Time of Flight using flight path and radius (Kepler's Equation)

In summary, the lunar probe will take 223.2TU to reach the vicinity of the moon, disregarding the moon's gravity. Flight path angle, specific angular momentum, eccentricity, and the general equation for a conic orbit radius were used to determine the time of flight. The parabolic eccentric anomaly was also taken into account.
  • #1
into space
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Homework Statement

A lunar probe is given just escape speed at an altitude of .25 DU and flight path angle 30°. How long will it take the probe to get to the vicinity of the moon (r = 60 DU), disregarding the moon's gravity?

Circular orbit, radius = 1.25 DU
θ = 30°
gravitational parameter = μ = 1 DU^3/TU^2
true anomaly = v
Eccentric Anomaly = E
eccentricity = e
t-to = Time of flight
a = semi-major axis

Homework Equations

Escape velocity = √(2μ/r) = 1.26 DU/TU
tan(θ) = (esin v)/(1 + ecos v)
tan (v/2) = √((1+e)/(1-e))tan(E/2)
t-to = √(a^3/μ)(2k(pi) + (E-esinE) - (Eo - esinEo))
θ(parabolic) = v/2

The Attempt at a Solution


The answer is 223.2TU
I just can't see how flight path relates to time of flight. The second equation in the relevant equations section seems like it would fit for equating flight path and true anomaly but I can't separate the true anomaly variable. I'm not sure if that last equation for a parabolic flight path applies (the orbit is parabolic but I don't know). Any help would be greatly appreciated.
 
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  • #2
So I'll post a solution I've been working on:

I'm not sure how to think of flight path, but I assumed the transfer maneuver was like a Hohmann Transfer with 60 DU forming the apoapsis radius:
e = (60-1.25)/(60 + 1.25) = .9592

Now I'm trying to find the semi-major axis from the periapsis radius
Rp = a(1-e)
a = Rp/(1-e) = 1.25/(1-.9592) = 30.625 DU

Assuming the probe reaches the Moon's vicinity at apoapsis radius, E = pi and:
t-to = √(30.625^3)(pi - sin pi) = 532 TU
 
  • #3
If the probe is given escape speed at some point, it will not have a closed orbit; it'll be parabolic.
 
  • #4
Okay, some hints.

If you know the radial distance, speed, and flight path angle at some instant then you can determine the specific angular momentum h of the probe. That, in turn, gives you the length of the latus rectum p (since you also know the gravitational parameter μ by assumption). The eccentricity is a given since you know that the probe has escape speed. That means you can write the general equation for the conic orbit radius:

## r = \frac{p}{1 + e cos(\nu)}##

and can solve for the two values of ##\nu## involved (at 1.25DU and 60DU). You should be able to work out the time of flight between those two values of true anomaly.

Extra hint: The parabolic eccentric anomaly will be helpful.
 
  • #5
Awesome, I got the right answer! Thanks so much, I had completely forgot eccentric anomaly was solved differently depending on the orbit/trajectory.
 

1) What is Time of Flight using flight path and radius?

Time of Flight using flight path and radius is a mathematical concept that calculates the time it takes for an object to travel from one point to another in a curved path, taking into account the object's flight path and the radius of its orbit. This concept is based on Kepler's equation, which relates the radius and speed of an object in orbit to its orbital period.

2) How is Time of Flight calculated using flight path and radius?

To calculate Time of Flight using flight path and radius, you first need to know the object's flight path and the radius of its orbit. Then, you can use Kepler's equation to determine the object's orbital speed. Finally, you can use the speed and the distance traveled along the flight path to calculate the time it takes for the object to complete its orbit.

3) What is Kepler's equation and how is it used in Time of Flight calculations?

Kepler's equation is a mathematical equation developed by Johannes Kepler in the 17th century to describe the motion of planets around the Sun. It relates the radius and speed of an object in orbit to its orbital period. In Time of Flight calculations, Kepler's equation is used to determine the orbital speed of an object, which is then used to calculate the Time of Flight.

4) What factors can affect Time of Flight using flight path and radius?

Time of Flight using flight path and radius can be affected by several factors, such as the object's mass, the shape of its orbit, and any external forces acting on the object. Additionally, changes in the object's flight path or the radius of its orbit can also impact the Time of Flight.

5) How is Time of Flight using flight path and radius useful in real-world applications?

Time of Flight using flight path and radius is a useful concept in many real-world applications, such as space travel, satellite communication, and navigation systems. It allows scientists and engineers to accurately predict the time it takes for an object to travel from one point to another, which is essential for planning and executing missions and operations.

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