Find eccentricity of orbit given position and velocity

1. The problem statement, all variables and given/known data



Hey, I need to find whether an asteroid is in a "bound orbit" around the Sun.

Furthermore, describe the shape of the orbit (elliptical or circular).

The only information I have been given is its velocity and position at a random point.



v_o=61m/s

r_o=13.75AU=2.06E12m



I also need to find its orbital energy.



2. Relevant equations

vis-viva eqn: v^2=GM((2/r)-(1/a))

angular momentum L=mvr

r_p=a(1-e) position at perihelion





3. The attempt at a solution



First i used the vis-viva eqn with my v_o and r_o but a (semi-major axis) is an unknown.

so a=((2/r_o)-(v_o^2)/GM)^-1

a=1.03E12m



Then i used momentum conservation with the observed position and the position at perihelion (where velocity is a maximum)



m(v_o)(r_o)=m(v_p)(r_p)

r_p=(v_o)(r_o)/(v_p) call this eqn 1



I computed the vis-viva eqn at perihelion, and i subbed in eqn 1 and a=1.03E12m



so i got a quadratic:

v_p^2=GM[((2v_p)/(v_o)(r_o))-(1/a)



(v_p)^2 - 2.11E6(v_p)+1.29E8=0



The solutions are 61m/s or 2.11E6m/s.



I used assumed 2.11E6m/s =v_p since it is larger.. and 61m/s is my original velocity.



then i found r_p using eqn 1

r_p=5.9E7m



finally i used the eccentricity eqn e=1-(r_p/a)

however r_p is of order 10^7 and a is of order 10^12, therefore i got e~1.

that implies my orbit is a straight line.... so where did i go wrong?

also, this feels like ALOT of steps so am i over thinking it??



Oh and for orbital energy I think the equation is E=-GM/2a. is that correct?