# General Astronomy/Mass

The mass of a star - along with its chemical composition - are key to its behavior. Once these two features of a star are determined, one can calculate the temperature of the star's core and the pressures of its gases as well as the strength of the gravitational force acting within the star itself. The star's mass and composition can, therefore, determine the star's size and luminosity.

How, then, do we begin to calculate a star's mass? Binary stars can help us determine the mass of stars.

Binary systems are composed of two stars that orbit each other. Mizar and Alcor, for instance, are two stars that are apparent (not actual) binaries; they appear to be very close together in the sky, they are actually distant from each other. Mizar, however, is a true binary, made up of Mizar A and Mizar B, with a period of 10,000 years.

Because binary stars' orbits depend on their gravitational attraction to each other, the measurement of their orbital distance and the length of time it takes them to complete one orbit can be used to calculate the summation of the stars' masses. Hence, the following equation can be used:

${\displaystyle (M_{1}+M_{2})P^{2}=(D_{1}+D_{2})^{3}=R^{3}}$

where ${\displaystyle M_{1}}$ and ${\displaystyle M_{2}}$ are the masses of the two stars, ${\displaystyle P}$ is the period of the orbit, ${\displaystyle D_{1}}$ and ${\displaystyle D_{2}}$ are the distances from the center of each star to the center of mass of the binary system, and ${\displaystyle R}$ is the total separation between the centers of the two stars, or ${\displaystyle (D_{1}+D_{2})}$.

Astronomers usually express the masses of astronomical objects in terms of our Sun's mass, where one solar mass is approximately equal to 1.99 x 1030 kilograms, or about 4.39 x 1030 pounds.

An astrometric binary is found by observing a star's proper motion. If it is wavy, that indicates the presence of a binary; non-binary stars have a straight line proper motion.

An eclipsing binary is detected by observing a star as it becomes occluded by its twin. An example of this is Algol (Beta Persei in Perseus).

There are three different types of binary systems of stars which provide information about the masses of those stars. Visual binaries refer to those stars which can each be visibly seen by astronomers through telescopes. The masses of these types of stars can be calculated when they are near enough to the Earth for astronomers to determine the size and tilt of the two stars' orbits about each other.

Another type of binary system that provides information about stellar masses is known as the spectroscopic binary. Using spectroscopic observations of the two stars, the doppler shifts of the two stars as they orbit each other can be measured and astronomers can collect sufficient data in order to find estimates for the masses of those binary stars. In this type of binary, astronomers are only able to conclude upon a lower limit of the masses because they must first know the orientation of the stars in their orbits. If the two stars do not form an eclipsing binary, then one of the stars never passes in front of the other, so astronomers are unable to calculate the orbit's orientation.

As stated previously, the use of binary systems to calculate star masses often leads to limited information. The above method of determining mass is very useful given that the other values are known, but obtaining these needed values can prove to be troublesome. When observing an astrometric binary, for example, we are unable to see the exact orbits. Moreover, even mapped orbits are problematic because they are two-dimensional representations of a truly three-dimensional motion of two stars. In either case, additional information is needed to calculate the orbits and masses of the stars more accurately. As a result, it is oftentimes easier to calculate the total mass of the binary system.

Astronomers have calculated that all stars that have been measured to date have masses that range from ${\displaystyle {\frac {1}{50}}}$ to 50 solar masses.

The mass of a star can then be used to determine its escape velocity: the velocity necessary for an object to escape the star's gravitational force. The escape velocity becomes greater as a star is more massive but decreases with the star's radius. The formula for calculating escape velocity is

${\displaystyle {\sqrt {\frac {2GM}{R}}}}$

where ${\displaystyle M}$ is mass, and ${\displaystyle R}$ is radius.

## Elements

Besides providing light, stars have another important role in the development of life: creation of elements! Stars produce many elements during their life cycles, which are thrown off through their solar wind, when they blow off their outer atmosphere and form planetary nebulae and most spectacularly, when they go supernova. Indeed, all elements, except for hydrogen and helium, were produced by stars. This means that the material making up plants, animals, even you, was at one time manufactured by a star! The famous astronomer Carl Sagan once observed, "We are made of star-stuff."

Main sequence stars are made up of hydrogen (H) and helium (He). Red giants are made of elements heavier than helium, up to and including iron (Fe). Iron is the most stable element, meaning that it's the hardest to transmute. Supernovas produce all elements heavier than iron, up to and including Californium (Cf).