apparent brightness).

Taking logarithms (to base 10) gives the equivalent form

m = -5/2 log(b/b0)

Absolute and apparent magnitudes

The human eye can detect a star of apparent magnitude not longer than about 6. With simple binoculars the limit is raised to stars of magnitude 9. The largest telescopes can record images of objects of apparent magnitude as faint as 27.

Absolute and apparent magnitudesTwo stars that have the same apparent magnitude are not necessarily equally bright intrinsically, since they may be at differente distances. To establish a system of absolute magnitudes that will tell us if one star is intrinsically brighter than another, we imagine that all stars are positioned at the same distance from Earth.

The apparent magnitude a star would have if placed at a distance of 10 pc from Earth is called the absolute magnitude M of the star.

Thus, it can be proved that:

Absolute and apparent magnitudesCommon Name Scientific Name Distance (light years) Apparent Magnitude Absolute Magnitude

Sun - -26.72 4.8Sirius Alpha CMa 8.6 -1.46 1.4Canopus Alpha Car 74 -0.72 -2.5Rigil Kentaurus Alpha Cen 4.3 -0.27 4.4Arcturus Alpha Boo 34 -0.04 0.2Vega Alpha Lyr 25 0.03 0.6Capella Alpha Aur 41 0.08 0.4Rigel Beta Ori ~1400 0.12 -8.1Procyon Alpha CMi 11.4 0.38 2.6Achernar Alpha Eri 69 0.46 -1.3Betelgeuse Alpha Ori ~1400 0.50 (var.) -7.2Hadar Beta Cen 320 0.61 (var.) -4.4Acrux Alpha Cru 510 0.76 -4.6Altair Alpha Aql 16 0.77 2.3Aldebaran Alpha Tau 60 0.85 (var.) -0.3Antares Alpha Sco ~520 0.96 (var.) -5.2Spica Alpha Vir 220 0.98 (var.) -3.2Pollux Beta Gem 40 1.14 0.7Fomalhaut Alpha PsA 22 1.16 2.0Becrux Beta Cru 460 1.25 (var.) -4.7Deneb Alpha Cyg 1500 1.25 -7.2Regulus Alpha Leo 69 1.35 -0.3Adhara Epsilon CMa 570 1.50 -4.8Castor Alpha Gem 49 1.57 0.5Gacrux Gamma Cru 120 1.63 (var.) -1.2Shaula Lambda Sco 330 1.63 (var.) -3.5

Example questions

Calculate the absolute magnitude of a star whose distance is 25 ly and whose apparent magnitude is 3.45

Calculate the distance to Sirius using that m = -1.43 and M = 1.4

Spectroscopic parallaxThe term spectroscopic parallax refers to a method of finding the distance to an star given the star’s luminosity and apparent brightness. The term is misleading in that no use of parallax is being made.

The question is then how to determine the luminosity of the star. This is done by examining its spectrum, from which the temperature can be deduced. Knowing the temperature and using the HR diagram allow us to determine the luminosity as well.

λ0T = 2.9 x 10-3 Km

Example question

A main sequence star emits most of its energy at a wavelength of 2.4 x 10-7 m. Its apparent brightness is measured to be 4.3 x 10-9 Wm-2. How far is the star?

The CepheidsCepheid variable stars are stars whose luminosity is not constant in time but varies from a minimum to a maximum periodically, the periods being typically from a 1 to 70 days. The brightness of the star increases sharply and then fades off more gradually. The reason for the periodic

behaviour of the brightness of Cepheid stars has to do with the interaction of radiation with matter in the atmosphere of the star. This interaction causes the outer layesrs of the star to undergo periodic expansions and contractions. The star is at its brightest when the surface of the star expands outward at high velocity. It is at its dimmest when the surface moves inward.

The CepheidsHenrietta Leavitt discovered a remarkable relationship between the peak luminosity of Cepheids and their period. The longer the period, the larger the luminosity.

Observing a Cepheid and finding its period allows the determination of its luminosity. This, in turn, allows the determination of its distance.

Range of efectiveness for each method

The ordinary parallax method allows the determination of distances up to about 100 pc. The spectroscopic parallax method extends distance measurements to about 10000 pc. The Cepheid variable star method extends the scale further to 15 Mpc.

Questions

Questions