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Appendix 1 Some Astrophysical Reminders Appendix 1 Some Astrophysical Reminders Marc Ollivier 1.1 A Physics and Astrophysics Overview 1.1.1 Star or Planet? Roughly speaking, we can say that the physics of stars and planets is mainly governed by their mass and thus by two effects: 1. Gravitation that tends to compress the object, thus releasing gravitational energy 2. Nuclear processes that start as the core temperature of the object increases The mass is thus a good parameter for classifying the different astrophysical objects, the adapted mass unit being the solar mass (written Ma). As the mass decreases, three categories of objects can be distinguished: ∼ 1. if M>0.08 Ma ( 80MJ where MJ is the Jupiter mass) the mass is sufficient and, as a consequence, the gravitational contraction in the core of the object is strong enough to start hydrogen fusion reactions. The object is then called a “star” and its radius is proportional to its mass. 2. If 0.013 Ma <M<0.08 Ma (13 MJ <M<80 MJ), the core temperature is not high enough for hydrogen fusion reactions, but does allow deuterium fu- sion reactions. The object is called a “brown dwarf” and its radius is inversely proportional to the cube root of its mass. 3. If M<0.013 Ma (M<13 MJ) the temperature a the center of the object does not permit any nuclear fusion reactions. The object is called a “planet”. In this category one distinguishes giant gaseous and telluric planets. This latter is not massive enough to accrete gas. The mass limit between giant and telluric planets is about 10 terrestrial masses. Remarks: – Contrary to hydrogen fusion, deuterium fusion does not play a role in de- termining the nature of the object. The 13 MJ limit between a planet and a brown dwarf results from a convention (and a consensus). – A planet is also (and first?) a body orbiting a star. Free floating brown dwarfs can be observed without necessarily being gravitationally bound to a central object. 550 Marc Ollivier 1.1.2 Gravitation and Kepler’s Laws Kepler’s three laws, even if they were formulated before Newton’s theory1,re- sult from the universal gravitation law that postulates that two masses m1 and m2 separated by a distance R induce, upon one another, an attractive force F parallel to the radius vector R between their respective centers of mass (the so-called central force). This force is given by the relation Gm m R F = 1 2 (1.1) R2 R where G is the universal gravitation constant = 6.67 × 10−11 Nm2 kg−2. Kepler’s laws are valid for two-body systems and remain valid for multiple systems (several planets) in the planetary low mass approximation (compared to the mass of the parent star). They were derived for the solar system but can be generalized to all the planetary systems. The first law, called the orbits law (1605) states that in the heliocentric referential, the orbit of each planet is an ellipse where the Sun is at one focus. The second law, called the areas law, states that whatever the position in the orbit, the movement of each planet is such that the segment joining the Sun and the planet covers similar areas during the same movement duration. The third law, called the periods law (1618) states that for each planet, the ratio between the orbit semi-major axis a cubed and the orbital period T squared is constant, with the relation a3 G(m + m ) Gm =const= star planet ≈ star (1.2) T 2 4 π2 4 π2 where mstar and mplanet are, respectively, the mass of the star and of the planet. 1.1.3 The Solar System In the classical presentation our solar system is composed of a star (the Sun) and nine planets:2 five telluric (Mercury, Venus, the Earth, Mars and Pluto) and four gaseous giants (Jupiter, Saturn, Uranus and Neptune). Table 1.1 summarizes the main characteristics of these planets. 1 Kepler’s (1571–1630) laws were empirically established beginning with detailed ob- servations of Venus, Mars, Jupiter and Saturn’s movements by Tycho Brahe (1546– 1601). The first two laws were published in 1609 in Astronomia Nova, the third was published in 1619 in Harmonices Mundi. They were mathematically demonstrated by Isaac Newton (1642–1727) 2 A more modern presentation would exclude Pluto from this list, because several sim- ilar objects have now been discovered near Pluto’s orbit. These objects are called “transneptunians”. With this new classification, Pluto would be the first and the most massive transneptunian. Observations and present solar system formation the- ories show that Pluto is more comparable to these objects, in terms of mass and orbit, than to Uranus and Neptune. 1 Some Astrophysical Reminders 551 Table 1.1. The main characteristics of the solar system planets Planet Equat. Diam. Mass Distance Period ∗ ∗∗ ∗∗∗ ∗∗∗∗ (km) (m⊕) (AU) (yr) Mercury 4850 (0.38) 0.0554 0.3871 0.2409 Venus 12140 (0.95) 0.815 0.7233 0.6152 Earth 12756 (1.0) 1.00 1.000 1.000 Mars 6790 (0.532) 0.1075 1.5237 1.8809 Jupiter 142600 (11.18) 317.83 5.2028 11.8623 Saturn 120200 (9.42) 95.147 9.5388 29.4577 Uranus 49000 (3.84) 14.54 19.1819 84.0139 Neptune 50200 (3.93) 17.23 30.0578 164.793 Pluto 6400 (0.52) 0.17 39.44 247.7 ∗ in brackets, the fraction of terrestrial diameter ∗∗ 24 m⊕=massoftheEarth=5.976 × 10 kg ∗∗∗ AU = astronomical unit = Earth-Sun mean distance = 1.49598 × 1011 m ∗∗∗∗ 1 yr = 365.25 terrestrialdays = 8766 hours = 31 557 600 s It is interesting to keep in mind several numbers from the previous table in order to compare exosystem characteristics, for instance: – The mass of Jupiter is the mass unit for giant planets. – The mass of the Earth is the mass unit for telluric planets. – The astronomical unit expresses the distance between an (exo)planet and its central star, and is one of the main physical characteristics. – An idea of the distance/period relation. This last point is fundamental in evaluating the observation times required to reconstruct an orbit. 1.1.4 Black Body Emission, Planck Law, Stefan–Boltzmann Law By definition, a black body is an idealized physical body, isolated, constituted of a medium in thermodynamic equilibrium and characterized by a single equi- librium temperature. It is a perfect absorbing medium, and therefore a perfect emitting medium. The radiation field of a black body is isotropic and depends only on the temperature. The spectral distribution of the radiation intensity is given by the Planck function that gives the monochromatic brightness of the black body at a frequency ν as a function of its temperature T 2 hν3 B = Wm−2sr−1Hz−1 (1.3) ν hν − c2 e ( kT 1) 552 Marc Ollivier where ν is the frequency in Hz, h is Planck’s constant = 6.62620 × 10−34 Js,c is the speed of the light = 2.9979 × 108 ms−1,andk is Boltzmann’s constant = 1.38 × 10−23 JK−1. One can plot the Planck function for several values of T (Fig. 1.1). This function reaches a maximum that depends on the temperature. At each temper- ature one can thus associate a color (wavelength of the maximum of emission) and reciprocally, after determination of the emission maximum, determine a so called black body temperature. By integration of the Planck law over all frequencies and all directions one can derive the total power (or flux) emitted by the black body at a temperature T . This relation is known as the Stefan–Boltzmann law F = σT4Wm−2 (1.4) where σ is Stefan’s constant = 5.66956 × 10−8 Wm−2 K−4. Reciprocally, to each source emitting a flux F (measured by a bolometer, for instance) one can associate a temperature called the effective temperature Teff given by the Stefan–Boltzmann law, starting from F . A direct application of the Stefan–Boltzmann law is the calculation of the effective temperature of a planet (radius Rpl, mean albedo A, distance from the star D) in radiative equilibrium with a star emitting a flux S. The equality between the flux received from the star and the flux emitted by the planet can be written as a function of the preceding parameters πR2 S(1 − A) pl =4πR2 σT4 . (1.5) D2 pl eff Fig. 1.1. Plot of the Planck law for several black body temperatures. The higher the black body temperature, the more the emission maximum moves towards short wavelengths 1 Some Astrophysical Reminders 553 In particular, one can see that the equilibrium temperature of the planet (Teff) does not depend on its size, but only on the flux from the star, the albedo and the distance from the star. 1.1.5 Hertzsprung–Russel Diagram, the Spectral Classification of Stars The Hertzsprung–Russel diagram (called later the HR diagram) was established in 1911 by the Danish astronomer Hertzsprung and independently rediscovered in 1913 by the American astronomer Russel. This diagram plots the absolute luminosity of a star (independent of its distance to the Earth) as a function of its effective temperature or another quantity linked to the temperature (for instance, the luminosity difference of the object through two different color filters (two different bandpasses)). This diagram is shown in Fig. 1.2. This diagram is a tool to visualize, not only the stellar morphology diversity, but also stellar evolution with time. Stars have a position on the HR diagram that evolves with time, from their birth to their death.
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