Supernova kicks have played rather a large role in my recent research. These kicks produce a variety of phenomena, from runaway stars to offset short gamma-ray bursts. In the past few years I've built up cogsworth, a Python package for self-consistent binary population synthesis and galactic dynamics simulations, with the aim of using the impact of supernova kicks on kinematics to better understand binary evolution. I thought it could be fun to interactively explore what the core collapse of a massive star does to its binary.
Specifically, this page walks you through the underlying physics and background of supernova kicks, provides an interactive interface to explore how kicks can change binary parameters, and guides you through some questions to better understand these kicks. You can either read through the background using the panels below, or skip straight to the interactive tool , enjoy!
Take some time to explore the physics behind supernova kicks and their impact on binary systems.
A core-collapse supernova plays out on the order of seconds, while the binary's orbital period is often days to years. The explosion is therefore effectively instantaneous: the two stars have no time to move significantly, so we can approximate their separation as frozen, such that only the masses and velocities change. We usually treat the supernova as a single impulse and calculate the new orbit immediately afterwards.
Two distinct effects contribute to the kick applied by the supernova:
Even a perfectly symmetric explosion changes the orbit, because it removes mass so quickly that the orbit cannot adjust (Blaauw 1961, Boersma 1961). The surviving stars keep the velocities they had, but the system is now less massive and therefore less tightly bound, so the orbit widens. The centre of mass also recoils: the ejecta is thrown off isotropically in the exploding star's frame, but that star was itself moving, so in the binary frame the ejecta carries away net momentum and the remaining binary must recoil to compensate.
For an initially circular orbit the new semi-major axis is simply
$$a_{\rm postSN} = a\,\frac{M_{\rm tot,postSN}}{2\,M_{\rm tot, postSN} - M_{\rm tot,preSN}},$$and the binary is unbound if more than half of the total mass is lost in the instant of explosion,
$$\Delta M = M_1 - M_{1,\rm rem} > \tfrac{1}{2}\,M_{\rm tot,preSN}.$$It's typically the case that this effect is subdominant compared to the natal kick. But in cases in which the mass loss is large and the natal kick is suppressed, the Blaauw kick can become significant.
In the tool, set the kick to zero (or toggle “Show Blaauw-only orbit”) to isolate this effect, and watch the orbit widen (or unbind!) as you lower the remnant mass.
Many effects break spherical symmetry in a supernova. Hydrodynamic instabilities during collapse (large-scale convection and the standing accretion-shock instability), together with anisotropic neutrino emission, launch the ejecta preferentially in one direction. By conservation of momentum the newborn compact object recoils in the opposite direction: leading to a natal kick that can be hundreds of km/s.
In the pre-supernova orbital frame, with the orbit in the \(x\)–\(y\) plane and its angular momentum along \(z\), the kick is written
$$\vec{v}_{\rm kick} = v_k\,\bigl(\cos\phi\cos\theta,\ \cos\phi\sin\theta,\ \sin\phi\bigr),$$where \(\phi\) is the angle out of the orbital plane and \(\theta\) is the azimthal angle. The kick is imparted to the collapsing star only, so it adds to the relative velocity of the two stars, while the centre of mass takes only a share of it (weighted by the remnant mass). Its direction can be just as important as its magnitude in many cases: the same kick can widen, shrink, tilt, or unbind the orbit depending on where it points.
Kick directions are usually assumed isotropic; the tool also lets you restrict them toward the poles with the kick opening angle.
Not all of the ejected material escapes. For more massive progenitors, a substantial fraction (up to 100%!) of the exploding envelope stays bound and falls back onto the collapsing core (e.g., Fryer et al. 2012). The fallback mass damps the natal kick for two reasons: the same explosion asymmetry now has to move a heavier remnant (so momentum conservation gives a smaller velocity), and the fallback material itself carries essentially no net momentum, diluting whatever kick was imparted. You'll note that in the case of complete fallback, the natal kick is completely suppressed.
There are two common prescriptions to capture this effect (you can try out both below):
This is why population-synthesis codes generally predict that black holes are born with much smaller velocities than neutron stars. Observationally, it's less clear whether black holes have significant kicks (e.g., Nagarajan+2025).
Okay you've made it this far and not been scared off, so let's get properly stuck in with the equations. The tool below is computing the standard impulsive-kick equations that have been discussed in maaaannny papers (e.g., Hills 1983; Kalogera 1996, check out the introduction of Wagg et al. 2025 for a more complete list). But I'm just going to re-explain them here in the vector form presented in Pfahl et al. 2002 because that's the version that personally makes the most sense to me. If my explanation is unclear, check out their paper for a more rigorous derivation!
Let's start by defining the system. We have a binary with masses \(M_1\) and \(M_2\), semi-major axis \(a\), eccentricity \(e\), and eccentric anomaly \(E\) (the eccentric anomaly defines the orbital phase at which the core collapse occurs). The total mass is \(M_{\rm tot} = M_1 + M_2\), and the separation is \(r = a\,(1 - e\,\cos E)\). The orbital angular momentum points perpendicular to the orbital plane and is \(\vec h = \vec r \times \vec v_{\rm rel}\), where \(\vec v_{\rm rel}\) is the relative velocity of the two stars. The eccentricity vector points in the direction of the periastron of the collapsing star and is \(\vec e = \frac{\vec v_{\rm rel} \times \vec h}{G\,M_{\rm tot}} - \frac{\vec r}{r}\).
Suddenly: an explosion! And I mean suddenly here, such that we treat the explosion as an impulse: meaning that the separation \(\vec r\) is unchanged. The collapsing star's mass drops \(M_1 \to M_{1,\rm rem}\), and its velocity gains the natal kick, so the relative velocity and total mass become
$$\vec v_{\,\rm rel, postSN} = \vec v_{\rm rel} + \vec v_{\rm kick}, \qquad M_{\rm tot,postSN} = M_{1,\rm rem} + M_2.$$First, we need to check whether the binary is disrupted by the supernova. We can calculate the post-SN vectors as follows and split (or should I say, disrupt?) into two cases
$$ \vec h_{\rm postSN} = \vec r \times \vec v_{\,\rm rel, postSN}, \qquad \vec e_{\rm postSN} = \frac{\vec v_{\,\rm rel, postSN} \times \vec h_{\rm postSN}}{G\,M_{\rm tot,postSN}} - \frac{\vec r}{r}.$$\(e_{\rm postSN} < 1\): the orbit is bound and the system recoils
Their relationship went through a rough patch, but this couple has clung together against the odds (based on a true story, 85% of binaries disrupt after the first supernova, e.g., Renzo et al. 2019). The post-supernova semi-major axis is now given by $$ a_{\rm postSN} = \frac{h_{\rm postSN}^{\,2}}{G\,M_{\rm tot,postSN}\,(1 - e_{\rm postSN}^{\,2})}. $$
The centre-of-mass (systemic) velocity of the binary recoils to compensate for the mass loss and the natal kick, and is given by $$\vec v_{\rm CM, postSN} = \underbrace{-\frac{M_2\,(M_1 - M_{1,\rm rem})}{M_{\rm tot,preSN}\,M_{\rm tot,postSN}}\,\vec v_{\rm rel}}_{\text{Blaauw kick}} \;+\; \underbrace{\frac{M_{1,\rm rem}}{M_{\rm tot,postSN}}\,\vec v_{\rm kick}}_{\text{Natal kick}}.$$ Note here that if you set $M_{1,\rm rem} = M_1$ (no mass loss) then the Blaauw kick term vanishes, and if you set $\vec v_{\rm kick} = 0$ then the natal kick term vanishes.
\(e_{\rm postSN} \geq 1\): the binary is disrupted and the companions are ejected
Alas, this stellar love story cannot handle the stress and ends in disaster, with the two companions going their separate ways.
In this case there is of course no $a_{\rm post SN}$ to speak of, the binary disrupts and the two stars separate with a relative velocity at infinity of $$ \vec v_\infty = v_\infty\ \left[- \frac{1}{e_{\rm postSN}} \hat{e_{\rm postSN}} + \left(1 - \frac{1}{e_{\rm postSN}^2} \right)^{1/2} \hat{h}_{\rm postSN} \times \hat{e}_{\rm postSN} \right], \qquad v_\infty = \sqrt{\frac{G\,M_{\rm tot,postSN}}{a_{\rm postSN}}}\,\sqrt{e_{\rm postSN}^{\,2} - 1},$$ which is shared between them based on their mass ratio (the typically lighter remnant carries most of it):
$$\vec v_{\rm remnant} = \frac{M_2}{M_{\rm tot,postSN}}\,\vec v_\infty + \vec v_{\rm CM, postSN}, \qquad \vec v_{\rm companion} = -\frac{M_{1,\rm rem}}{M_{\rm tot,postSN}}\,\vec v_\infty + \vec v_{\rm CM, postSN}.$$So far I've only discussed this from a theoretical perspective (spot the simulator...🙃), but there is good observational evidence that compact remnants receive large kicks. Radio pulsars have precisely measured proper motions, and their distances (from parallax or dispersion measure) convert these into transverse velocities. Crucially, young pulsars (ages \(\lesssim\) a few Myr) have not yet been deflected much by the Galactic potential, so their velocities still reflect their birth kicks.
Many different studies have attempted to characterise the distribution of these velocities, here are just a few of these:
You'll note that these measurements probe the kicks of neutron stars. For black holes, the situation is more complex and less well-constrained (e.g., Nagarajan+2025).
Use the sliders to adjust the pre-supernova binary and the natal kick, and watch how the orbit changes. You can also click and drag the 3D view to rotate the orbit, or use the buttons to zoom and reset the view.
The background above answers most of these questions, but you can also test your intuition with the tool, whatever floats your boat :D
Wider binaries, with less massive companions, are easier to disrupt. The binding energy of a binary is $$ E_{\rm bind} = -\frac{G\,M_1\,M_2}{2\,a},$$ so the wider the separation \(a\) and the smaller the masses, the less energy is required from a kick to unbind it.
Try it: fix the kick and slide the separation up, or the companion mass down (or both!) and watch the binary fly apart.
A kick almost always raises the eccentricity: it changes the speed at one point of the orbit, which can no longer trace a circle. The separation can go either way, and it's exactly because of the direction. A kick roughly along the direction of orbital motion widens the orbit (or unbinds it); a kick opposing the motion often removes energy and tightens it. The polar angle can matter too, a kick out of the plane tilts the orbit and can more often cause a disruption.
Try it: First look at how the eccentricity changes with different kick strengths. Then you can also start to explore the effects of directions; try adjusting the polar angle and azimuthal angle relative to the orbital velocity (shown by arrows on the orbits).
This one is a classic misconception: it's important to remember that the kick is applied to the remnant, not the companion.
When a large kick disrupts the binary, the companion is essentially just released from its orbit and so it flies off with the velocity it already had. This velocity is its pre-supernova orbital velocity $$ v_{\rm orb}= \sqrt{\frac{G M_{\rm tot,preSN}}{a}}, $$ with only a small difference. Indeed, once the kick is large enough to cause a disruption, the faster the remnant leaves the system, the closer the companion's ejection velocity is to its pre-supernova orbital velocity. I discuss this in more detail in Wagg et al. 2025 if you're curious.
Try it: set up a disruption with a large kick (go for maybe 500 km/s), compare the companion's ejection velocity to its pre-supernova orbital velocity, then increase the kick and see how the companion's speed changes. You could also try changing the pre-supernova separation and compare how much more of a difference that makes.
For a circular orbit and no natal kick, the binary disrupts once more than half of the total mass is lost in the instant of explosion:
$$\Delta M = M_1 - M_{1,\rm postSN} > \tfrac{1}{2}\,M_{\rm tot,preSN}.$$As usual, if you lose less than half, the orbit simply widens. But once you cross the half-mass line, it flies apart. Eccentric orbits complicate this threshold, since it starts to matter at what point during the orbit the mass loss occurs.
Try it: Set the kick to zero on a circular orbit, then lower the remnant mass until the ejected mass passes half of the total (you may need a larger collapsing star mass to get this difference large enough).
The larger the companion mass, the smaller the systemic velocity. The kick is given only to the remnant, but the whole bound system shares the momentum, so a heavier companion means the same kick produces a smaller systemic velocity.
Try it: hold the kick fixed for a system that remains bound, increase the companion mass, and watch the systemic velocity drop.
And that's all I've got on kicks, I hope you enjoyed it and learn a little about how supernovae can shape the lives of binaries. If you have any questions, comments, or suggestions, please reach out!