There are no active interstellar missions. JHUAPL is investigating what such a mission would look like. This presentation goes into a bit more detail.
As part of the studies for the next Heliophysics Decadal Survey JHUAPL is studying a probe to get to 1,000 AU within 50 years. ...
The study tries to answer these questions:
- Focus on the time frame in the next Decadal: 2023 – 2032: Can we fly then?
- Assess the science: Does the case remain compelling?
- Approach for Decadal: Assess technical readiness for a launch NO LATER THAN 2030
Conclude this effort in nine months, reporting back to NASA by February 2019
A weight target for the probe has been set:
The probe will weight 300-800 kg of which 40 kg will be instruments. By comparison New Horizons is a 478 kg with 30.4 kg of instruments, Parker Solar Probe 685 kg with 50 kg of instruments, while Pioneer 10/11 and Ulysses were lighter.
Various propulsion options and gravity assists are examined.
In-Space Propulsion Does Not Solve the Problem – Underpowered for the Mass. SEP, RTG with ion engine, nuclear electric system studied.
The only other option is a really big rocket, SLS combined with gravity assist.
Even then, it'd be difficult. Goal is a speed of 20 AU/year. a STAR 48 upper stage gets you 4 km/s, you need 14 km/s during the approach to the Sun at a distance of 4 solar radii. 10 km/s at 2 solar radii.
The Oberth-Kuiper maneuver requires close approach to the Sun, which requires heat shielding which adds weight.
The current mission plan is for an SLS launch to Jupiter, reverse gravity assist to reduce speed, plunge into the Sun at 4 solar radii (Parker Solar Probe will reach 8 solar radii) and use an Oberth maneuver to achieve 8 AU per year or twice the heliospheric escape speed of Voyager 1. Since they will be in the area, they are also proposing that a Kuiper Belt target is also target for a fast flyby.
Ulysses used an IUS plus a PAM-D for 20 tons worth of upper stages to get the delta-V needed here. STAR-48 is about 2 tons. SLS is not big enough to get 20 tons to the required orbit.