Dynamical Parallax
Encyclopedia
In astronomy
, the distance
to a visual binary star may be estimated from the mass
es of its two components, the size of their orbit
, and the period of their revolution around one another. A dynamical parallax is an (annual) parallax
which is computed from such an estimated distance.
To calculate a dynamical parallax, the angular semi-major axis
of the orbit of the stars around each other is observed, together with their apparent brightness. By using Newton's
generalisation of Kepler's
Third Law, which states that the total mass of a binary system multiplied by the square of its orbital period
is proportional to the cube
of its semi-major axis
, together with the mass-luminosity relation, the distance to the binary star can then be determined.
In this technique, the masses of the two stars in a binary system are estimated, usually as being the mass of the sun. Then, using Kepler's laws of celestial mechanics
, the distance between the stars is calculated. Once this distance is found, the distance away can be found via the arc subtended in the sky, giving a preliminary distance measurement. From this measurement and the apparent magnitude
s of both stars, the luminosities can be found, and by using the mass–luminosity relationship, the masses of each star. These masses are used to re-calculate the separation distance, and the process is repeated. The process is iterated many times, and accuracies as high as 5% can be achieved. The mass–luminosity relationship can also be used to determine the lifetime of stars by noting that lifetime is approximately proportional to M/L. One finds that more massive stars live shorter. A more sophisticated calculation factors in a star's loss of mass over time.
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, the distance
Distance
Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
to a visual binary star may be estimated from the mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...
es of its two components, the size of their orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
, and the period of their revolution around one another. A dynamical parallax is an (annual) parallax
Parallax
Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines. The term is derived from the Greek παράλλαξις , meaning "alteration"...
which is computed from such an estimated distance.
To calculate a dynamical parallax, the angular semi-major axis
Semi-major axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
of the orbit of the stars around each other is observed, together with their apparent brightness. By using Newton's
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
generalisation of Kepler's
Johannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
Third Law, which states that the total mass of a binary system multiplied by the square of its orbital period
Orbital period
The orbital period is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.There are several kinds of...
is proportional to the cube
Cube (arithmetic)
In arithmetic and algebra, the cube of a number n is its third power — the result of the number multiplying by itself three times:...
of its semi-major axis
Semi-major axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
, together with the mass-luminosity relation, the distance to the binary star can then be determined.
In this technique, the masses of the two stars in a binary system are estimated, usually as being the mass of the sun. Then, using Kepler's laws of celestial mechanics
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...
, the distance between the stars is calculated. Once this distance is found, the distance away can be found via the arc subtended in the sky, giving a preliminary distance measurement. From this measurement and the apparent magnitude
Apparent magnitude
The apparent magnitude of a celestial body is a measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the atmosphere...
s of both stars, the luminosities can be found, and by using the mass–luminosity relationship, the masses of each star. These masses are used to re-calculate the separation distance, and the process is repeated. The process is iterated many times, and accuracies as high as 5% can be achieved. The mass–luminosity relationship can also be used to determine the lifetime of stars by noting that lifetime is approximately proportional to M/L. One finds that more massive stars live shorter. A more sophisticated calculation factors in a star's loss of mass over time.
See also
- Photometric parallax methodPhotometric parallax methodThe photometric parallax method is a method of data analysis used in astronomy that uses the colours and apparent brightnesses of stars to infer their distances...
- Spectroscopic parallaxSpectroscopic parallaxSpectroscopic parallax is an astronomical method for measuring the distances to stars. Despite its name, it does not rely on the apparent change in the position of the star . This technique can be applied to any main sequence star for which a spectrum can be recorded...