Chegg Textbook Solutions for An Introduction to Modern Astrophysics 2nd Edition: Chapter 18

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  • Step 1 of 4

    A star system that consists of two stars orbiting around a common center of mass is known as a binary star system. It is also known as multiple star systems when two or more stars orbit the common center of mass. In a binary star system the star with heavier mass is known as Primary star and the star with the lower mass is known as secondary star or the companion star of the primary star.

    The density along the surface with constant effective gravitational potentialof these binary stars would be expected to be constant because for these binary stars the equipotential surfaces are level surfaces.

    From equation 10.6, the equation for the hydrostatic equilibrium is,

    Here, is the universal gravitational constant, is the mass inside the sphere, is the density of the gas, and is the radius of the sphere.

  • Step 2 of 4

    The general equation the relates the acceleration due to gravity and universal gravitational constant is,

    Here, is the mass of any celestial object andis its radius.

    Relate the acceleration due to gravity of spherical mass with.

    Substitute forin equation.

    From equation 18.8, the equation for the gravitational potential is,

    The equation that relates the effective gravitational potential and gravitational potential energy for test massis,


  • Step 3 of 4

    According to Newton’s second law, the force acting on an object is equal to the product of mass and acceleration.

    Substituteforin equationand solve for.

    Substitute forin equation.

    So, the density of the binary system is constant.

  • Step 4 of 4

    Since the density along the surface of the binary star system is constant, the pressure and the gravitational potential vary spatially. Thus the pressure along the equipotential surface remains to be constant.

    From equation 10.11, the equation for the ideal gas law is,

    Here, is the mass of the hydrogen atom, is the mean molecular weight, is the density of the gas, is the temperature of the gas, and is Boltzmann’s constant.

    Assume that the mean molecular weight is constant. Thus from the equation of ideal gas law, it is clear that for a constant pressure and density along the equipotential surface, the temperature of the surface also remains to be constant. Thus, the argument for the constant temperature of the star’s surface holds good.

    The proximity of the other star could heat up each star on its sides that face the other star.

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