ASEN 5050 SPACEFLIGHT DYNAMICS Two-Body Motion Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 3: The Two Body Problem 1.

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


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.


Concept Quiz #2

• Great job! 1/3 of the class missed one problem; 2/3 got ‘em all right.


Concept Quiz #2


Energy is a scalar

Concept Quiz #2


3 position, 3 velocity per body

6 per body X 3 bodies = 18

Concept Quiz #2


TOTAL angular momentum is always conserved in a conservative system.

Concept Quiz #2


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!


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.


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


Kepler’s 1st Law

“Conic Section” is the intersection of a plane with a cone. m is at the primary focus of the ellipse.


Conic Sections


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)?


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

Lecture 3: The Two Body Problem 15(Vallado, 2013)


Elliptical Orbits 0 < e < 1

Sometimes flattening is also used

a = semimajor axisb = semiminor axis


Elliptic Orbits

p = semiparameter or semilatus rectum

Earth Sun Moon

ra apoapsis apogee aphelion aposelenium

etc.

rp periapsis perigee perihelion periselenium

etc.



Elliptical Orbits 0 < e < 1

Check: what’s

What is

Hmmmm, so what is


Elliptic Orbits

• What is the velocity of a satellite at each point along an elliptic orbit?



Parabolic Orbit


Parabolic Orbit

Note: As 180r ∞

v 0

A parabolic orbit is a borderline case between an open hyperbolic orbit and a closed elliptic orbit



Hyperbolic Orbit


Hyperbolic Orbit


Hyperbolic Orbit

• Interplanetary transfers use hyperbolic orbits everywhere– Launch

– Gravity assists

– Arrivals

– Probes


Properties of Conic Sections


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)


Flight Path Angle

Another useful relationship is:


Specific Energy

Recall the energy equation:

Note at periapse h=rpvp rp=a(1-e)


Vis-Viva Equation

The energy equation:

Solving for v yields the Vis-Viva Equation!

or


Vis-Viva Equation

The energy equation:

Solving for v yields the Vis-Viva Equation!

or


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.


Proving Kepler’s 2nd and 3rd Laws

Expression of Kepler’s 3rd Law

Proves 2nd Law




Proves 2nd Law




Proves 2nd Law




Proves 2nd Law



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


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


ASEN 5050 SPACEFLIGHT DYNAMICS Two-Body Motion Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 3: The Two Body Problem 1.

Documents

body problem

body x

example problem

scalarconcept quiz

amconcept quiz

week descent

survey orbit

homework assignment