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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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    $\begingroup$ @DavidHammen Does a hyperbolic escape trajectory still count as an orbit? $\endgroup$ – Padarom Apr 25 '14 at 9:55
  • $\begingroup$ Albeit in cooperation with ESA, there's SOHO $\endgroup$ – nos Apr 25 '14 at 19:53
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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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  • $\begingroup$ @JCRM - I mentioned without naming them that there were / are a number of vehicles in pseudo orbits about the linear Sun-Earth Lagrange points. Besides, LISA Pathfinder was launched on 3 December 2015, which was over 19 months after I wrote this answer. $\endgroup$ – David Hammen May 22 '18 at 13:44
  • $\begingroup$ I have no Idea how I missed that :( $\endgroup$ – JCRM May 23 '18 at 5:49
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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.

Wikipedia has a nice graphic of how that orbit was managed:

2nd orbit of Ulysses around the Sun

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  • $\begingroup$ +1 for Ulysses callout, I'm a big fan of that mission's trajectory! $\endgroup$ – uhoh Mar 4 '17 at 0:52

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