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Binary star systems serve as laboratories for the measurement of star masses through the gravitational effects of the two stars on each other. Three observational types of binaries – namely, visual, eclipsing, and spectroscopic – yield different combinations of parameters describing the binary orbit and the masses of the two stars. We consider an example of each type – respectively, α Centauri, β Persei (Algol), and φ Cygni.

Kepler described the orbits of solar planets with his three laws. They are grounded in Newton's laws. The equation of motion from Newton's second and gravitational force laws may be solved to obtain the elliptical motions described by Kepler for the case of a very large central mass, M ≫ m. The results can then be extended to the case of two arbitrary masses orbiting their common barycenter (center of mass). The result is a generalized Kepler's third law, a relation between the masses, period, and relative semimajor axis. We also obtain expressions for the system angular momentum and energy. Kepler's laws are useful in determining the orbital elements of a binary system.

The generalized third law can be restated so that the measurable quantities for a star in a spectroscopic binary yield the mass function, a combination of the two masses and inclination. This provides a lower limit to the partner mass. […]

What we learn in this chapter

A normal star is basically a ball of hot gas held together by gravity. Processes that underlie the stability of a star begin when the stellar matter is still part of the diffuse interstellar medium (ISM). A portion of the ISM can not begin condensation to higher densities unless its size exceeds the Jeans length. Its gravitational potential must be sufficient to prevent the escape of individual atoms with thermal kinetic energies.

A star is in hydrostatic equilibrium when the inward pull of gravity on each mass element of the star is balanced by the upward force due to the pressure gradient at the location of the element. The potential and kinetic energies of the mass elements summed over an entire star in hydrostatic equilibrium yield the virial theorem. The theorem states that the sum of twice the kinetic energy and the (negative) potential energy equals zero. Its application to clusters of galaxies indicates they are bound by a preponderance of dark matter.

Several time constants characterize a star. A star would radiate away its current thermal content at its current luminosity in the Kelvin–Helmholtz or thermal time. In the dynamical time, a mass element at radius r without pressure support would fall inward a distance r under the influence of the (fixed) gravitational force at r. A photon will travel from the center of the star to its surface through many random scatters in the diffusion time. […]

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A hot plasma of ionized atoms emits thermal bremsstrahlung radiation through the Coulomb collisions of the electrons and ions. The electrons experience large accelerations in the collisions and thus efficiently radiate photons, which escape the plasma if it is optically thin. The energy Q radiated in a single collision is obtained from Larmor's formula. The characteristic frequency of the emitted radiation is estimated from the duration of the collision, which, in turn, depends on the electron speed and its impact parameter (projected distance of closest approach to the ion). Multiplication of Q by the electron flux and ion density and integration over the range of speeds in the Maxwell–Boltzmann distribution yield the volume emissivityjν(ν) (W m−3 Hz−1), the power emitted from unit volume into unit frequency interval at frequency ν as a function of frequency. It is proportional to the product of the electron and ion densities and is approximately exponential with frequency. A slowly varying Gaunt factor modifies the spectral shape somewhat. Most of the power is emitted at frequencies near that specified by hν ≈ kT.

Integration of the volume emissivity over all frequencies and over the volume of a plasma cloud results in the luminosity of the cloud. […]

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Albert Einstein postulated that the speed of light has the same value in any inertial frame of reference or, equivalently, that there is no preferred frame of reference. The consequence of this postulate is the special theory of relativity, which yields nonintuitive relations between measurements in different inertial frames of reference. We demonstrate the Lorentz transformations for space and time (x, t) and the compact and invariant four-vector formulation. From this, the four-vectors for momentum-energy (p, U) and wave propagation-frequency (k,ω) are formed, and these in turn yield the associated Lorentz transformations. The transformations for electric and magnetic field vectors are also presented. Examples of each type of transformation are given. The relativistic Doppler shift of wavelength or frequency is derived from time dilation and also directly from the k, ω transformations. The latter yield the transformation of radiation direction (aberration) from one inertial frame to another. Stellar aberration explains the displaced celestial positions of stars due to the earth's motion about the sun.

Astrophysical jets often emerge from objects that are accreting matter such as protostars, stellar black holes in binary systems, and active galactic nuclei (AGN) of galaxies. With our special-relativity tools, we study three aspects of the jet phenomenon: the beaming of radiation from objects traveling near the speed of light, the associated Doppler boosting of intensity, and superluminal motion. […]

What we learn in this chapter

A photon gas in perfect thermal equilibrium with its surroundings at some temperature T will exhibit an energy spectrum of a specific amplitude and shape known as the blackbody spectrum, which was first proposed by Max Planck in 1901. In its form as a specific intensityI(ν) (W m−2 Hz−1 sr−1), the blackbody spectrum peaks at a frequency proportional to its temperature. At low frequencies (the Rayleigh–Jeans approximation), it increases linearly with temperature and quadratically with frequency. At high frequencies (the Wien approximation), it decreases quasi-exponentially. The energy density, ∝ T4, and photon number density, ∝ T3, follow directly from I(ν). The former is closely related to the pressure of a photon gas, whereas the latter is closely related to the distribution function, the number density in six-dimensional phase space. Calculation of the average photon energy yields 2.70 kT.

The total energy flux (W) passing in one direction through a unit surface is proportional to T4. A normal gaseous (spherical) star emits a spectrum that approximates (roughly) that of a blackbody, which allows the luminosity to be expressed in terms of the stellar radius and an effective temperature. Momentum transfer by the photons to a hypothetical surface yields a pressure that is one-third the energy density. The blackbody flux is the maximum intensity that can be obtained from a thermal body. […]

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The normal Compton effect involves the collision of a photon with a nearly stationary electron. In the inverse process, a high-energy electron gives energy to a photon. This process, known as inverse Compton (IC) scattering, is important in the jets of active galactic nuclei and in clusters of galaxies.

Momentum and energy conservation yield the energy and scattered angle of the electron in the normal effect. The increase of the photon energy in the IC process follows from the application of the normal effect in the rest frame of the energetic incident electron. The final result is that a relativistic electron with Lorentz factor γ = U/mc2 will increase the photon energy by a factor of ∼γ2. In jets, for example, this can propel x-ray photons up to extreme gamma-ray energies detected by TeV gamma-ray astronomers. The modification of photon spectra due to single or multiple IC scatters is called Comptonization.

Energetic electrons in a nebula containing magnetic fields will radiate by the synchrotron process. The synchrotron photons may then interact with the energetic electrons that created them via the IC process and thus be boosted to extremely high energies. This is known as the synchrotron self-Compton (SSC) process. […]

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Radio waves traversing the interstellar medium (ISM) reveal a great deal about the medium because the waves are modified by plasmas and magnetic fields during their transit through the Galaxy. Dispersion is the variation with frequency of the group velocity of a radiation pulse. The measured spread of arrival times is directly related to the dispersion measure (DM), which is the integral of the electron density ne along the line of sight.

Faraday rotation is the rotational displacement of the electric vector of linearly polarized (LP) radiation about the propagation direction. The measured quantity, the rotation measure (RM), is the integral along the propagation path of the product neBz, where Bz is the component of the galactic magnetic field B parallel to the propagation direction.

The relations between the interstellar quantities and these two phenomena follow from Maxwell's equations as they apply to a dilute plasma. Their solution leads to a frequency-dependent phase velocity in terms of the dielectric constant of the medium. The square root of the latter is the frequency-dependent index of refraction. This in turn leads to the dispersion relation – a relation between the index of refraction, frequency, and wavelength. […]

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The structure of a star may be modeled with the aid of the four fundamental equations of stellar structure – namely, mass and luminosity distributions, hydrostatic equilibrium and radiative (or convective) transport – together with the equation of state (EOS) and several secondary equations. Each of these describes some essential applicable physics that must be obeyed at each point of the stellar interior. Given a stellar mass and elemental composition, these equations can be used to create a model of a star that yields the radial distribution of mass density, pressure, temperature, and luminosity within the star. The adiabatic constraint leads to a condition for convective energy transport in stars.

Hydrogen-burning stars of different masses lie along the main sequence on a Hertzsprung–Russell (H-R) diagram, which is a plot of luminosity versus temperature. Stars of high mass burn their hydrogen rapidly, in millions of years, whereas stars of low mass can burn hydrogen stably for billions of years. After leaving the main sequence, a star becomes a red giant or a supergiant and eventually evolves to become a white dwarf, neutron star, or black hole. A group of stars created together from a single collapsing cloud at a given time is known as a globular cluster or an open cluster. H-R diagrams of cluster stars dramatically exhibit the differing ages of clusters and the results of stellar evolution.