asen 5050 spaceflight dynamics two-body motion prof. jeffrey s. parker university of colorado –...
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ASEN 5050SPACEFLIGHT DYNAMICS
Two-Body Motion
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 3: The Two Body Problem 1
Announcements• Homework #1 is due Friday 9/5 at 9:00 am
– Either handed in or uploaded to D2L– Late policy is 10% per school day, where a “day” starts at
9:00 am.
• Homework #2 is due Friday 9/12 at 9:00 am
• Concept Quiz #3 will be available starting soon after this lecture.
• Reading: Chapters 1 and 2
Lecture 3: The Two Body Problem 2
Space News
• Dawn is en route to Ceres, following a beautiful visit at Vesta. It will arrive at Ceres in January.
• Left: Dawn’s 1-month descent from RC3 to Survey. Center: Dawn’s expected 6-week descent from the survey orbit to HAMO.
• Right: Dawn’s 8-week descent from HAMO to LAMO.
Lecture 3: The Two Body Problem 3
Concept Quiz #2
• Great job! 1/3 of the class missed one problem; 2/3 got ‘em all right.
Lecture 3: The Two Body Problem 4
Concept Quiz #2
Lecture 3: The Two Body Problem 6
3 position, 3 velocity per body
6 per body X 3 bodies = 18
Concept Quiz #2
Lecture 3: The Two Body Problem 7
TOTAL angular momentum is always conserved in a conservative system.
Challenge #1
• The best submission for the “New Earth” observations of the alternate Solar System earned Leah 1 bonus extra credit point.– What’s that worth? Something more than zero and less
than a full homework assignment – I’ll only mention names when given permission.
– I’ll try to have more of these challenges in the future!
Lecture 3: The Two Body Problem 9
Homework #2
• This homework will use information presented today and Friday, hence its due date will be a week from Friday.
• HW2 has a lot of math, but this is the good stuff in astrodynamics. An example problem:– A satellite has been launched into a 798 x 816 km orbit (perigee
height x apogee height), which is very close to the planned orbit of 795 x 814 km. What is the error in the semi-major axis, eccentricity, and the orbital period?
• I suggest starting to build your own library of code to perform astrodynamical computations. We will be doing this a lot in this course.
Lecture 3: The Two Body Problem 10
Today’s Lecture Topics
• Kepler’s Laws
• Properties of conic orbits
• The Vis-Viva Equation! You will fall in love with this equation.
• Next time: Converting between the anomalies• Then: More two-body orbital element computations
Lecture 3: The Two Body Problem 11
Kepler’s 1st Law
“Conic Section” is the intersection of a plane with a cone. m is at the primary focus of the ellipse.
Lecture 3: The Two Body Problem 12
Challenge #2
• For those of you who are very familiar with the properties of conic sections:
• Consider planar orbits (elliptical, parabolic, hyperbolic)• What do you get if you plot vx(t) vs. vy(t)?
Lecture 3: The Two Body Problem 14
vx
vy
Send me an email with the subject: Challenge #2No computers (no cheating!)What do you get if you plot this for:•Circles•Ellipses•Parabolas•Hyperbolas
Geometry of Conic Sections
Elliptical Orbits 0 < e < 1
Sometimes flattening is also used
a = semimajor axisb = semiminor axis
Lecture 3: The Two Body Problem 16
Elliptic Orbits
p = semiparameter or semilatus rectum
Earth Sun Moon
ra apoapsis apogee aphelion aposelenium
etc.
rp periapsis perigee perihelion periselenium
etc.
Lecture 3: The Two Body Problem 17
Geometry of Conic Sections
Elliptical Orbits 0 < e < 1
Check: what’s
What is
Hmmmm, so what is
Lecture 3: The Two Body Problem 18
Elliptic Orbits
• What is the velocity of a satellite at each point along an elliptic orbit?
Lecture 3: The Two Body Problem 19
Parabolic Orbit
Note: As 180r ∞
v 0
A parabolic orbit is a borderline case between an open hyperbolic orbit and a closed elliptic orbit
Lecture 3: The Two Body Problem 21
Hyperbolic Orbit
• Interplanetary transfers use hyperbolic orbits everywhere– Launch
– Gravity assists
– Arrivals
– Probes
Lecture 3: The Two Body Problem 24(Vallado, 2013)
Flight Path Angle
This is also a good time to define the flight path angle, fpa, as the angle from the local horizontal to the velocity vector.
+ from periapsis to apoapsis
- from apoapsis to periapsis
0 at periapsis and apoapsis
Always 0 for circular orbits
Only elliptic orbits
(h=rava=rpvp)
Lecture 3: The Two Body Problem 26
Specific Energy
Recall the energy equation:
Note at periapse h=rpvp rp=a(1-e)
Lecture 3: The Two Body Problem 28
Vis-Viva Equation
The energy equation:
Solving for v yields the Vis-Viva Equation!
or
Lecture 3: The Two Body Problem 29
Vis-Viva Equation
The energy equation:
Solving for v yields the Vis-Viva Equation!
or
Lecture 3: The Two Body Problem 30
Additional derivables
And any number of other things. I’m sure I’ll find an interesting way to stretch your imagination on a quiz / HW / test.
Lecture 3: The Two Body Problem 31
Proving Kepler’s 2nd and 3rd Laws
Expression of Kepler’s 3rd Law
Proves 2nd Law
Lecture 3: The Two Body Problem 32
Proving Kepler’s 2nd and 3rd Laws
Expression of Kepler’s 3rd Law
Proves 2nd Law
Lecture 3: The Two Body Problem 33
Proving Kepler’s 2nd and 3rd Laws
Expression of Kepler’s 3rd Law
Proves 2nd Law
Lecture 3: The Two Body Problem 34
Proving Kepler’s 2nd and 3rd Laws
Expression of Kepler’s 3rd Law
Proves 2nd Law
Lecture 3: The Two Body Problem 35
Proving Kepler’s 2nd and 3rd Laws
Mean angular rate of change of the object in orbit
Shuttle (300km) 90 minEarth Obs (800 km) 101 minGPS (20,000 km) ~12 hrsGEO (36,000 km) ~24 hrs
Lecture 3: The Two Body Problem 36
Final Statements• Homework #1 is due Friday 9/5 at 9:00 am
– Either handed in or uploaded to D2L– Late policy is 10% per school day, where a “day” starts at
9:00 am.
• Homework #2 is due Friday 9/12 at 9:00 am
• Concept Quiz #3 will be available starting soon after this lecture.
• Reading: Chapters 1 and 2
Lecture 3: The Two Body Problem 37
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