2.6 Modelling Binary Stars
2.6.1 System Geometry
One of the most accurate, and most commonly used, models for binary stars is the Roche model (Kallrath & Milone, 2009). This model is based on a number of simple assumptions, namely that both the stars behave as point masses surrounded by massless envelopes, there is no differential rotation, and the stars do not have appreciable radial pulsations (this is to ensure that the shape of star is, at any time, defined solely by the instantaneous gravitational force). The Roche model attempts to explain the morphology and interaction of the stars from this tidal
Figure 2.6: Roche potential lobes and morphology of a detached binary system. From a set of examples provided by Bradstreet (1993), from Kallrath & Milone (2009)
force. An expression for this potential was initially derived by Kopal (1959), and generalised by Wilson (1979) to apply to elliptic orbits.
We obtain an expression for the potential which can be solved iteratively for all points, with the equipotential surfaces determining the morphology of the star. This is most commonly computed by calculating the potential at the stellar pole, and using this as the value on equipotential surface.
Both objects will have a tear-drop shape with the apex pointing towards the other star, an equipotential shape known as the Roche Lobe. The filling of this Roche Lobe determines the morphological classification the system. In the case that the Roche Lobe reaches the first Lagrange point, L1 mass transfer will occur (symbiotic binaries). In the case ofζAurigae both stars are bound within separate equipotential surfaces, and their evolution is more-or-less independent from one another, a configuration known as a detached binary system, see Fig 2.6.
This distortion of the stellar surface gives rise to a orbital modulation of the flux, as we see a larger surface cross-section at quarter-phases. This effect is known as ellipsoidal variation, and has the effect of introducing a fluctuation in the light curve from the binary system, even in non-eclipsing systems (though clearly systems viewed pole-on would not display this effect).
With the Roche model determining the shape of the objects in the system, the dynamical orbit of the system is the classic Kepler two-body problem. In the case of doubly-lined spectroscopic binaries (i.e. ζ Aurigae) our most important observable is the radial velocity curves of the two objects, which is the velocity component along the line-of-sight for both objects obtained from measurements of line Doppler shifts. From this radial velocity curve we define the semi-amplitude of the radial velocity for each object,
Ki = 2π T aisini √ 1−e2 (2.138)
whereT is the period, e is the eccentricity,i is the inclination of the orbit, and ai
is the semi-major axis of the orbit of i. The semi-amplitude of the radial velocity is half of the amplitude of the measured radial velocity curve for a given object. Since the distance of the objects from the barycentre will, at each instant, be in direct ratio to their masses, we can determine the mass ratio as
M1 M2 = a2 a1 = K2 K1 (2.139) Using Kepler’s third law we can determine the sum of the masses,
T2 = 4π
2
G(M1+M2)
a3 (2.140)
wherea=a1+a2. Substituting in the expression for the semi-major axis, in terms of the semi-amplitude of the radial velocity we obtain
M1+M2 =
T(1−e2)3/2
2πGsin3i (K1+K2)
3 (2.141)
Now that we have determined the sum and the ratio of the masses we can compute the individual masses of the components. This approach is only possible if the inclination is known, and this is difficult to ascertain, however for eclipsing object close attention to the light-curve can determine the inclination. Of course if the objects are eclipsing the inclination must be almost π/2 with respect to the plane-of-sky. The best determined stellar masses come from this type of binary analysis, as this method is not dependent on the (often not well known) distance
Figure 2.7: In this diagram the orbital plane is coloured yellow, and intersects a reference plane which is grey, with the orbital elements annotated. Image Credit: Wikimedia
to the system.
In order to study the orbits in detail we must define a number of important parameters, the six Keplerian orbital elements in the reduced mass frame. These can be seen graphically in Fig. 2.7.
The two which determine the shape of the orbit, and which were important in calculating the masses, are
• The semi-major axis a, the length of the longest diameter of the ellipse divided by two (a=a1+a2).
• The eccentricity is defined ase= 1−(rp/a) whererp is the minimum distance
between the objects, the point of periastron.
The two which determine the orientation of the orbit (with respect to the observer) are
• Inclination, i, the offset of the orbit from the reference plane.
• The longitude of the ascending node, Ω, is the angle between where the orbit passes upward through the reference plane and a reference direction. In general the reference plane is the plane-of-sky and so, as can be seen in our diagram, rotations about the axis connecting the barycentre and the reference direction do not have any physical significance, hence Ω can be arbitrarily fixed.
The final elements are
• The argument of periapsis,ω, is the angle measured from the ascending node to the point of periastron.1
• The true anomaly, ν, which introduces a time varying property (ignoring apsidal motion), is the angle between the point of periastron, the focus of the orbit, and the position of the orbiting object at any given time (see Fig. 2.8).
In order to compute ν we must define two auxiliary anomalies, the mean anomaly, M, and the eccentric anomaly, E. The mean anomaly is not in fact an angle, it simply varies linearly over the orbit from 0 to 2π
M(t) = 2π((t−t0) mod T) (2.142) where t0 is some reference time, usually periastron or mid-eclipse.
The eccentric anomaly is the angle between the major axis and a line connecting the centre of the orbit, to the point where a line perpendicular to the major axis, passing through the object, intersects a circle inscribing the orbit (Fig. 2.8). The eccentric anomaly is related to the mean anomaly by
M =E−esinE (2.143)
1This value may change, a phenomenon known as apsidal motion, as a result of tidal inter-
action, perturbations, the stellar quadrupole, or general relativistic effects (famously so in the case of Mercury)
Figure 2.8: The eccentric anomaly,E, is the angle between major axis and a line connecting the centre of the orbit, to the point where a line perpendicular to the major axis, passing through the object, intersects a circle inscribing the orbit. The true anomaly is the angle between the point of periastron, the focus, and the orbiting object.
This equation, being transcendental, must be solved by numerical methods, and WD uses the iterative method of Pad´e approximants
xi+1 =xi+ ∆xi, ∆xi =− f(x)f0(x) f0(x)2−1 2f 0(x)f00(x) (2.144) ∆Ei = (Ei−esinEi−M)(1−ecosEi)
(1−ecosEi)2− 12(Ei−esinEi−M)esinEi
(2.145) Once we have solved for E, ν can be computed by
tanν 2 = r 1 +e 1−etan E 2 (2.146)
With the true anomaly computed we can now specify the coordinates of the object as it orbits, given the instantaneous separation of the objects
D= a(1−e)
1 +ecosν (2.147)
We have now fully specified the relative positions of the objects as a function of time.
Figure 2.9: Synthetic spectra from Kurucz. We can see that the spectra differ from a simple blackbody spectrum, especially at the Balmer jump. Image credit:
http: // kurucz. harvard. edu/