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Does NASA have projects with spacecraft which are specifically made for orbiting somewhere around the Sun? If yes, what can they be used for?

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Everything humankind has put into space is orbiting the Sun. Orbit is not a mutually exclusive word. A satellite in low Earth orbit is orbiting the Earth and the Sun. An object on a hyperbolic trajectory (five of them); they're still orbiting the Sun as well. None of the has come anywhere close to another star. The last, New Horizons, has yet to reach Pluto's orbit. –  David Hammen Apr 25 at 1:05
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@DavidHammen Does a hyperbolic escape trajectory still count as an orbit? –  Padarom Apr 25 at 9:55
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Also everything humankind has NOT put into space is orbiting the sun. A complete orbit of the Earth around the Sun occurs every 365.256363004 mean solar days –  James Jenkins Apr 25 at 16:17
    
@DavidHammen: While what you say is true, that does not mean that everything humanity has put into space has been designed specifically to orbit the Sun. The question seems to be asking about something which does in fact orbit the Sun and no other planet. The fact that everything else does too as a consequence is unrelated to something being designed for that explicit purpose, just as the fact that cars are designed to be driven by humans is unrelated to the fact that some are designed to be raced. –  Magus Apr 25 at 17:00
    
@Magus - My comment was made before Zoltan refined the question. Originally he asked "Do NASA have projects with spacecraft which are orbiting somewhere around the Sun?" The words "specifically made for" were added later. –  David Hammen Apr 25 at 17:16
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I suspect you mean objects that haven't left the solar system (and what is the boundary?) and that aren't orbiting another planet.

Every object en route to another planet that has left the Earth's sphere of influence and has not yet entered the target planet's sphere of influence is orbiting the Sun. Whether the Pioneer 10, Pioneer 11, Voyager 1, and Voyager 2 missions count is a matter of debate. Dawn is en route to Ceres. Rosetta is en route to 67P/Churyumov–Gerasimenko; the encounter will be later this month. New Horizons, while on an escape trajectory, hasn't reached Pluto's orbit yet. India's Mars Orbiter Mission and the US's MAVEN satellite are both en route to Mars.

There are also a number of vehicles in pseudo orbits about the linear Sun-Earth Lagrange points (SEL1 and SEL2). These Lagrange point orbits are becoming the go-to place to put orbiting observatories.

In addition to these, a number of satellites have intentionally been placed into heliocentric orbits with the intent to study either interplanetary space or the Sun itself. These include

  • Pioneer 5, to study interplanetary space between the Earth and Venus;
  • Pioneer 6, 7, 8, and 9, to study the solar wind, solar magnetic field, and solar and cosmic rays;
  • Helios 1 and 2, to study various solar processes;
  • Ulysses, to study the Sun from a very high inclination orbit;
  • Genesis, a sample return mission of solar wind particles;
  • STEREO A and B, which monitor the Sun.

STEREO A and B are the only two that are currently active.

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In addition to the definition given by David H, I would add Ulysses, which is specifically observing the sun from out of the plane of the ecliptic and might mean what the questioner is asking for.

Ulysses made two passes around the sun, one over each pole, and its mission is officially over after the second pass. In order to get out of the plane of the ecliptic, it did a gravity assist at Jupiter, so it could come back towards the sun, over the top/bottom. Pretty amazing orbital mechanics management to do that.

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