3.3 Supernova explosion

A supernova explosion in a massive binary system occurs on a timescale much shorter than the orbital period, so the loss of mass is practically instantaneous. This case can be treated analytically (see, e.g., [39107Jump To The Next Citation Point25939214246043180397]).

In general, the loss of matter and radiation is non-spherical, so that the remnant of the supernova explosion (neutron star or black hole) acquires some recoil velocity called kick velocity w⃗. In a binary, the kick velocity should be added to the orbital velocity of the pre-supernova star.

The usual treatment proceeds as follows. Let us consider a pre-SN binary with initial masses M1 and M2. The stars move in a circular orbit with orbital separation ai and relative velocity ⃗Vi. The star M1 explodes leaving a compact remnant of mass Mc. The total mass of the binary decreases by the amount ΔM = M1 − Mc. It is usually assumed that the compact star acquires some additional velocity (kick velocity) ⃗w (see detailed discussion in Section 3.4). Unless the binary is disrupted, it will end up in a new orbit with eccentricity e, major semi axis af, and angle 𝜃 between the orbital planes before and after the explosion. In general, the new barycentre will also receive some velocity, but we neglect this motion. The goal is to evaluate the parameters af, e, and 𝜃.

It is convenient to work in an instantaneous reference frame centered on M2 right at the time of explosion. The x-axis is the line from M2 to M1, the y-axis points in the direction of ⃗Vi, and the z-axis is perpendicular to the orbital plane. In this frame, the pre-SN relative velocity is ⃗V = (0,V ,0) i i, where ∘ ---------------- Vi = G (M1 + M2 )∕ai (see Equation (13View Equation)). The initial total orbital momentum is J⃗i = μiai(0,0,− Vi). The explosion is considered to be instantaneous. Right after the explosion, the position vector of the exploded star M1 has not changed: ⃗r = (ai,0, 0). However, other quantities have changed: ⃗ Vf = (wx, Vi + wy,wz ) and ⃗ Jf = μfai(0, wz,− (Vi + wy )), where ⃗w = (wx, wy,wz ) is the kick velocity and μf = McM2 ∕(Mc + M2 ) is the reduced mass of the system after explosion. The parameters af and e are being found from equating the total energy and the absolute value of orbital momentum of the initial circular orbit to those of the resulting elliptical orbit (see Equations (10View Equation, 14View Equation, 12View Equation)):

V 2f GMcM2 GMcM2 μf--- − -------- = − --------, (38 ) ∘ --2-------ai---- ∘ 2af------------------- μfai w2 + (Vi + wy )2 = μf G(Mc + M2 )af(1 − e2). (39 ) z
For the resulting af and e one finds
af [ ( w2x + w2z + (Vi + wy )2)] −1 -- = 2 − χ -----------2--------- (40 ) ai Vi
( ) 2 ai w2z + (Vi + wy )2 1 − e = χ a- ------V-2------- , (41 ) f i
where χ ≡ (M1 + M2 )∕(Mc + M2 ) ≥ 1. The angle 𝜃 is defined by
⃗ ⃗ cos𝜃 = -Jf-⋅Ji-, |⃗Jf| |⃗Ji|

which results in

V + w cos𝜃 = ∘-----i----y-----. (42 ) w2z + (Vi + wy)2

The condition for disruption of the binary system depends on the absolute value Vf of the final velocity, and on the parameter χ. The binary disrupts if its total energy defined by the left-hand-side of Equation (38View Equation) becomes non-negative or, equivalently, if its eccentricity defined by Equation (41View Equation) becomes e ≥ 1. From either of these requirements one derives the condition for disruption:

∘ -- Vf- 2- Vi ≥ χ . (43 )
The system remains bound if the opposite inequality is satisfied. Equation (43View Equation) can also be written in terms of the escape (parabolic) velocity Ve defined by the requirement
2 μfV-e − GMcM2--- = 0. 2 ai

Since χ = M ∕(M − ΔM ) and V 2 = 2G (M − ΔM )∕ai = 2V 2∕χ e i, one can write Equation (43View Equation) in the form

Vf ≥ Ve. (44 )
The condition of disruption simplifies in the case of a spherically symmetric SN explosion, that is, when there is no kick velocity, ⃗w = 0, and, therefore, V = V f i. In this case, Equation (43View Equation) reads χ ≥ 2, which is equivalent to ΔM ≥ M ∕2. Thus, the system unbinds if more than a half of the mass of the binary is lost. In other words, the resulting eccentricity
M − M e = --1-----c (45 ) Mc + M2
following from Equations (40View Equation, 41View Equation), and w⃗ = 0 becomes larger than 1, if ΔM > M ∕2.

So far, we have considered an originally circular orbit. If the pre-SN star moves in an originally eccentric orbit, the condition of disruption of the system under symmetric explosion reads

1 r ΔM = M1 − Mc > ----, 2 ai

where r is the distance between the components at the moment of explosion.

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