Hitting The Sun

From the Earth, rhe 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 Earth's orbit velocity, you just orbit near the Earth, perhaps to crash into it 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

An even easier method is a "bank shot" around Jupiter. Jupiter's semimajor axis is 778.57e6 km, which is a circular orbit velocity of 13.056 km/s. Jupiter's escape velocity is 59.5 km/s, and it can modify a trajectory by as much as twice that velocity. A Hohmann from Earth to Jupiter has a semimajor axis of 464.08e6 km, so the perihelion velocity is 38.58 km/s, 9.43 km/s more than Earth orbital speed, or a Earth launch velocity of 14.63 km/s. That is all the delta V we need; the rest we steal from Jupiter.

The vehicle arrives near Jupiter with a prograde aphelion velocity of 7.41 km/s, which is retrograde relative to Jupiter by 5.65 km/s. A hyperbolic orbit around Jupiter can deflect the orbit straight down at the Sun, with no tangential velocity and a sunwards radial velocity of 5.65 km/s. The descent will take about 1.5 Earth years. The trip to Jupiter will take about 2.7 Earth years, and the synodic period is 1.092 earth years, so the average wait to do this is 0.55 years.

On average, the Jupiter bank shot into the Sun takes 4.8 years and requires 14.6 km/s, compared to stop and drop, which takes only 65 days, but requires 29.2 km/s.

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.