Hitting The Sun

From the Earth, the Sun may be the most difficult-to-reach object in the Solar System. Some folks natter about dropping nuclear waste into the Sun. Nope. Unless you escape Earth, AND lose ALL of Earth's orbit velocity, you just orbit near the Earth's orbit, perhaps to crash back into the Earth eventually.

The Sun's standard gravitational parameter is 1.32712440018e11 km³/s². At 1 AU (149597870.7 km) from the Sun, Earth's circular orbit velocity is 29.78 km/s. The Sun's radius is approximately 696,000 km, so a Hohmann from Earth radius to within Sun radius implies a maximum aphelion velocity of 2.866 km/s, which is retrograde from the Earth's orbit by 26.91 km/s. Vector-summed with Earth's 11.186 escape velocity, that is a 29.15 km/s launch directly from Earth. The trip would take around 65 days. We'll call this orbit stop and drop.

Another way to get there is to launch from Earth to near solar escape, travelling hundreds of AU from the Sun, where a relatively small thrust can kill the angular momentum for a plumet straight down to the Sun. This would take decades, but the delta V will be a little less. Escape prograde means leaving the Earth's orbit at ( √2 - 1 ) of Earth, 12.34 km/s, again vector-summed with escape, or 16.652 km/s.

Bank shot around Jupiter

A gravity assist maneuver is vaguely like a "gravitational collision"; the incoming velocity vector relative to the target body is inverted, then rotated in some arbitrary direction around the planet. Arriving at the same relative velocity as Jupiter's orbital velocity means arriving at Jupiter's radius with a kinetic energy equal to that velocity, presumably outbound and skimming the inside of Jupiter's orbit for a hyperbolic deflection past the front, then directly retrograde relative to Jupiter's orbit, equal and opposite to Jupiter's 13.056 km/s circular orbital velocity.

The velocity energy added to Jupiter's orbital energy is the same as the escape energy, so this is also requires 16.652 km/s launch from Earth, but the encounter with Jupiter takes less time than a deep-space escape (or a "bank shot" around Saturn, Uranus, or Neptune; Mars, Pluto the asteroids, and other dwarf planets with escape velocities smaller than solar orbit velocities can't be used like this). The time calculation is a little complicated. We will compute using μ = 4π² (AU³/year²), vp = √8 π AU/year, r = 5.2 AU, cos(ν) = -0.6153846, ν = 127.97897° D = 2.04939 = √4.2 and Barker's equation to compute the time from Earth to Jupiter as 0.55 years. The descent will take sqrt( 5.2³/32) = 2.1 years. The synodic period is 1.1 years, and on average we will wait 0.55 years for a launch opportunity, so the total time will be 3.2 years on average.

Venus Gravity Assist

More complex missions to the inner solar system involve multiple passes by Venus and Earth to shed angular momentum; The MESSENGER, Bepi Colombo and Parker Solar Probe missions use multiple gravity assists to approach orbits around Mercury or very close to the Sun. These missions require on the order of a decade to reach their destinations.

It is not clear how much additional delta V at planetary periapsis is needed to plunge from the last pass around Venus or Earth directly into the Sun. Mercury is probably too small to accomplish this last step directly.

So on average, the Jupiter bank shot into the Sun takes 3.2 years and requires 16.65 km/s launch velocity, compared to stop and drop, which requires only 0.18 years, but requires 29.15 km/s. The Venus Gravity Assist approach may require the least total delta V, but requires years.

Either way, if the goal is to get rid of something nasty, the Sun is difficult to reach. Consider dropping it on Venus (11.5 km/s, 0.4 years) instead, which is already nasty.