1. Chapter 4 – Radiation Essentials

2. Reconsider TE Consider a closed box (a ‘cavity’ at equilibrium) with no heat loss or gain, uniform T throughout. The radiation field inside it will be isotropic and unchanging. Its nature depends only on T. Consider a patch of the wall. What will its emission look like?

3. The Functional Form Reconsider the dimensionality and units!

4. The Planck Function Two representations:

5. Some History Planck the man: a tragic life Planck the function: its importance and origin Planck the telescope: Planck crater:

6. A Unique Temperature In such bodies, a single T describes the radiation field and the material; that is, it is in equilibrium. This means that the radiation must interact with the material: the material is opaque. Light cannot stream through it.

7. An Awkward Name Hence the name “blackbody” – an object that absorbs all radiant energy falling onto it (= ‘black’). Some are shown here:

8. And Some More

9. More Picturesque

10. Immediate Implications Hotter bodies emit more radiation per unit area than do cooler bodies. Hotter bodies give off characteristically shorter wavelengths (‘bluer colours’) as the temperature is increased.

11. Wien Displacement Law

12. Cool vs Hot

13. Are All Objects Blackbodies? They must be dense enough to be opaque at the wavelength in question. We must consider the light emitted by the object, ignoring any inconsequential reflected light. (Your blue jeans are not 20,000K!)

14. Red = Cool? Yes and No

15. Nebular Emission

16. Mars

17. One Great Example

18. Is the Sun a ‘Black Body’?

19. Obviously Not Perfect Note the depressed flux in the UV, caused by absorption lines in the solar atmosphere – the less dense (not opaque!) outer regions into which we can peer to some depth. Not LTE!

20. The Brightness Temperature We can define a T Bν at each frequency, a number that (when plugged into the Planck formula) yields the observed specific intensity at that frequency.

21. Variations For non-LTE non-blackbodies, T Bν will vary with ν. Here it is for the Sun in the visible part of the spectrum. (There are huge differences elsewhere: for example, the Sun is an X-ray source!)

22. Limiting Cases Rayleigh-Jeans Law Consider long wavelengths (e.g. the radio part of the spectrum) - large λ, small ν, low-energy photons. So hν << kT, and

23. In other words: Consider a blackbody of given T. At the long-λ end of the spectrum, the intensity is growing like ν 2. This is often referred to as the Rayleigh-Jeans tail.

24. Conversely, at Short λ Wien’s Approximation: Here exp (hν / kT) >> 1, so and at high frequencies (short λ) the exponential term dominates.

25. The Observed Behaviour

26. The Wien Displacement Law Back to the hot poker! What’s the peak of the blackbody curve? λ max T = 0.290 (careful with units!! - cgs) and ν max = 5.88 x 10 10 T (larger T  shorter λ but higher ν, of course)

27. Everday Example “room temperature” (20 o C) ~ 290 K So λ max T = 0.290 implies λ max (room) = 0.001 in cgs = 10 -5 metres = 10 microns This also applies to you and me.

28. Estimating Stellar Temperatures Compare the B magnitude (the flux through the B filter, centred at 440 nm) to the V magnitude (V filter centered at 550 nm). As before, this yields a colour, B – V, through B – V = 2.5 log (flux through V / flux through B) (zero-point arbitrarily defined by Vega-like stars)

29. B-V Calibrated in Terms of Temp Things to note: - the stars are not blackbodies, so no single brightness temperature will yield the precise intensity at every wavelength - the filters are broad and have their own ‘response curves’ (the convolution of the spectral flux with the filter throughput)

30. See Table G.7 (p 394) One interesting note: as T  ∞, B-V goes asymptotically to -0.33 So stars hotter than about 35000K all have the same B-V colour.

31. Stefan-Boltzmann Law (Not unique to astronomy!) What is the total flux emitted by a blackbody? Simply integrate under the Planck curve. We find: B(T) = 2 (kπ) 4 / (15 c 2 h 3 ) T 4 That is, it is a strong power (4) of the temperature.

32. Notice the Growth!

33. Furthermore The flux from any part of the star comes out into 2π steradians (the “hemisphere” that patch faces) so the astrophysical flux of a BB is given by F BB = σ T 4 where σ is π times the constant on the previous page. This is the Stefan-Boltzmann law.

34. Stars Aren’t Perfect BBs… …so we write F star = σ T eff 4 thereby defining T eff as the “effective temperature” -- the temperature that a BB would need to be to produce the observed flux flowing from the star.

35. The Total Luminosity of a Star… … is determined by two things: the flux flowing out per unit area, and the total radiating area (in the simplest case, the surface area of the assumed spherical star). Thus: L = 4 π R 2 σ T eff 4

36. The “Theoretical” HR Diagram From L = 4 π R 2 σ T eff 4 log L = constants + 2 log R + 4 log T eff The Luminosity is related to magnitudes; the temperature term determines colours. If all stars were the same radius (and acted like blackbodies!), there would be a monotonic linear relationship between them.

37. The Original HR Diagram for Stars This plots absolute magnitude versus spectral type (the latter is a proxy for temperature)

38. Colour-Magnitude Diagrams (CMDs) Closely related to the HR diagram:

39. What We Notice No single monotonic relationship. Why? Because stars of the same colour may be of very different sizes (so white dwarfs, red giants)

40. To Be Specific Notice that there is a radius dependence even along the Main Sequence.

41. Fitting Theory to Observation People who build ‘stellar models’ work out the emerging luminosity, and the effective temperature. The data sit in a “theoretical HR diagram.” But exactly how these translate to observables is not easy to determine! There can be complex calibration issues in comparing theory to reality.

42. A Theorist’s View

43. 47 Tuc

44. Within a Star A true BB has isotropic radiation within, and is in LTE with a unique temperature. In stars, there is a temperature gradient, but over reasonable scales the temperature is locally constant, and LTE holds. (Obvious exception: the outermost parts, in the stellar atmosphere.)

45. Energy Density Within the star, the energy density owing to radiation alone is u = a T 4 with a (the radiation constant) = 7.57 x 10 -15 erg cm -3 K -4 (Try T = 10 5 ! Then u ~ 10 6 erg cm -3 )

46. Radiation Pressure The radiation pressure is similarly dependent on temperature: in fact, P rad = (1/3) a T 4 This aids in the pressure support of the star, and must be included in theoretical stellar structure calculations.

47. Very Hot Stars The radiation pressure can be extremely large and drive mass outflows. The spectroscopic signature is emission lines from outflowing gas but with a strong blue absorption component (a P Cygni profile) from gas along the line of sight.

48. The “Eddington Limit” Stars greater than ~130 or so Solar Masses have no stable configurations. The temperature required so that the kinetic pressure + radiation pressure maintains hydrostatic equilibrium is so high that the radiation drives mass outflows. Note: this depends on composition (because of opacity effects that couple the radiation to the material and drives the outflows).

49. “Population III” An important question: Just how massive can a pure H+He star be? (as in the early universe) The answer: probably quite massive… It is important because it determines the earliest chemical enrichment, energy input via early supernovae, etc. No clear answers yet.

50. “Grey Bodies” (Imperfect blackbodies) Most objects reflect some light, so are not black bodies. If that behaviour is independent of wavelength, then the body is said to be “grey” The reflectivity is the albedo

51. Some Albedos

52. Mosgt Objects Aren’t Grey!

53. Vision and Albedo We see things by reflected visible light, but any absorbed light contributes to the heating (as do internalsources like metabolism, radioactivity, residual heat of formation, differentiation, etc. An amount 1-A of the incoming light is absorbed (“A” is the albedo) and heats the object to some equilibrium that may not reflect the spectrum of incoming light.

54. We Saw This Earlier with Mars

55. Other Factors Internal Heat!

56. Greenhouse Effect! The Greenhouse Effect Venus - Runaway Greenhouse, Warning for Earth

57. Directly Detecting Extrasolar Planets How? There are problems. -How much sunlight does a planet reflect? (Think about Jupiter) -How far is it from the parent star? (easier to pick out, but less bright) Analogy: can we see the moons of Jupiter with the unaided eye?

58. Some Solutions Superb image quality (adaptive optics, etc) Coronagraphs (to mask the central star) Use wavelengths that heighten the planet’s contrast against the star

59. Coronagraphs

61. Direct Detection Work in the IR, preferably in wavebands where the star has strong absorption features!

62. Better Ways Transits (Kepler) Radial velocity perturbations