Compact Binaries Part I Frank Verbunt this version: March 13, 2015 these lecture notes will be extended and adapted as the course proceeds. please check the date above to see whether your download is the most recent one. Astronomical and physical constants Physical constants are constant, but the values we assign to them improve with the quality of measuring apparatus. That is why the date must be given at which the value one uses is taken. The National Institute of Standards and Technology in Maryland, U.S.A., keeps track of new measurements of physical constants, and every few years updates the list of recommended values. The information is collected on a website: http://physics.nist.gov/cuu/Constants This site also gives the paper in which the (updates of) the constants are described. From the December 2007 preprint of this paper we copy Table XLIX below, which is based on the 2006 adjustment. based on the 2006 adjustment. Relative std. Quantity Symbol Numerical value Unit uncert. ur 1 speed of light in vacuum c, c0 299 792 458 m s− (exact) 7 2 magnetic constant µ0 4π 10− N A− × 7 2 = 12.566 370 614... 10− N A− (exact) 2 × 12 1 electric constant 1/µ0c ǫ0 8.854 187 817... 10− F m− (exact) Newtonian constant × 11 3 1 2 4 of gravitation G 6.674 28(67) 10− m kg− s− 1.0 10− × × 34 8 Planck constant h 6.626 068 96(33) 10− J s 5.0 10− × 34 × 8 h/2π ¯h 1.054 571 628(53) 10− J s 5.0 10− × 19 × 8 elementary charge e 1.602 176 487(40) 10− C 2.5 10− × 15 × 8 magnetic flux quantum h/2e Φ0 2.067 833 667(52) 10− Wb 2.5 10− 2 × 5 × 10 conductance quantum 2e /h G0 7.748 091 7004(53) 10− S 6.8 10− × × 31 8 electron mass me 9.109 382 15(45) 10− kg 5.0 10− × 27 × 8 proton mass mp 1.672 621 637(83) 10− kg 5.0 10− × × 10 proton-electron mass ratio mp/me 1836.152 672 47(80) 4.3 10− 2 3 × 10 fine-structure constant e /4πǫ0¯hc α 7.297 352 5376(50) 10− 6.8 10− 1 × × 10 inverse fine-structure constant α− 137.035 999 679(94) 6.8 10− × 2 1 12 Rydberg constant α mec/2h R 10 973 731.568 527(73) m− 6.6 10− ∞ 23 1 × 8 Avogadro constant NA, L 6.022 141 79(30) 10 mol− 5.0 10− × 1 × 8 Faraday constant NAe F 96 485.3399(24) C mol− 2.5 10− 1 1 × 6 molar gas constant R 8.314 472(15) J mol− K− 1.7 10− 23 1 × 6 Boltzmann constant R/NA k 1.380 6504(24) 10− J K− 1.7 10− Stefan-Boltzmann constant × × 2 4 3 2 8 2 4 6 (π /60)k /¯h c σ 5.670 400(40) 10− W m− K− 7.0 10− × × Non-SI units accepted for use with the SI 19 8 electron volt: (e/C) J eV 1.602 176 487(40) 10− J 2.5 10− (unified) atomic mass unit × × 1 12 27 8 1u= mu = 12 m( C) u 1.660 538 782(83) 10− kg 5.0 10− 3 1 × × = 10− kg mol− /NA Table C.1 Physical Constants, from http://physics.nist.gov/cuu/Constants Note that the first three constants are defined and therefore have no measurement errors! i In addition to the constants given in Table C.1, we list in Table C.2 some derived physical constants (from the same source, Table L), and also some astronomical constants, taken from Section K6 of the Astronomical Almanac for the year 2006. The astronomical constants are defined by the Interational Astronomical Union at values close to the actual value, and therefore have no errors, unlike the measured values. In the exercises throughout this lecture, use 3 digits; except in the computer exercises, where the full accuracy is used. quantity symbol numerical value (w. error) Additional physical constants −27 neutron mass mn 1:674927211(84) × 10 kg = 1.00866491597(43) amu a ≡ 4σ=c a 7:56591(25) × 10−1? J cm−3K−4 2 2 −15 electron radius re = e =4πomec 2.817940325(28)×10 m −13 e Compton wavelength ~=mec = re/α 3.861592678(26)×10 m 2 2 2 −10 Bohr radius a1 = 4πo~ =mec = re/α 0.5291772108(18)×10 m wavelength at 1 keV hc=keV 12.3984191(11) A˚ keV−1 2 2 Rydberg energy mec α =2 13.6056923(12) eV 2 Thomson cross section σT = 8πre =3 0.665245873(13) barn a a ≡ 4σ=c 7.56577(5)×10−16 J m−3 K−4 −3 Wien constant b = λmaxT 2.8977685(51)×10 m K Astronomical constants 20 3 −2 solar mass GM 1.32712442076×10 m s 30 solar mass M 1:9884 × 10 kg 8 solar radius R 6:96 × 10 m astronomical unit A:U: 1:49597871 × 1011 m Julian year yr 365:25 × 86400 s unofficial: 26 −1 solar luminosity L ' 3:85 × 10 J s parsec pc ' 3:086 × 1016 m Solar temperature Teff '5780 K −1 −1 Hubble constant Ho 70 km s Mpc Table C.2 Additional physical constants from http://physics.nist.gov/cuu/Constants Table L and astronomical constants, from Astronomical Almanac for 2006, K6 Some remarks on angles A circle is divided in 360 degrees (◦), each degree in 60 minutes (0), and minute in 60 arcseconds (00). 2π radians thus correspond to 360 × 60 × 60 arcseconds, i.e. 100 corresponds to ' 4:848 × 10−6radians. The circle of right ascension is divided into 24 hours (h), each hour in 60 minutes (m), each minutes in 60 seconds (s). The arcsecond (00) has a fixed size on the celestial dome, whereas the second (s) has an extent which depends on the declination. At the equator 1h corresponds to 15◦, at declination δ, 1h corresponds to 15◦ cos δ. ii Chapter 1 Historical Introduction This chapter gives a brief historical overview of the study of binaries in general and compact binaries in particular, and in doing so explains some of the terminology that is still used. 1.1 History until 19001 It has been noted long ago that stars as seen on the sky sometimes occur in pairs. Thus, the star list in the Almagest of Ptolemaios, which dates from ±150 AD, de- scribes the 8th star in the constellation Sagittarius as `the nebulous and double (diplouc~ ) star at the eye'. After the invention of the telescope (around 1610) it was very quickly found that some stars that appear single to the naked eye, are resolved into a pair of stars by the telescope. The first known instance is in a let- ter by Benedetto Castelli to Galileo Galilei on January 7, 1617, where it is noted that Mizar is double.2 Galileo observed Mizar himself and determined the distance between the two stars as 1500. The discovery made its way into print in the `New Almagest' by Giovanni Battista Riccioli in 1650, and as a result Riccioli is often credited with this discovery. In a similar way, Huygens made a drawing showing that θ Orionis is a triple star (Figure 1.1), but the presence of multiple stars in the Orion nebula had already been noted by Johann Baptist Cysat SJ3 1618. The list of well-known stars known to be double when viewed in the telescope includes the following: year star published by comment 1650 Mizar (ζ UMa) Riccioli found earlier by Castelli 1656 θ Ori Huygens triple, found earlier by Cysat 1685 α Cru Fontenay SJ 1689 α Cen Richaud SJ 1718 γ Vir Bradley 1719 Castor (α Gem) Pound 1753 61 Cygni Bradley 1This Section borrows extensively from Aitken 1935 2see the article on Mizar by the Czech amateur astronomer Leos Ondra on leo.astronomy.cz 3SJ, Societatis Jesu, i.e. from the Society of Jesus: a Jesuit. Jesuits attached great importance to education and science, and during the counter-reformation trained good astronomers. Examples are the first European astronomers in China: Ricci (1552-1610) and Verbiest (1623-1688); and the rediscoverers of ancient Babylonian astronomy: Epping (1835-1894) and Kugler (1862-1929) 1 Figure 1.1: Drawing of the Orion nebula made by Huygens (left) compared with a modern photograph (from www.integram.com/astro/Trapezium.html, right). All these doubles were not considered to be anything else than two stars whose apparent positions on the sky happened to be close. Then in 1767 the British astronomer John Michell noted and proved that this closeness is not due to chance, in other words that most pairs are real physical pairs. An important consequence is visual that stars may have very different intrinsic brightnesses. Michell argues as follows binary (for brevity, I modernize his notation). Take one star. The probability p that a single statistical other star placed on an arbitrary position in the sky is within x◦ (=0.01745x rad) from the first star is given by the ratio of the surface of a circle with radius of x degrees to the surface of the whole sphere: π × (0:01745x)2=(4π) ' 7:615 × 10−5x2, for p 1. The probability that it is not in the circle is 1 − p. If there are n stars with a brightness as high as the faintest in the pair considered, the probability that none of them is within x degrees is (1 − p)n ' 1 − np, provided np 1.