How did the Kepler Telescope reach its orbit?

Question

The Kepler Telescope was (is) in a heliocentric Earth trailing orbit, but I want to figure out how it was able to reach this orbit exactly.

It was launched on a Delta II with 3 stages and I studied the different burns and the parking orbit (picture below). After coasting in the parking orbit, the second and third stage put the telescope in its appropriate orbit. Still I find it difficult to reconstruct how exactly it went from LEO Parking orbit to a heliocentric orbit approximately the same as Earth and trailing it. How exactly did those 2 burns put it in this orbit?

A professor helped me by stating it made several widening ellipses from LEO to its orbit (she gave me this picture). But how does this work with just 2 burns?

Can somebody show me how the trajectory from LEO went and where the burns took place?

Thanks!

Answer 1

Going from the diagram you presented, it seem that after the first of those two burns (the second burn of the second stage) if was in an elliptical Earth orbit ("94 x 1180 n.mi."). A minute or so later, still pretty much at the perigee of that orbit, the third stage lit up and burned for 90 seconds. At the end of that it was in a hyperbolic Earth orbit with perigee 99 n.mi. On that orbit it escapes from Earth, slowing as it gets further away. Had Earth been all alone in the Universe, it would have approached a velocity of $\sqrt{0.61} = 0.78 km/s$ as the distance to Earth approached infinity. In fact though, the influence of the Sun becomes the dominant consideration once you get more than a million km or so from Earth, and from that perspective, it was now in a similar orbit to Earth's, modified by that 0.78 km/s, and the direction was chosen so that this was an ellipse a little larger than Earth's orbit, which takes a little longer, so that it slowly dropped behind Earth.

Answer 2

If they were indeed quoted correctly here, then in that case your professor is not exactly right.

A plot of the first few days (not shown) shows a featureless departure from Earth to heliocentric orbit, no spiraling.

Nonetheless, there be spirals!

Depending on what you choose for the center of your plot and whether your frame rotates once per year or is "inertial¹" it can absolutely look like a spiral around the Earth!

And I can see why one might assume, having seen the spirally pattern and not realize it always looks like that in some frame, that one might think that it looks like a spacecraft spiraling away from Earth, perhaps under ion propulsion.

But it's just the way to bodies in similar orbits look if their periods are very slightly different and one or both are slightly elliptical.

Plots of the two STEREO spacecraft will look something like this too, except there's a pair of counter-spiraling orbits since one leads and one lags.

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In the plots distances are in kilometers, and the data is from JPL's Horizons every 6 hours since launch 2009-03-07 04:57.

¹not a rotating frame but still accelerating because origin moves with the Earth

import numpy as np
import matplotlib.pyplot as plt
class Body(object):
    def init(self, name):
        self.name = name
def rotz(vecs, th):
    x, y, z = vecs
    cth, sth = np.cos(th), np.sin(th)
    xrot = x * cth - y * sth
    yrot = y * cth + x * sth
    return np.vstack((xrot, yrot, z))
fnames = ('Kepler Sun full horizons_results.txt',
          'Kepler Earth full horizons_results.txt',
          'Kepler Kepler full horizons_results.txt')
bodies = []
for fname in fnames:
with open(fname, 'r') as infile:
    lines = infile.read().splitlines()

iSOE = [i for i, line in enumerate(lines) if &quot;<span class="math-container">$$SOE" in line][0]
iEOE = [i for i, line in enumerate(lines) if "$$</span>EOE&quot; in line][0]

print(iSOE, iEOE, lines[iSOE], lines[iEOE])
lines = [line.split(',') for line in lines[iSOE+1:iEOE]]
JD  = np.array([float(line[0]) for line in lines])
pos = np.array([[float(item) for item in line[2:5]] for line in lines])
vel = np.array([[float(item) for item in line[5:8]] for line in lines])

b = Body(fname.split()[0])
b.JD = JD
b.pos = pos.T.copy()
b.vel = vel.T.copy()

bodies.append(b)


sun, earth, kepler = bodies
days = kepler.JD - kepler.JD[0]
x, y, z = earth.pos - sun.pos
theta = np.arctan2(y, x)
for body in bodies:
    body.posr = rotz(body.pos, -theta)
if True:
    plt.figure()
    plt.subplot(2, 2, 1)
    x, y, z = kepler.pos - earth.pos
    plt.plot(x, y, '-r', linewidth=1.0)
    plt.plot([0], [0], 'ob')
    plt.gca().set_aspect('equal')
    plt.title('Kepler minus Earth "inertial"')
plt.subplot(2, 2, 2)
x, y, z = kepler.posr - earth.posr
plt.plot(x, y, '-r', linewidth=1.0)
plt.plot([0], [0], 'ob')
plt.gca().set_aspect('equal')
plt.title('Kepler minus Earth rotating')

plt.subplot(2, 2, 3)
x, y, z = earth.pos - sun.pos
plt.plot(x, y, '-b', linewidth=1.5)
x, y, z = kepler.pos - sun.pos
plt.plot(x, y, '-r', linewidth=1.0)
plt.plot([0], [0], 'oy')
plt.gca().set_aspect('equal')
plt.title('Kepler &amp; Earth  minus Sun &quot;inertial&quot;')

plt.subplot(2, 2, 4)
x, y, z = earth.posr - sun.posr
plt.plot(x, y, '-b', linewidth=2.5)
x, y, z = kepler.posr - sun.posr
plt.plot(x, y, '-r', linewidth=1.0)
plt.plot([0], [0], 'oy')
plt.gca().set_aspect('equal')
plt.title('Kepler &amp; Earth  minus Sun rotating')
plt.show()