Question

As the winter passes, the Earth spins as usual around the Sun. On a cold February night just past midnight, a lone astronomer spots an unusual object in the starlit sky. Near the far end of the constellation of Draco, north of the star HD 91190, there appeared a faint reflective anomaly. After careful observation over the next few hours, the astronomer noticed that the object was very close to Earth. Jotting down the coordinates and times, the astronomer came up with spherical coordinates

Feb 28^th -> (x,y,z) = (1.16 x 10⁸ km, 6.05 x 10⁸ km , 38.54 x 10⁸ km )

After many days of careful observation, the astronomer found that the object was indeed moving! By the middle of May, the astronomer was observing the object at:

May 15^th -> (x,y,z) = (1.05 x 10⁸ km, 5.73 x 10⁸ km , 38.28 x 10⁸ km )

Now, with this information, we must decide if this object will come close enough to Earth that it could collide. There are some other equations that are needed, in particular, the orbit of Earth:

r = 1.52 x 10⁸ / (1 + 0.0167 * cos(θ)) km

where θ is in degrees, found from Earth's rotation around the Sun. Thus, 0⁰ is Dec 21^st, the Winter Solstice, and 180⁰ is June 21^st, the Summer Solstice.

Find a set of parametric equations, using the Earth's position as the origin for each of the object's observations (it will change for each date).

Given this information, and assuming near-linear travel for the unknown object

1. How fast is the object moving?

2. In what direction is the object moving?

3. All of the planets in the Solar System are moving in on a nearly flat plane. How long until this object enters into that plane?

4. Does it seem like this object will hit Earth? Why or why not?

Answer 1