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This all implies that in order for the binary to be able to form a terrestrial planet in
its HZ, its eccentricity cannot have large values. In a binary with a small eccentricity,
the deviation of the planet's orbit from circular is also small and appears in the form
of secular changes with long periods (see, e.g. Eggl et al.
2012
). Therefore, to use
Eq. (
13.2
), one can approximate the orbit of the planet by a circle without the loss
of generality.
The habitability of a planet in a binary system also requires long-term stability
in the HZ. For a given semimajor axis a
Bin
, eccentricity e
Bin
and mass ratio of the
binary, there is an upper limit for the semimajor axis of the planet beyond which
the perturbing effect of the secondary star will make the orbit of the planet unstable.
This maximum or
critical
semimajor axis .a
Max
/ is given by (Rabl and Dvorak 1988;
Holman and Wiegert
1999
)
D
a
Bin
0:464
0:38
0:631e
Bin
C
0:586e
Bin
C
0:15e
Bin
0:198e
Bin
:
(13.7)
a
Max
In Eq. (
13.7
),
D
m
2
=.m
1
C
m
2
/ where m
1
and m
2
are the masses of the
primary (planet hosting) and secondary stars, respectively. One can use Eq. (
13.7
)
to determine the maximum binary eccentricity that would allow the planet to have a
stable orbit in the HZ (l
out
a
Max
). For any smaller value of the binary eccentricity,
the entire HZ will be stable.
13.8
Examples of the Habitable Zone of Main Sequence
S-Type Binaries
As mentioned earlier, we assume that the orbit of the planet around its host star
is circular. Without knowing the exact orbital configuration of the planet, one can
only estimate the boundaries of the binary HZ by calculating the maximum and
minimum additional flux from the secondary star at its closest and furthest distances
from a fictitious Earth-like planet, as a first-order approximation. Note that using the
maximum flux of the secondary onto the planet for calculating the new binary HZ
overestimates the shift of the HZ from the single star's HZ to the binary HZ due to
the secondary because the planet's atmosphere can buffer an increase in radiation
temporarily. This shift is underestimated when one uses the minimum flux received
from the secondary star onto the planet. To improve on this estimation, one needs to
know the orbital positions of the planet as well as the stars in the binary. That way
one can determine the exact flux over time reaching the planet as well as the number
of planetary orbits over which the secondary's flux can be averaged. This depends
on the system's geometry (both stellar and planetary parameters) and needs to be
calculated for each planet-hosting S-type system, individually.
To explore the maximum effect of the binary semimajor axis and eccentricity on
the contribution of one star to the extent of the HZ around the other component,
we consider three extreme cases: an M2-M2, an F0-F0 and an F8-M1 binary.
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