Didymos and Dimorphos: Kinetic Impact (2D)

A kinetic impactor and a two-body orbit

Dimorphos (about 160 m across) circles Didymos (about 780 m) in roughly 11.9 hours. Seen from Didymos the pair is a two-body problem with gravitational parameter μ = G (M_D + m_d). A spacecraft of mass m_sc arriving at speed v hits Dimorphos, which is the one body that gets pushed. If nothing were thrown off the surface, momentum conservation would give Δv = m_sc v / m_d. The crater ejecta leaves with extra momentum in the opposite direction, so the real push is larger by the momentum enhancement factor β:

The relative velocity of the two bodies changes by exactly that Δv (Didymos is not touched), so the new orbit follows from the state vector with the vis-viva equation and Kepler's third law:

For a retrograde push along a circular orbit this reduces to ΔP/P ≈ −3 Δv / v_orb. With the public DART numbers (580 kg, 6.1 km/s, m_d ≈ 4.3×10⁹ kg, β ≈ 3.6) the orbital speed is only about 17 cm/s, so a push of a few millimetres per second shortens the period by about half an hour — the shift that was measured in 2022.

How the period change is seen

The orbit is almost edge-on to Earth, so every orbit Dimorphos passes in front of and behind Didymos (mutual events) and the combined brightness dips. The graph uses the overlap area of two discs of radii R_D and R_d with equal albedo: brightness = 1 − A_overlap / (π (R_D² + R_d²)). A shorter period makes each dip arrive earlier than predicted, and the timing error grows linearly with the number of orbits — the “timing shift” graph. That is why a small change in speed becomes an easily measured shift in clock time.

What the page computes

• Orbit elements from the state vector before and after the hit; positions at any time from the Kepler equation (Newton solver), with a separate velocity-Verlet integrator used for the checks.
• The direction of the push is the spacecraft velocity direction, tilted by the “off the track” angle towards the inward radial direction, so only the along-track part changes the period to first order.
• Ejecta: sampled grains with a power-law speed distribution inside a cone, each given an equal mass so that their summed axial momentum equals (β−1) m_sc v exactly.

Honest simplifications

• Point-mass gravity: no J₂ of the two lumpy bodies, no spin–orbit coupling, no libration, no solar tides or radiation pressure. Dimorphos is drawn tidally locked, and the real body librates and may have been reshaped by the hit.
• The kick is applied to the relative velocity instantly and in the orbit plane; out-of-plane components, the shift of Didymos and Dimorphos’s changed shape are ignored.
• β, m_d and the geometry are linked in reality (a heavier target means a smaller Δv for the same β); here they are independent sliders. The “DART as flown” preset uses rounded public values (about −33 min, β about 3.6) and is not a re-analysis of the data.
• Ejecta are drawn on straight inertial paths (no gravity, no radiation pressure), spacecraft size is exaggerated about 5 times, and the approach is a short cinematic with compressed distances.
• Mutual-event brightness uses equal albedo and ignores shadows (eclipses) and the phase angle; real light curves are the sum of both occultations and eclipses.

This is an independent educational model of a planetary-defence technology test; it is not affiliated with any space agency.

• Press Skip to impact and watch the approach, the flash and the ejecta cone, then the orange new orbit appear beside the grey old one.
• Run Pure momentum transfer (beta 1) then DART as flown: the same spacecraft changes the period by about 9 minutes or about 36 minutes — the ejecta recoil is most of the push.
• Open the Telescope view (3D) or look at the sky inset (2D) and the brightness graph: with the observer 4° above the plane there are two dips per orbit; raise the elevation above ~20° and the mutual events disappear.
• Try Impact at periapsis, e = 0.2 versus apoapsis: a kick at different places changes e very differently while the period change is similar.
• Watch the “timing shift” graph: after two weeks the events are several hours early.