The Question
(Submitted June 09, 1997)
I wish to know how we go about calculating the mass of
our Sun and other stars. Are there other ways to measure
bodies in space rather than by gravitational means? Do we
estimate the mass of our Sun by measuring it against the
combined mass of all the planets in our solar system? I'm
interested in the equations and procedures.
The Answer
You've asked about one of the fundamental issues in astronomy,
namely determining the mass of objects such as the Sun and other
stars. The short answer is that there is no other way to *directly*
measure the mass of the Sun or any other star than by observing
the gravitational effects of one object on another.
You can estimate the mass of the Sun one of two ways. Both
use Newton's Laws of motion. The first way uses Newton's revision of
Kepler's third law, which states that the period squared or any body
orbiting the Sun is proportional to its average distance from the Sun
cubed. Newton generalized this for all gravitating systems. In the
case of the Sun, the equation we can use is:
Mass_of_Sun=((4*pi^{2})/G) (a^{3}/P^{2}),
where pi=3.14159, G is a fundamental constant,
a is the radius of the Earth's orbit about the Sun, and P is the orbital
period of the Earth about the Sun.
Another method uses Newton's second law and gravitation. In this case,
one starts with F=m*a (where F is the force on an object, m is the mass
of the object and a is the acceleration of the object due to the force).
Since the gravitational force can be expressed as
F = G(M_{Sun})(M_{earth})/(R^{2})
where M_{Sun} and M_{earth} are the masses of the Sun and Earth, and R is the
distance between the two, and the acceleration for a circular orbit is
equal to the velocity^{2}/R, Newton's second law can be rewritten in this
case to give
Mass_of_Sun=velocity^{2}*R/G
In both cases, putting in the values for Earth's orbital velocity,
distance from Sun and the value for G gives a value of about 2 x
10^{30} kg
for the mass of the Sun. (That's 2 with 30 zeroes after it.)
Since you are obviously very interested in astronomy, I
encourage you to locate a textbook on introductory astronomy at
the college level. It will have discussions of mass determination,
and many of your other questions at about the same mathematical level
as this answer. Some textbooks are not very mathematical at all.
Try 'Astronomy The Cosmic Perspective' by M. Zeilik and J. Gaustad,
or just look around your school or community library for astronomy
texts. In any such text, you will find the values of the Earth's orbital
parameters, and can go about determining the mass of the Sun on your own!
Regards,
Padi Boyd for the Ask an Astrophysicist
