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# Dimensions of Stars

Dimensions of Stars. Having determined that stars are at large distances, high temperatures and very luminous, the next question to ask is how large they are. An Example:. Dimensions of Stars. The star Betelgeuse ( a Orionis) has the following parameters: Distance 150 pc

## Dimensions of Stars

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1. Dimensions of Stars • Having determined that stars are at large distances, high temperatures and very luminous, the next question to ask is how large they are. • An Example:

2. Dimensions of Stars • The star Betelgeuse (a Orionis) has the following parameters: • Distance 150 pc • Apparent magnitude 0 • Absolute magnitude -6 • Relative luminosity 104 L¤ • Temperature 3000 K (I.e., a class M star).

3. Dimensions of Stars • The luminosity depends on the surface area of the star (µR2) and T 4 (from Stefan’s law). • Hence, the ratio of the radii of Betelgeuse and the Sun is: • R / R¤ = (T¤ / T )2 (L / L¤)1/2 = 370 • This corresponds to almost 2 AU – I.e., larger than the orbit of Mars (1.5 AU)!

4. 106 1000 R¤ 100 R¤ Supergiants 104 10 R¤ 1 R¤ Main sequence 102 Giants 0.1 R¤ Luminosity (L¤) 1 Sun 0.01 R¤ 10-2 0.001 R¤ Red dwarfs White dwarfs 10-4 40,000 20,000 10,000 5,000 2,500 Temperature (K) Dimensions of Stars • Applying the same technique to other stars allows lines of constant radii to be plotted on the HR diagram.

5. Dimensions of Stars • Note: • Most stars on the main sequence have radii similar to the Sun, but that, as we have shown, the luminous red stars have large radii. • White dwarfs have very small radii – comparable to that of the Earth!

6. Luminosity and Spectral Class • Can the luminosity be deduced directly from the spectral class? • No - e.g. Betelgeuse and Barnard’s Star are both class M, but the latter is 18 stellar magnitudes fainter • Details of the spectrum provides a handle:

7. Luminosity and Spectral Class • Absorption line shapes: • Narrow lines Þ Low density, hence large dimensions • Broad lines Þ High density, hence a compact object Hg Rigel, a B8 supergiant Hd The B8 main sequence component of Algol

8. 106 I 104 II III 102 IV V 1 Luminosity (L¤) 10-2 10-4 2,500 20,000 5,000 10,000 40,000 Temperature (K) Luminosity and Spectral Class • Such studies lead to the idea of Luminosity Classes • Labelled I to V in order of decreasing luminosity • Provides another handle on distance • “Spectroscopic Parallax”

9. Spectroscopic Parallax • Example: • The star a Leonis (Regulus) has an apparent magnitude of 1.4 and its spectrum shows it to be a B7V star. • This gives an absolute magnitude of -0.6. • The distance modulus is therefore 2, giving a distance of 25 pc.

10. Spectroscopic Parallax • Note: • The luminosity classes are broad and poorly defined. Hence, a distance determined by this method should be regarded as an estimate • This method has nothing to do with Parallax!

11. Masses of Stars • We have now established methods of measuring : • Distance • Brightness • Composition • Temperature • Size

12. Masses of Stars • we require the mass to gain a complete picture of the properties of stars. • For example: • are giant and dwarf stars proportionatly more massive than the Sun, or are they about the same? • How does mass vary along the main sequence?

13. Masses of Stars • It is impractical to measure the mass of an isolated star. • Many stars occur in gravitationally bound multiple systems – in particular, binary stars. • By observing the orbits of such stars (and knowing the distance), the total mass can be found

14. a M1 M2 Masses of Stars • Application of Kepler’s Third Law • Miin Solar Masses, a in AU and P the orbital period in years

15. Masses of Stars • Example: • The orbit of the binary star 70 Ophiuchi (parallax 0.2 arc sec) has been plotted over many years. • The period is 87.7 years • The semi-major axis is 4.5 arc sec. • What is the sum of the masses of the two stars?

16. Masses of Stars • 70 Ophiuchi • Distance = 5pc • ~ 106 AU. • Hence, the semi-major axis is ~ 22 AU. • The period is 87.7 years • The total mass is ~ 1.4 M¤

17. a2 a1 M1 M2 Masses of Stars • The ratio of the masses: • The two stars orbit a common centre of mass • The major axes of the two ellipses are in the inverse ratio of the masses. M1/M2 = a2/a1

18. Masses of Stars • Together with the sum of the masses, the individual masses can now be obtained. • Orbital period Þ Total mass • Ratio of major axes Þ Ratio of masses

19. Masses of Stars • Binary systems containing red giants: • show masses of red giants fall into the same range as main sequence stars • confirming our conjecture they have low densities) • White dwarfs are found to have masses about 1 M¤ • indicating these are very dense objects!

20. Luminosity, L (Solar Luminosities) L = M 7/2 Mass, M (Solar Masses) Mass-Luminosity Relationship • Is there a relationship between luminosity and mass?

21. Mass-Luminosity Relationship • L = M7/2 • Approximate relationship • Valid for low mass stars up to about 1 solar mass • L = M3 gives better fit for high mass stars • Note for Main Sequence stars only

22. Stellar Interiors • Basic Ideas • Hydrostatic Equilibrium • Classical and Quantum Gases • Electron and photon gases • Ionisation • The Saha Equation • Heat Transfer Mechanisms • Fusion

23. Hydrostatic Equilibrium • Since main sequence stars are stable, and are neither expanding or contracting, they are clearly in equilibrium • Internal pressure supports against gravity

24. Hydrostatic Equilibrium • We wish to examine: • The relationship between mean pressure and potential and kinetic energies • The Virial Theorem • The consequences of equilibrium of non-relativistic and ultra-relativistic gases

25. Dr R g(r)DM (P+DP)DA P.DA r Mass DM Hydrostatic Equilibrium • Consider a volume element in a spherical system of mass M and radius R

26. Hydrostatic Equilibrium • Net force due to the pressure gradient • Noting that the mass, DM, of the element is: DM = r(r)Dr DA, and including the gravitational force, g(r) we have the acceleration of the volume element:

27. Hydrostatic Equilibrium • In equilibrium, net acceleration is zero. • Hence: • We can write g(r) as Where m(r) is the mass contained within a radius r

28. Hydrostatic Equilibrium • Therefore, if the star is to be in hydrostatic equilibrium, the pressure gradient at any distance r from the centre is:

29. = dm Hydrostatic Equilibrium • We can use this expression to relate the average pressure to the total gravitational potential • Multiply both sides by 4pr3 • Integrate from 0 to R

30. Hydrostatic Equilibrium • The right hand side is simply the gravitational potential: • The left hand side can be evaluated using integration by parts:

31. = 0 = dV = -3<P >V Hydrostatic Equilibrium • The left-hand side then becomes:

32. Hydrostatic Equilibrium • Hence, the average pressure in a system in hydrostatic equilibrium and with gravitational potential Egis: • The Virial Theorem

33. Hydrostatic Equilibrium • Non-relativistic (p << mc)vs. ultra-relativistic (p >> mc) gases. • Fundamental relationships:

34. Hydrostatic Equilibrium • Non-relativistic gases: • p.v. = mv2 • Hence: • Pressure = 2/3 kinetic energy density

35. Hydrostatic Equilibrium • Consequences: • Hence:

36. Hydrostatic Equilibrium • Since: • i.e., the system is bound with binding energy -Etotal = internal kinetic energy

37. Hydrostatic Equilibrium • Implications: • Tightly bound systems are hot • Energy loss from a gas cloud causes contraction and heating • 1/2 loss of gravitational energy goes into radiation, 1/2 into raising the temperature • Increase in temperature provides pressure to counter gravity • Excess energy production in a star causes expansion and cooling

38. Hydrostatic Equilibrium • Ultra-relativistic gases: • p.v. = pc • P >>mc, so EKE= pc • Hence: • Pressure = 1/3 kinetic energy density

39. Hydrostatic Equilibrium • Consequences: • EKE= -Eg • System only in equilibrium if Etotal = 0 • Hydrostatic equilibrium of ultra-relativistic particles (e.g., photon gas, high energy electron gas) easily disrupted.

40. Next Lecture: • Classical and Quantum Gases • The Saha Equation • Derivation • Consequences for ionisation and absorption

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