Presets
Playback
Orbit
Excess speed and perihelion set the hyperbola.
Brightness and activity
Absolute magnitude, activity exponent and sublimation radius.
Layers
Model checks
The science
Unbound orbits
A body whose speed at great distance, v∞, is above zero has positive orbital energy and follows a hyperbola around the Sun rather than an ellipse. With μ = GM☉:
ε = v2/2 − μ/r = v∞2/2, a = μ/v∞2, e = 1 + q v∞2/μ > 1, vq = √(v∞2 + 2μ/q)The path turns through the angle δ = 2 arcsin(1/e) and has impact parameter b = a√(e2−1). Position in time follows the hyperbolic Kepler equation e sinh H − H = n t with n = √(μ/a3), x = a(e − cosh H), y = a√(e2−1) sinh H. The first two known visitors had v∞ of about 26 km/s ('Oumuamua, 2017) and 32 km/s (2I/Borisov, 2019); an object with e well above 1 cannot have been bound to the Sun.
Brightness and activity
The apparent magnitude of a comet is modelled as m = H + 5 log10Δ + 2.5 n log10 r, where r is the distance from the Sun, Δ the distance from Earth and n the activity exponent (2 for an inert body that only reflects light, about 4 for a typical active comet). Water-ice sublimation switches on inside roughly 2.5-3 AU: inside that radius the page draws a coma and a tail pointing away from the Sun.
Honest simplifications
- Pure two-body gravity: no planetary perturbations, no non-gravitational acceleration from outgassing (which is what produced the small extra acceleration seen for 'Oumuamua), no solar radiation pressure.
- Earth moves on a circular 1 AU orbit with an adjustable starting phase; the observing geometry is therefore schematic and the magnitudes are illustrative, not predictions for a real object.
- Planar geometry: the real objects approach from out of the ecliptic plane; only the heliocentric distance and the hyperbola shape are meant to be quantitatively right.
- The 'Oumuamua-like and Borisov-like presets use the published v∞ and perihelion distance rounded, with generic H values.
Checks run in the page
(1) An RK4 integration started at 200 AU on the analytic hyperbola reaches the perihelion distance q. (2) Along that whole flight the energy and the angular momentum stay constant. (3) The measured turning angle between the incoming and the outgoing velocity equals 2 arcsin(1/e).
What to try
- Choose 'Oumuamua-like: it crosses the inner solar system in a few hundred days (graph 1) and fades quickly; the speed (graph 2) peaks at perihelion far above v∞.
- Choose Borisov-like: a larger perihelion, an active tail inside the green ring and a different brightness curve with exponent 4.
- Lower v∞ to 0.5 km/s: the path approaches a parabola, e approaches 1 and the deflection angle approaches 180 degrees.
- Set v∞ to 80 km/s: the path is nearly straight and the object is gone in weeks.
- Compare the speed curve with the escape-speed curve: the object is always faster than escape speed, which is the definition of unbound.