How can we calculate the intercept time for a NEO on a hyperbolic trajectory?

[Imagin_e]

Homework Statement

Hi!
I was wondering how one can calculate the time we have to intercept and destroy a NEO?
I have the semimajor axis, eccentricity, true anomaly, (v) inclination and r (a negative value) for the object, but I don't know where to begin).

Homework Equations

All the orbital parameter equations (not in vector forms)

The Attempt at a Solution

I know that it is a hyperbolic orbit since e is greater than 2. Its position and velocity are related to the semi-major axis, which we have a value for. We have to take into account that the Earth has a speed, and that the speed at the equator (in units of km/s) can be calculated with:
Ve=2*pi*R/(24*3600)

The orbital speed can be calculated from conservation of energy. The total energy of the comet is
E=(1/2)mv

The period of the NEO can be calculated with:

P^2=((4*pi^2)/(G*Me))*a^3 (Me = mass of Earth, we can neglect the NEO's mass). I am sure that I need to include the period of both the Earth and the NEO

I know that I have many equations regarding the orbital parameters, but I don't know where to begin. Can anyone guide me (with explanation and/or what equations to use)?

Thanks in advance!

 

[gneill]

A number of things are not clear (at least to me) regarding this problem. Perhaps you can flesh out the details?

a) You have orbit parameters but you don't say what frame of reference they pertain to. Heliocentric-ecliptic? Or is it an "osculating" orbit in an Earth-centered reference frame that pertains only to when the object is in the Earth's sphere of influence?

b) You say that the orbit is hyperbolic but it has a period? Hyperbolic orbits are open, non-repeating.

c) What does it mean to intercept and destroy the object? Does it count if you intercept it at the Earth's surface?

I think you need to make the "mission" statement more precise before you can expect to make progress.

 

[Imagin_e]

I might have been a little vague when it came to the details. Well, I assume that since the eccentricity is 2.8 The following are also given for this exercise:
v= 20 deg , a=-2790.44 km , i=22 deg , v=247.9 deg , r= 2,105,101.331 km

No, the interception should happen over over planet. I assume that we need to calculate the velocity needed for us to reach the asteroid. After that, we can probably use an equation for the time. I assume it is:
p=2*pi*sqrt(a^3/my) ; my= gravitational parameter.

I guess that it shouldn't be in the ECI coordinate system since it is orbiting the Sun

 

[Imagin_e]

I assume that the intercept with the target only can happen when the phase angle between Earth and the NEO are the same. I know that one can calculate the velocity of the S/C with (my=gravitational param.): v=sqrt((2*my/r)-(my/a)) If we calculate the escape velocity of the Earth with Ve=sqrt(2*G*Me/r) , we will get the difference in velocity. With this information, we can calculate the time needed. But my assumptions does not include all the orbital parameters, so it must be wrong

 

[gneill]

I might have been a little vague when it came to the details. Well, I assume that since the eccentricity is 2.8 The following are also given for this exercise:
v= 20 deg , a=-2790.44 km , i=22 deg , v=247.9 deg , r= 2,105,101.331 km

Can you spell out the names of the parameters you've given? You've used "v" twice, and that value for r corresponds to about 330 Earth radii, or about 0.014 AU. In a Heliocentric-ecliptic coordinate system that would place the object rather close to the Sun.

It is still not clear to me what the scope of your intercept mission is. Are you planning to leave from the Earth's surface, achieve a low Earth orbit, then depart from orbit to intercept the object? All of these things require maneuvers that could be time consuming if you have to wait for appropriate alignment conditions. You'll be dealing with at least some portion of the trajectory within the Earth's sphere of influence, then some portion under the Sun's influence. Are you going to "meet" the object as it approaches, or try to catch it as it departs? What is the Earth's location in its orbit when all this is going on?

Have you made a sketch of the object's orbit along with the Earth's orbit?

 

[Imagin_e]

Oh, I saw that forgot to change that error. We have a (semi-major axis)
e= eccentricity i=inclination , ν=true anomaly r=Radius it has from Earth's center of mass . The v=20 should not be there.
Exactly, that is what I want to do. I want to know how much time we have to intercept the NEO (and when we are there, destroy it. But that is not necessary to put emphasis on I guess) . I was looking at this problem again and came to the conclusion that the NEO is in orbit around the Sun, so we have to do some kind of orbit maneuver (they NEO is probably in the same plane all the time, so it may be easier to calculate this move as it has a coplanar orbit). It can be a simple Hohmann transfer, but then again, I can be wrong. Here are my new solution :

a for the intercept is equal for a for the target, (a_int=a_target) which is a value we have. Next, we can find the mean motion with:
ω=sqrt(μ/a³ ) (μ=gravitational parameter). Then, we can let the NEO complete one orbit, i.e. k=1 .
Thus, the phase can be calculated with; τ_phase=(2πk+ν)/ω . The interceptor will complete one revolution on the transfer orbit (k = 1). This value should be
larger than the original orbit for this example. - - > a_phase=(μ*(τ_phase/2πk)²)^1/3
Next, find the initial change in velocity for the interceptor to enter the higher orbit. Note the negative sign indicates firing opposite the velocity vector.
ΔV=|(sqrt(2μ/a)-sqrt(μ/a_phase)-sqrt(μ/a_int)|= ... m/s . The total Δv to rendezvous will be twice this amount, or Δv = 2(ΔVinitial) . And now, we should be able to calculate the time? But I don't know if it is correct.

 

[Imagin_e]

I don't think this is valid for this particular orbit, maybe for a circular one. I'm pretty much stuck at this moment

 

[gneill]

The NEO is on a hyperbolic trajectory. It will not "complete one orbit": It swings through the solar system once and once only.

The best option is to intercept the NEO on approach to Earth, as trying to catch it as it departs will require too much ΔV and time (It's like placing a road block in place rather than trying to get up to speed to chase a speed vehicle).

You give the NEO an orbital inclination of 22 degrees, which is considerable. The Earth's orbital inclination with respect to the ecliptic is defined to be zero. So a plane change for the interceptor is required unless the point of interception happens to coincide with the NEO's passage through the ecliptic plane (ascending or descending node). This would be an energy and time-efficient place to make the interception.

A simple Hohmann transfer would be able to take you to the location. Timing would require reaching that location at the appropriate instant. The time-of-flight on the transfer would be half the period of the transfer orbit. You need to time the departure from low-Earth orbit to place the apogee of the transfer orbit at the target location at the correct time. So you'll probably want to work the problem backwards starting with finding the ascending and descending nodes of the NEO and choosing which one is appropriate, then transform that location into an Earth-centered frame of reference to design your transfer orbit. Because the Earth is also moving on its own orbit you'll likely need to do some trial and error calculations to find the "sweet spot" for timing the departure. This type of mission planning is usually done with computer simulations to find those sweet spots.