and have all the position vectors of the orbit.
I have the orbital elements of a satellite orbit around earth. It is an inclined elliptical orbit. I have plotted the orbit in 3D and have all the position vectors of the orbit.I am now trying to plot a ground track of the orbit on MATLAB and I am unsure where to start. I am looking for guidance on the steps required to achieve this
If someone built a solid sphere of Iron 300 meters in diameter at some convenient location on the surface of the Earth and built a vacuum chamber around it, would it then be possible to put something small (say, a 0.1 mm diameter bb) into orbit around it? In other words, would this large Iron sphere have a Hill Sphere? If not, how large would such an Iron sphere have to be to have a Hill Sphere? Would Tungsten be a better (though probably not cheaper) material, thanks to its density, to build something like this with? I know that due to the 3-body problem this orbit will not be stable. For the purposes of my question, once around would be fine to be an orbit. I know it's wildly impractical idea and probably not possible to build from an engineering standpoint. Think of this as a grand science experiment in gravity and orbital mechanics, and it is paid for and agreed upon by all nations that this is a good idea.
Lets say you put your sphere 1 foot above the Earth. For an object to make it once around your sphere, in the part of the orbit between the sphere and the Earth you would have to have a more powerful gravitational pull from your sphere than the gravitational pull of the Earth. If your sphere had the same density as the Earth, that could only happen if your sphere was larger than the Earth.Of course, if you make it super-dense, you can get away with a much smaller mass -- a tiny black hole will be able to orbit something in a tiny orbit and small mass. But if you're talking reasonable density, it's on the order of the density of the Earth, and that means a sphere larger than the Earth to get something to orbit it while it's very close to the Earth's surface.Things can orbit the Moon even though the Moon is smaller than the Earth only because the Moon is far enough away from the Earth that Earth's gravity drops off a lot. On the surface of the Earth, you need something roughly Earth-sized.Of course, having something Earth-sized right next to the surface of the Earth is not only impractical, but it would immediately rip the whole planet to shreds.
Note that if you have your iron sphere orbiting the Earth instead of on the surface of the Earth, then it becomes more practical to have something orbit it. At that point, you only need to worry about the tidal force, which is the difference between Earth's gravity on one side of your sphere and Earth's gravity on the other. If your sphere's gravity is greater than that difference, you're in business.
The relevant equations are in the Wikipedia article for Hill Sphere if you want to calculate it exactly:https://en.wikipedia.org/wiki/Hill_sphereTo summarize, as other commenters have noted, sadly, it won't work for any reasonable sized iron sphere anywhere below LEO.