FORMULAE FOR BINARY STARS
------------------------
John Pazmino
NYSkies Astronomy Inc
www.nyskies.org
nyskies@nyskies.org
2019 November 9
Introduction
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I give here a collection of formulae relating to binary star
astronomy. They also are applicable to other sectors of astronomy.
While i explain them here, derivation as such is skipped. Due to the
variety of formula, details are dispersed in astronomy litterature
Parallax and distance
-------------------
The distance to a star is
(distance in parsecs) = 1 / (parallax in arcsec)
This defines the 'parsec' as that distance for which the parallax
is one arcsecond, period. It is about 3.08 lightyears.
because parallax is always a very small number, the small angle
rules are sufficient to find the distance for any other parallax,
being merely the reciprocal of the parallax.
Note well that parallax on the stars is based on the Earth orbit
radius while parallax within the solar system is bamked off of the
Earth's global radius.
Linear separation
---------------
The linear separation between stars of a binary system is
(lin se in AU) = (ang sep in arcsec) * (distance in parsec)
This is for stars in a binary system, both stars at the same
distance away. It can not apply to stars in a constellation or a line-
of-sight double star.
The separation is the frontal, face-on, projected, value. it
ignores dilution by orbit inclination to the line of sight. It is a
minimum value.
Combined mass
-----------
The combined mass of stars in a binary system is
(mass1 + mass2) = (SMA^3) / (period^2)
SMA is the semimajor axis of the dinary orbit. Binary star orbits
are Tupcily very excentric. tHEY HAVE NO OBVIOUS 'RADIUS' LIKE eARTH'S
ALMOST CIRCULAR ORBIT.
The units are solar mass, Earth orbit radius, Earth years. Other
sets of units require reduction to these three.
Kepler found this formula as (orbit radius^3) = (orbit period^2)
for the solar system. The term (mass1+mass2) is very nearly unity.
because the Sun, mass1, is so very much greater than for any planet,
mass2. his. Newton worked out the full formula from his theory of
gravity.
= =newton's gravity theory was at first only pertinent to the solar
system,, there in his time the only place of orbital motion. Herschel
discovered binary stars in 1790s, showing tat gravity is truly
universal.
Individual masses
---------------
The individual masses of stars in a binary system are
mass1 / mass2 = RV2 / RV1 = deviation2 / deviation1
RV1 and EV2 are the Doppler radial velocities of the two stars at
a given moment, typicly the maximum values.
deviation1 and deviation2 are the excursion of the stars from
linear proper motion, also at the same moment, again tyicly at maximum
value
Both RV and deviation are face-on or frontal parameters. They may
be diluted by orbit inclination against our line of sight. The masses
are minimum values, as if the orbit was edge-on to us. .
The radial velocity is generally eas/ier to observe than proper
motion excursion.
The more massive star has the lower radial velocity and deviation.
Telescope resolution
------------------
The nominal resolution of a telescope is
(resolution in arcsec) = (120 arcsec.mm) / (aperture in mm)
\
This is an ideal limit for two stars of 6th magnitude, stable air,
good optics, good eyesight. Individual achievement vary widely.
For unequal comites of more than two magnitudes difference, the
resolution may fall short of the value in this formula.
Home telescopes rarely achieve the ideal resolution because the
atmosphere is hardly ever calm and stable. The image of a star is
swelled-out dot. These dots for a double star overlap and blend while
still farther apart than the ideal limit.
Spectrometric distance
--------------------
The spectrometric distance of the star us
log(distance in pc) = (Mapp - Mabs + 5) / 5
Mapp and Mabs are the apparent and absolute magnitudes of the
star. The absolute magnitude is f a standard Hertzsrung-Russell
Diagram. From examination of the star's spectrum the spectral class
and Main Sequence residence are determined.
The star is marked on the Main Sequence against its spectral
class, From this point the absolute magnitude is read out and put into
the formula.
This procedure is dome for both stars of the binary system. The
formula should give the same distance for both.
According as the spectrometer sued, the star's temperature or
color index may be captured in the place of the actual spectral class.
These can locate the star on the Main Sequence equally well.
This distance method is good when the star really is a Main
Sequence star. For stars off of the Main Sequence the plot is more of
a guesstimate since such stars have no definite place on the HRD. The
uncertainty carries into the spectrometric distance.
Radial velocity
-------------
The radial velocity of a star in a binary system is
EV = SOL * (wavelengthR - wavelengthE) / wavelengthE
WeavelengthR is that received for a given spectral line in the
star's spectrum.
WavelengthE is that emitted for the same line at rest, like in a
laboratory spectrum.
SOL is the speed of light, almost always rounded to 300,000 KPS.
Wavelength R is the Dooppler-shifted wavelengthE caused by the
line-of-sight motion of the star.
A positive RV, wavelengthR greater than wavelenhthE, is a
recession from us. A negative EV, wavelengthR less than wavelenghtE,
is an accession toward us.
RV is only the radial, line-of-sight component of the star's full
spatial motion. The transverse or tangential motion is found by
astrometry on the star's proper motion.
Total magnitude
-------------
The total magnitude of a double star is
Mtot = -2.5 * log(1 + alg(-0.4 * Mdiff)))
1Mtot is the combined brightness of the two stars seen as a single
ybut with no separation between them.
Mdiff is the difference of the two magnitudes of the stars, in the
order Mdim - Mbright. This is a positive number.
alg is the inverse log, alg(X) = 10^(X). Sci/tech calculettes have
a specfic alg feature, else use the power of 10.
Mtot is always brighter than Mbright.
Sun's brightness
--------------
The brightness of our Sun as seen from the star is
(Sun app magn) = (5 * log(pc distance)) - 0.2
This is based on an absolute magnitude of Sun of +4.8 magnitude.
The formula is a disguised inverse square law, comparing the Sun
at 10 parsec to that at its actual distance from the star.
Luminosity
--------
The luminosity of the star is
log(luminosity = 0.4 * (((Sun magn) - (star magn))
The magnitudes are both absolute or both apparent, which ever pair
is to hand.
The Sun apparent magnitude is that seen from the star, not Earth.
The absolute magnitude of Sun is +4.8.
Mind the subtraction order, Sun-minus-star. Reversing it yields
silly answers or an error. To help reduce mistakes the formula is
written to remove minus signa.
The luminosity is typicly greater than the Sun's because most
stars are very remote from us. The Sun appears from the star much
fainter than the star appears to us.
Conclusion
--------
Most of the formulae here are first studied by home astronomers
while exploring double stars. With some star smarts they find
application in other fields of astronomy.
Some are jump off points for further investigation. The formula
for the Sun's brightness in the star's sky can show how the people on
an exoplanet see our Sun in their sky. Comparing that to our view of
the central star of that exoplanet gives us insight to the standing of
our solar system among the many others so far known.
All of these formulae employ simple algebra, handled by the
sci/tech calculette in the home astronomer's toolbox.Yet it is
sometimes amazing just how much deeper into the workings of the
universe they can lead us.