ASEN 5050 SPACEFLIGHT DYNAMICS Two-Body Motion

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ASEN 5050SPACEFLIGHT DYNAMICSTwo-Body Motion

• Prof. Jeffrey S. Parker
• University of Colorado Boulder

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Announcements

• Homework 1 is due Friday 9/5 at 900 am
• Either handed in or uploaded to D2L
• Late policy is 10 per school day, where a day
  starts at 900 am.
• Homework 2 is due Friday 9/12 at 900 am
• Concept Quiz 3 will be available starting soon
  after this lecture.
• Reading Chapters 1 and 2

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Space News

• Dawn is en route to Ceres, following a beautiful
  visit at Vesta. It will arrive at Ceres in
  January.
• Left Dawns 1-month descent from RC3 to Survey.
  Center Dawns expected 6-week descent from the
  survey orbit to HAMO.
• Right Dawns 8-week descent from HAMO to LAMO.

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Concept Quiz 2

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

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Concept Quiz 2
Energy is a scalar
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Concept Quiz 2
3 position, 3 velocity per body 6 per body X 3
bodies 18
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Concept Quiz 2
TOTAL angular momentum is always conserved in a
conservative system.
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Concept Quiz 2
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Challenge 1

• The best submission for the New Earth
  observations of the alternate Solar System earned
  Leah 1 bonus extra credit point.
• Whats that worth? Something more than zero and
  less than a full homework assignment ?
• Ill only mention names when given permission.
• Ill try to have more of these challenges in the
  future!

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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.

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Todays Lecture Topics

• Keplers 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

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Keplers 1st Law

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

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Conic Sections
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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)?

• Send me an email with the subject Challenge 2
• No computers (no cheating!)
• What do you get if you plot this for
• Circles
• Ellipses
• Parabolas
• Hyperbolas

vy
vx
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Geometry of Conic Sections
(Vallado, 2013)
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Geometry of Conic Sections
a semimajor axis b semiminor axis

• Elliptical Orbits 0 lt e lt 1
• Sometimes flattening is also used

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Elliptic Orbits

• p semiparameter or semilatus rectum
• Earth Sun Moon
• ra apoapsis ? apogee ? aphelion ? aposelenium ?
  etc.
• rp periapsis ? perigee ? perihelion ? periseleni
  um ? etc.

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Geometry of Conic Sections

• Elliptical Orbits 0 lt e lt 1
• Check whats
• What is
• Hmmmm, so what is

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Elliptic Orbits

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

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Geometry of Conic Sections

• Parabolic Orbit

(Vallado, 2013)
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Parabolic Orbit
Note As n ? 180? r ? 8 v ? 0 A parabolic
orbit is a borderline case between an open
hyperbolic orbit and a closed elliptic orbit
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Geometry of Conic Sections

• Hyperbolic Orbit

(Vallado, 2013)
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Hyperbolic Orbit
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Hyperbolic Orbit

• Interplanetary transfers use hyperbolic orbits
  everywhere
• Launch
• Gravity assists
• Arrivals
• Probes

(Vallado, 2013)
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Properties of Conic Sections
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Flight Path Angle

• This is also a good time to define the flight
  path angle, ffpa, 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

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Flight Path Angle

• Another useful relationship is

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Specific Energy

• Recall the energy equation
• Note at periapse hrpvp rpa(1-e)

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Vis-Viva Equation

• The energy equation
• Solving for v yields the Vis-Viva Equation!
• or

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Vis-Viva Equation

• The energy equation
• Solving for v yields the Vis-Viva Equation!
• or

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Additional derivables

• And any number of other things. Im sure Ill
  find an interesting way to stretch your
  imagination on a quiz / HW / test.

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Proving Keplers 2nd and 3rd Laws
Expression of Keplers 3rd Law
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Proving Keplers 2nd and 3rd Laws
Expression of Keplers 3rd Law
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Proving Keplers 2nd and 3rd Laws
Expression of Keplers 3rd Law
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Proving Keplers 2nd and 3rd Laws
Expression of Keplers 3rd Law
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Proving Keplers 2nd and 3rd Laws
Shuttle (300km) 90 min Earth Obs (800 km) 101
min GPS (20,000 km) 12 hrs GEO (36,000 km) 24
hrs
Mean angular rate of change of the object in orbit
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Final Statements

• Homework 1 is due Friday 9/5 at 900 am
• Either handed in or uploaded to D2L
• Late policy is 10 per school day, where a day
  starts at 900 am.
• Homework 2 is due Friday 9/12 at 900 am
• Concept Quiz 3 will be available starting soon
  after this lecture.
• Reading Chapters 1 and 2