The above image comes from a 600-kilogram, refrigerator-sized robot travelling at about 15 miles per second, around 170 million miles from Earth. They can be glad that the asteroid is also around 170 million miles from earth, travelling at 15 miles per second and this even in the same direction.
Relative motion is smaller though:
The gravitational force between the probe (660kg) and asteroid (4.5 * 10^11kg) is just 0.1 N. The probe has therefore to be circling the asteroid with about 0.25m/s or just 1km/h (0.57 mph for american readers).
And how else could it be: the asteroid was assumed to be a point mass and the probe in a circular orbit of 450m around it.
The hayabusa spacecraft is not in orbit about the asteroid. It's in orbit about the Sun, in an orbit very, very close to the orbit the asteroid is in, so they stay close to each other for a long time (about a year and a half IIRC).
I couldn't find any information on the trajectory. The sphere of influence in which the gravitational attraction of the asteroid dominates over that of the sun has a radius of about 6km though. So as long as the probe moves inside this area, it has to orbit or spend fuel of the reaction control system (the ion thrusters are inactive in the vicinity of the asteroid, they'd also have only enough thrust to allow hovering in a distance of hundreds of meters (in the tens of milli-newtons)).
> The sphere of influence in which the gravitational attraction of the asteroid dominates over that of the sun has a radius of about 6km though.
It depends on what you consider "dominates" to mean. For example, the corresponding value for the Earth and the Sun is 924,000 km, and the Moon is only 400,000 km from the Earth, but the Moon's orbit is still always concave towards the Sun.
If Earth wouldn't dominate the space in which the Moon moves, the Moon wouldn't be in orbit around Earth at all and the Sun couldn't perturbe this orbit (as you even stated yourself)?
> If Earth wouldn't dominate the space in which the Moon moves
My point is that "dominate" here could mean different things, and the Earth does not dominate all of them.
Specifically, the fact that the Moon's orbit is always concave to the Sun means that the "acceleration due to gravity" of the Sun on the Moon is larger than the "acceleration due to gravity" of the Earth on the Moon. In other words, the net "acceleration due to gravity" of the Moon is always towards the Sun, not towards the Earth. So the Earth does not dominate in this way.
The "sphere of influence" you mention is based on tidal forces: the tidal effect of the Earth on the Moon is larger than the tidal effect of the Sun on the Moon. So the Earth does dominate in this way.
> the Moon wouldn't be in orbit around Earth at all and the Sun couldn't perturbe this orbit (as you even stated yourself)?
No, that's not what I said. Whether you think of the Moon as orbiting the Earth or the Sun depends on how you define "orbit" and what you are trying to do. If you define "orbit" according to which body the Moon is accelerating towards, on net, then the Moon is orbiting the Sun, not the Earth (see above); in this sense, the Moon and the Earth are in two closely matched orbits about the Sun.
And my point about the Hayabusa spacecraft and the asteroid is that, if you calculate the "acceleration due to gravity", you find that it's similar to the Moon's--the spacecraft's net acceleration is towards the Sun, not towards the asteroid.
The disclaimer was meant to say that it's just an idealized system of two masses at a given distance. I picked these values to show what speed an object orbiting around a 'light' object like an asteroid approximately has.
So sorry: it's not in that orbit. I should have made that clear instead of talking about the probe, circling, ... which sounds pretty concrete.
(The asteroid is not spherical or homogenous and is even supposed to be a "rubble pile" with large holes inside which makes it highly questionable at best to treat it as point mass (you could if the probe were far enough away from it ... which it is not at all), but hey ... ¯\_(ツ)_/¯)
I assume you calculated the average force and relative velocity experienced in the 450m circular orbit. At the perigee of the polar orbit (when the picture was taken) the force and the relative speed should be bigger. Also due to the small size of the asteroid the gravity should be quite uneven during the approach.
Relative motion is smaller though: The gravitational force between the probe (660kg) and asteroid (4.5 * 10^11kg) is just 0.1 N. The probe has therefore to be circling the asteroid with about 0.25m/s or just 1km/h (0.57 mph for american readers).
And how else could it be: the asteroid was assumed to be a point mass and the probe in a circular orbit of 450m around it.