Find the Lead Pursuit Line to Reach a Planet

[GrantB]

Suppose you are at a point (x₀,y₀) in space. There is a planet at position (x₁,y₁) orbiting in a circle a distance r away from the orbit center (x₂,y₂).

The planet has constant angular velocity as it orbits. You move at a constant speed towards the planet, and want to move in a straight line.

How do you determine how to reach the planet, traveling only in a straight line?

Thanks.

 

[Bobbywhy]

GrantB, when you say "You move at a constant velocity from the planet,..." does that mean you are moving AWAY from the planet? Then you want to determine how to reach the planet, so I am confused.

So, I am guessing you may get some insight on the problem by considering this:

"Basic fighter maneuvers (BFM) are tactical movements performed by fighter aircraft during air combat maneuvering (also called ACM, or dogfighting), in order to gain a positional advantage over the opponent."

http://en.wikipedia.org/wiki/Basic_fighter_maneuvers

This article includes the Lead Persuit attack course, which you mention in the thread title, but do not mention in the post itself.

 

[GrantB]

Sorry, it was a mistype.

It should say:

The planet has constant angular velocity as it orbits. You move at a constant speed towards the planet, and want to move in a straight line.

So, you want to get to the planet, and you are at an arbitrary point a distance d from the planet, moving only in a straight line.

Thanks, and sorry for the mistype.

 

[Bobbywhy]

I think your problem of intercepting a moving planet can be solved by using the methods of missile guidance trajectories. Here is an article you may be able to use. See:

www.jhuapl.edu/techdigest/TD/td2901/Palumbo_Homing.pdf

“Figure 2. Planar engagement geometry. The planar intercept problem is illustrated along with most of the angular and Cartesian quantities necessary to derive modern guidance laws.”

 

[pmacias]

To reach the planet, you will need to find the lead pursuit line, which is the straight line that connects your current position (x0,y0) to the position of the planet (x1,y1). This line will guide you towards the planet in the most efficient way, allowing you to reach it while traveling in a straight line.

To determine the lead pursuit line, you will need to consider the relative positions, velocities, and accelerations of both your spacecraft and the planet. Since the planet is orbiting at a constant angular velocity, its position and velocity can be determined using basic equations of circular motion.

Next, you will need to calculate the relative velocity between your spacecraft and the planet. This can be done by subtracting the velocity of the planet from your spacecraft's velocity. This relative velocity will be the direction and speed at which you need to move in order to reach the planet.

Finally, you can use this relative velocity to calculate the lead pursuit line using the concept of vector addition. By adding your spacecraft's velocity vector to the relative velocity vector, you will get the direction and speed of the lead pursuit line. This will guide you towards the planet in the most efficient way, allowing you to reach it while traveling in a straight line.

It's important to note that the lead pursuit line will change as your spacecraft moves closer to the planet, so it will need to be continuously recalculated to ensure you are on the most efficient path towards the planet. As a scientist, you can use mathematical equations and principles to determine and adjust the lead pursuit line, making your journey to the planet as efficient and accurate as possible.