HomeWork 2: Ay 125 Questions 1–4 are due by COB May 7, 2010. Question 5 is due by COB May 11, 2010.

April 28, 2010

1 Brown Dwarf-Main Sequence Transition

On the lower main sequence, the radius is approximately proportional to mass. Using the Sun as the normalization compute the mean temperature of the low mass main sequence stars. Compare the mean pressure for two cases1: ideal gas and degenerate electron gas (assuming full ionization). Explain and account for the transition from main sequence to the brown dwarf sequence. [25pt]

2 Specific Heat of Solids

As a part of our study of the cooling of white dwarfs we reviewed the ba- sic physics of solids of pure substances. Here, we apply our newly acquired knowledge to sodium. At room temperature, Sodium is a solid with the fol- lowing physical properties: molar volume of 23.7 cm3 and the crystal struc- ture is a cubic body centered (lattice parameter, a = 4.29 A˚ and an inferred Debye temperature of 134 ± 2 K; see Field & Medlin 1974, Acta Crystl. A30, 234). The boiling point is 1156 K, the melting point is 371 K and the specific heat at room temperature is 1230 J/kg/K. Using the discussion in Shapiro & Teukolsky compute the boiling point, the melting point, the Fermi energy (assume one free electron per atom) and the Debye temperature. Compare these to the measured properties. [25 pt]

1Assume that the mass fraction of hydrogen and helium is X = 0.6 and Y = 0.4.

1 3 White Dwarf: Thermal History

Consider three pure Carbon white dwarfs: 0.6 M , 0.9 M and 1.2 M . As- sume that they all are born at 105 K. Ignoring the outer non-degenerate layer (electrons) compute the thermal energy in ions and electrons at the time of birth.2 [10 pt] The Mestel cooling model has two simplifying assumptions: (i) the pri- mary store of energy is in ions and whose equation of state approximated by the perfect gas law; (ii) the cooling luminosity is transported via radiation. Assume that the outer (non-degenerate) layer is pure Helium mass. De- rive the Mestel cooling formula. Plot the resulting luminosity assuming Kramer’s opacity law. [10 pt] Compute the thermal energy in the non-degenerate layer (and compare to the luminosity and derive thermal timescale of the outer layer). [10 pt] At what age does convection become important? (i.e. use the mid-point in the outer layer to compute the temperature gradient). [10pt] At this phase, assume the entire layer is convective. Compute the lumi- nosity curve. [10pt] Next derive the formula for melting and Debye temperatures. Estimate the timescale when these temperatures are reached. Compare your estimate of the timescale to those shown in Figure 12 of the Hansen review. [10 pt] For those more ambitious: include in the thermal energy budget the latent heat and the increase in cV when the lattice is formed. Continue the cooling calculation for

4 Finite Temperature Correction

−2 The standard Chandrashekar limiting mass of 5.84µe M is computed for a zero temperature Fermi (electron) gas. Show that the lowest-order finite- 2 temperature corrections increase it by a factor of [1 + (πkBT/F ) ]. Can you physically explain why the limiting mass is increased? [15 pt]

2The basic physics can be found in Shapiro & Teukolsky. Compare your results to that stated in Hansen (2004) review paper (Physics Reports, 399, pp1-70, 2004). Please derive the necessary formulae. The point is to learn by deriving and then appreciate by plotting.

2 5 Seven Famous White Dwarfs

Write a succinct paper on ”Seven Famous White Dwarfs”.3 [100 pt] Ideally this would include extremities (most massive, least massive, most magnetized, most rapidly rotating, youngest, oldest, with interesting diag- nostics (seismology), most compact binary, type Ia progenitor, historical, with weird composition and whatever else catches your imagination). The audience I have in mind are other first year astronomy graduate students. Thus you should provide clear background and at the same time explain what makes your choice interesting. It is best to justify statements you make e.g. do not simply state that “according to Prof. X. Idiot the mass of the white dwarf is 1.35 M ”. Evaluate the result as best as you can. Many times being suspicious in the simplest manner serves you well. You may wish to talk to your TA regarding the use of BibTeX (so that you do not have to type in references by hand). Please follow the Phinney rule: never quote a reference which you have not at least made an attempt to read.

A Seven Famous White Dwarfs

As usual when I need advice I go for nothing but the best.4 Here are some responses.

A.1 From E. Sterl Phinney 40 Eri B: first WD to have spectrum, which determined that it was an A star, but too faint → small radius. [First seen by Herschel 1825, it’s peculiarity compared to ordinary stars first noted by Russell 1914 in his first Hertzsprung Russell diagram’s 1914] Sirius B: -suspected astrometrically by Bessel [of the functions!] 1844, first seen by Alvan Clark 1862, but but no spectrum to determine small radius

3This is not a typical homework problem. I would like you to read the literature and write a succinct paper, using ApJ style file. Writing concisely is a skill and usually acquired with great effort. Some suffer at the sword of perfection; others wallow around in verbiage. Practice and a good mentor can help one improve one’s writing skills. I will be reading your report with considerable interest and plan to give extensive feedback. 4An observation: first rate interact with other first rate people, without fear or favor. Second rate people prefer to interact with third rate people.

3 until 1915 by Adams; first gravitational redshift to test GR: Adams 1925). Closest white dwarf to earth. WZ Sagittae -first nova shown to be a binary star [P=82 min] and first invocation of gravitational radiation in the ‘real world’: i.e. first CV shown to be a binary (Kraft, Mathews & Greenstein 1962 [all Caltech]). AM CVn (=HZ-29, discovered by Humason & Zwicky 1947; Greenstein & Matthews 1957 showed it had no H lines, only He; Paczynski 1967 and Faulkner, Flannery & Warner 1972 proposed the current model: low mass He WD accreting onto another WD) Grw +70◦824 -first magnetic white dwarf (strange spectrum noted by Minkowski 1938; circular polarization discovered by the unusual Mr. Kemp 1970, leading to identification of magnetism). U Gem -first CV (discovered 1855, first proposed to be caused by disk instability [Yoji Osaki 1974] after Kraft et al showed novae were in binaries -see WZ Sag above) ZZ Ceti -prototypical oscillating white dwarf DQ Her -prototypical ”polar” -accreting magnetic binary RXJ0806.3+1527 (shortest orbital period binary system known: Pb=5 min) EUVE J0317-85.5 -rapidly rotating (725s), 1.35Msun (log g=9.3), mag- netic (450MG) single white dwarf. Vennes et al 2003.

A.2 From James Liebert, University of Arizona I suggest seven since Snow White had seven white dwarfs. Historically, Sirius B is numero unno. My colleague Jay Holberg has just published a book ”Sirius: the Brightest Diamond in the Sky”. This covers its prediction (F. Bessel in the 1830s) due to the wobbled path of Sirius A, followed by observers trying to see it. Finally, Alvin Clark found it. The first gravitational redshift at Mt. Wilson. The redshift was apparently detected but was the wrong value. Heatherington in the 1970s claimed that the Direc- tor of Mt. Wilson, Walter Adams, fudged the data to prove Einstein’s theory. Jesse Greenstein and Bev Oke wrote a paper disputing Heatherington. Then there is its cultural history, the tribe in darkest Africa that somehow knew Sirius had a companion, the records claiming that the star was red two thou- sand years ago. Then the astrophysics – it has a well measured mass, radius and other parameters (both A and B). Hertzsprung a hundred years ago first published that it shared a motion with the Ursa Major moving group, and

4 coined the term Sirius supercluster. A few years ago a group of us fit the position of A in the HRD to get an age (237 Myr +/- maybe at worst 10%). This does not agree with the age of the Ursa Major stars – 400 - 500 Myr (J. King). So, it must not be a member of the Sirius supercluster. The important historical white dwarfs, also among the brightest known, include 40 Eri B (type DA) and van Maanen 2. The latter has a temperature like the Sun (one of the first cool white dwarfs), and shows heavy metals in the spectrum. But it has no detected hydrogen. Procyon B, 40 Eri B are among the few with astrometric masses. Procyon B is very different than Sirius B – much cooler and with a hybrid atmospheric abundances (DQZ, helium rich with traces of carbon and heavy metals). That gets us to 4. LHS4033 (from the Luyten Catalog) is probably the most massive, well- studied single white dwarf (2004 ApJ 605, 400). Near 1.32 solar masses, wouldn’t require much a push to go over the edge if it were accreting. SDSS has given us some amazing cases in quadrupling so far the known sample of WDs. ApJ 2004, 606, got us SDSS123410.37-022802.9 (sorry for long phone number) is one of the least massive, about 0.18-0.19 solar, though this value requires believing modelling of helium core WDs. Even more famous is the similar companion to the millisecond pulsar J1012+5307 from your work at Caltech. I think this is best known pulsar companion with very low mass. We have a radio project to test the best SDSS cases for radio emission. This get’s to Snow’s 7. Now I get even more arbitrary. LHS2534 was the first magnetic DZ, with split heavy metal features. Na I ”D” lines split into a nice triplet. ApJ 550 with Reid and Schmidt. Zeeman (1897 ApJ, 5, 332) first reported the Zeeman effect measuring sodium in the lab. This was the first time it was seen in the sky. SDSS J2346+3853 (abbreviated for sanity) in 2005 AJ, 130, 734 has the highest measured field strength near 109 Gauss, in weak neutron star terri- tory. Finally, for number 10, I nominate Feige 7, a rotating magnetic showing for the first time both hydrogen and helium features (1977 ApJ, 214, 457). Discovery of this star probably got me a job at Steward Observatory!

5 A.3 From Lars Bildsten, University of California at Santa Barbara ZZ Ceti, pulsating white dwarf G29-38, odd white dwarf with IR excess (asteroid?) Pleiades White Dwarf Helium White Dwarfs. I suggest the following recent classics: Marsh, Dhillon and Duck (MNRAS, 1995), Fontaine, Brassard and Bergeron (PASP, 2001), Kepler & Bradley (1995, Baltic Astronomy)

A.4 From Marten van Kerkwijk, Moore Scholar in Res- idence, Professor of Astronomy, University of Toronto Sirius B: the original puzzle of hotter but smaller. (Well, that’s what I thought until I read Sterl’s list – clearly, don’t know my history; so maybe 40 Eri B instead.) PSR J1012+5307: how a white dwarf can be much older than one would guess. Still one of the lowest-mass ones known. U Sco: one of the most massive, with predictable recurrent explosions. The one that will tell us what the ratio of ejected over accreted mass is (at least for one system). GD 358: asteroseismology’s first success, yet also one that remains most enigmatic (as when all modes are replaces by a single strong one). RX J0806: shortest binary known, so tight that mass impacts the accretor directly. AE Aqr: only system I know where we see mass being ejected by a propellor. Also the fastest known rotator. LHS 3250: one of the coolest WDs, with opacity due to H2 CIA in NIR and wing of Ly alpha in B. Just bizarre.

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