Binary Stars (Part III)

To start, we apologize for the delay…some of the equations weren’t working, and we didn’t want everyone to be disappointed (and it could have been worse…if this was in binary it would’ve taken longer).  So now that we’ve discussed binary stars at length, we must also address some very necessary calculations for dealing with binary systems — but don’t worry, it’s nothing too complicated. We will not show derivations for the equations we use; they may be found in the sources/links below that cover orbital mechanics in greater depth.

If you’ll remember, Kepler’s Third Law showed that there was a direct relation between the square of a planet’s period and the cube of its orbit’s semi-major axis. We can use a slightly modified version of Kepler’s Third Law  to solve for the total mass of any binary system:

\frac { { a }^{ 3 } }{ { p }^{ 2 } } ={ M }_{ 1 }+{ M }_{ 2 }

  • a = average separation of the two components
  • p = orbital period (doesn’t matter which component — their periods are the same)
  • Note that Kepler’s Third Law in this form requires specific units — AUs, Earth years, and solar masses.

If our problem is written with standard MKS metric units, we turn instead to Newton’s Form of Kepler’s Third Law. Yes, this form of it looks strange, but don’t worry, it IS equivalent to Kepler’s Third Law (multiplied by a constant):

{ p } ^ { 2 } = \frac { 4 { \pi } ^ { 2 } { a } ^ { 3 } } { G ( { M }_{ 1 } + { M }_{ 2 } ) }

  • G = universal gravitational constant, 6.67 * 10^-11 m^3/s^2/kg
  • a = average separation between the components
  • p = orbital period

And now for something completely different. In our last post about binaries, we mentioned that it was possible to use the shifts in spectra to determine the velocity of stars in a spectroscopic binary system. But how exactly would we do this?

For a spectroscopic binary system, we can use the non-relativistic Doppler shift formula:

{ \lambda }_{ obsv }={ \lambda }_{ emit } ( 1 \pm \frac { v }{ c } )

  • λ_obsv = observed wavelength of a given spectral line
  • λ_emit = “normal” wavelength of the same spectral line (i.e. what it would be in a laboratory)
  • v/c is positive when the source is moving away from us, and negative when it is moving towards us

When a star’s spectrum is at its most redshifted, the star is moving away from us at its fastest rate; similarly, maximum blueshift indicates the greatest velocity with which the star is approaching us. A greater difference between the observed and emitted wavelength translates to a more pronounced Doppler effect and a higher velocity for the star. We must apply this equation to both stars, since one is (almost always) more massive than the other and therefore they travel at different speeds.

Now that we have the components’ tangential velocities, we can calculate other vital information through Newtonian mechanics (all circular motion equations are valid, if you are willing to approximate the binary system orbits as circular). There are, however, a few equations you probably won’t see in a normal physics book, such as:

\frac { { M }_{ 1 } }{ { M }_{ 2 } } = \frac { { d }_{ 2 } }{ { d }_{ 1 } } = \frac { { v }_{ 2 } }{ { v }_{ 1 } }

Center of mass (barycenter) is where:

{ { M }_{ 1 } }{ { d }_{ 1 } } = { M }_{ 2 } { d }_{ 2 }

  • d_1 is the distance from component 1 to the center of mass
  • d_2 is the distance from component 2 to the center of mass

Thus, if we can find both the sum of the masses and the ratio of the masses, we can determine the individual mass of both components of the binary system!

But as we said, we have yet to introduce some general circular motion (which is where we can get some of Kepler’s laws stated).   This includes mainly orbital velocity:

v=\sqrt{GM/r}

  • v=orbital velocity
  • r=radius

This in a sense comes from v=C/T , meaning circular velocity is just the distance around a circle (circumference) divided by the time taken (period).  But this applies with bodies orbiting another (like the Earth around the Sun, the Moon around the Earth).

Let’s review another common diagram, which we would like to call the binary star velocity graph:

Time to get you all moving again. From: http://www.astronomynotes.com/starprop/s10.htm

We may have shown this little bugger in the past, but we shall now apply some of the math we learned in this post.  We may not go too in depth, but here the motions of both stars can clearly be seen and plotted.  These curves can tell us whether the radial velocity (or as stated how fast a star is moving) of stars are moving away or towards us (positive means away, negative means towards).

This graph can be in part constructed by finding the doppler shift of a spectrum, finding the velocities of stars, and plotting this graph over time.  Afterwards, more data can be collected about temperature, stellar radii, period of orbit, and orbital separation to further calculations.  But knowing the recessional velocity and orbital velocities of binary stars can be useful to finding mass and other aspects of a binary system.

Up until now we assumed aspects of binaries where we calculated mass were viewed face-on (the plane of orbit is perpendicular to our line of sight), and that our spectroscopic binary systems have been viewed edge-on. But of course, this is almost never the case in reality, and things become more complicated when we take into account the possible inclination of the binary system, not to mention elliptical instead of circular formulas.  For now, let’s just talk about inclination.

Just as anything in physics (kind of) they can be summarized by some formulas!  Ah, life made easy (maybe).  Here is a visual demonstration (the first is no inclination, the second is with a binary inclined at an angle from our view of i):

and

Before this is further explained, an inclination of 90 degrees gets the lower limit of the sum of the masses (when the inclination angle is irrelevant essentially).  Now let’s explain why that’s important by showing what happens when we get an inclination=i.

{v}_{r} = v\sin(i)

{v}_{r} = velocity measured by the doppler shift (it is measured along our line of site, and we know we can use the doppler effect to calculate velocities from spectra)

and this can be plugged into our formula relating period and velocity (and this is orbital velocity, which again brings up the importance of the binary star velocity graph):

\frac{P { ( {v}_{1,r}+{v}_{2,r} ) }^{3} }{2 \pi G {sin}^{3} ( i ) } = {M}_{1}+{M}_{2}

This takes into account the orbital velocities for each star to get the total mass of the system.  In addition, increasing the angle increases the speed seen.

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Sources and links for further reading:

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