EQUATION SUMMARY
Because of the relatively large
number of defined quantities and mathematical relationships in this unit's
reading, I thought it might help to summarize the parameters associated with
measurements of relatively nearby stars. These nearby stars represent a
data-base that is the foundation for stellar astronomy, especially those for
which we can determine stellar mass.
1. The only direct method for measuring distance to near-by stars is
the method of geometric parallax - the apparent shift of stars relative to the
background of very distant stars when Earth moves through its orbit of the Sun.
The present data-base encompasses about 1 million stars out to a distance of
about 200 pc (almost 700 light-years) from our solar system. You'll see many
other distance estimation methods in future weeks, and they all depend upon this
slender data-base. The simple-looking relation between distance and the
measured parallactic angle "p" is
distance = 1/p, with p expressed in arc-seconds
and distance in parsecs (pc)
2. From light spectrum measurements of individual stars (whether or
not they are in the data-base of relatively nearby stars), astronomers can
estimate a star's surface temperature and apparent brightness (B).
3. For stars in the nearby star data-base (known distance), one can
then calculate luminosity (L) with the relation
L (is proportional to) {B x (distance)2}.
Remember that L has units of power, or watts. While the
luminosity of a star is a fixed quantity determined by the mass (and history) of
the star, the apparent brightness depends on its distance from us.
4. Given the luminosity from step 3, and the surface temperature (T)
from step 2, you can calculate the radius from the
relation
L (is proportional
to) {R2 x T4}, or (solving for R2),
R2 (is proportional to)
{L/T4}
5. To determine the mass of a star
directly, it must be in a multiple-star system (or have planets orbiting it in
known orbits). Fortunately, for stars in that important nearby star data-base,
most of them are in multiple-star systems, and most of those systems are binary
pairs. So our Sun is in a minority of single-star systems. Three-star systems
are less numerous than binaries, and four-star systems are less numerous still.
The following description of mass determination is for stars in a
binary pair. In a binary pair, each star
orbits in an elliptical path about the center of mass of the 2-star
system.
Definition of terms:
a1 = semi-major axis of orbit of star number 1
(units of astronomical units, or A.U.)
a2 = semi-major axis of orbit of star number
2
a = a1 +
a2
M1 = mass of star number 1
(units of solar masses)
M2 = mass of star
number 2
P = period of mutual orbit (units of
years)
Equations used to find
mass:
1. This is a generalized form of Kepler's laws
(first formulated for planets around a star (our Sun) far more massive than any
of the planets):
P2 =
a3/(M1 + M2)
2. The
second equation is a consequence of the definition of the term "center of mass"
for the binary system:
M1/M2
= a2/a1
If the astronomer can
measure a1, a2, and P, she can compute the two (unknown)
masses using the 2 equations. In effect, if one can measure everything about
the mutual orbit, calculation of the individual masses looks easy. Knowing
a1 and a2 also involves knowing the distance from observer
to the star system, so in addition the astronomer must measure that distance.
For our nearby star data-base, this means measuring the geometric parallax and
calculating distance. Naturally, almost nothing in astronomy is really easy,
and there are other complications involved in mass measurements, most of which
astronomers are able to overcome in many cases. This results in a subset of the
nearby star data-base for which we know mass, in addition to the other
quantities like luminosity, size, and surface temperature. Using this data-base
subset, we could then search for correlations between mass and the other
quantities, and we would notice strong correlations for stars on
the main sequence. (i.e.,
not for stars such as white dwarfs and red
giants).
6. Astronomers measure the component of a star's
motion along our line of sight to the star (called radial velocity
component) using the Doppler shift of the star's spectral absorption lines.
This obviously requires measurement of the star's light spectrum using a
telescope and spectrometer.
7. Astronomers obtain the
component of a star's motion perpendicular to our line of sight to the star
(called transverse velocity component) by repeatedly measuring a star's
location versus locations of much more distant stars. This clearly can become
entangled with measurement of stellar parallax, which involves the same kind of
measurement at 6-month intervals. Astronomers are very clever and can sort out
these two different motions - one real (actual transverse motion) and one
apparent (parallax).