page_title: Binary mass function
section_title: Introduction
hierarchical_section_title: Binary mass function / Introduction
caption_attribution_description: English: Two bodies with a slight difference in mass orbiting around a common barycenter (red cross) with circular orbits. Polski: Dwa ciała o zbliżonych masach krążące wokół wspólnego barycentrum (np. Pluton i Charon)
context_page_description: In astronomy, the binary mass function or simply mass function is a function that constrains the mass of the unseen component in a single-lined spectroscopic binary star or in a planetary system. It can be calculated from observable quantities only, namely the orbital period of the binary system, and the peak radial velocity of the observed star. The velocity of one binary component and the orbital period provide information on the separation and gravitational force between the two components, and hence on the masses of the components.
context_section_description: The binary mass function follows from Kepler's third law when the radial velocity of one (observed) binary component is introduced.
  Kepler's third law describes the motion of two bodies orbiting a common center of mass. It relates the orbital period (the time it takes to complete one full orbit) with the distance between the two bodies (the orbital separation), and the sum of their masses. For a given orbital separation, a higher total system mass implies higher orbital velocities. On the other hand, for a given system mass, a longer orbital period implies a larger separation and lower orbital velocities.
  Because the orbital period and orbital velocities in the binary system are related to the masses of the binary components, measuring these parameters provides some information about the masses of one or both components. But because the true orbital velocity cannot be determined generally, this information is limited.
  Radial velocity is the velocity component of orbital velocity in the line of sight of the observer. Unlike true orbital velocity, radial velocity can be determined from Doppler spectroscopy of spectral lines in the light of a star, or from variations in the arrival times of pulses from a radio pulsar. A binary system is called a single-lined spectroscopic binary if the radial motion of only one of the two binary components can be measured. In this case, a lower limit on the mass of the other (unseen) component can be determined.
  The true mass and true orbital velocity cannot be determined from the radial velocity because the orbital inclination is generally unknown. (The inclination is the orientation of the orbit from the point of view of the observer, and relates true and radial velocity.) This causes a degeneracy between mass and inclination. For example, if the measured radial velocity is low, this can mean that the true orbital velocity is low (implying low mass objects) and the inclination high (the orbit is seen edge-on), or that the true velocity is high (implying high mass objects) but the inclination low (the orbit is seen face-on).
caption: Two bodies orbiting a common center of mass, indicated by the red plus. The larger body has a higher mass, and therefore a smaller orbit and a lower orbital velocity than its lower-mass companion.