ASEN 5050 SPACEFLIGHT 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 5 Energy is a scalar

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.

Concept Quiz #2 Lecture 3: The Two Body Problem 8

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 1 st 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

Conic Sections Lecture 3: The Two Body Problem 13

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 v x (t) vs. v y (t)? Lecture 3: The Two Body Problem 14 vx vy Send me an with the subject: Challenge #2 No 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)

Geometry of Conic Sections Elliptical Orbits 0 < e < 1 Sometimes flattening is also used a = semimajor axis b = semiminor axis Lecture 3: The Two Body Problem 16

Elliptic Orbits p = semiparameter or semilatus rectum EarthSunMoon r a apoapsis  apogee  aphelion  aposelenium  etc. r p 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

Geometry of Conic Sections Parabolic Orbit Lecture 3: The Two Body Problem 20 (Vallado, 2013)

Parabolic Orbit Note: As 180  r ∞r ∞ 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

Geometry of Conic Sections Hyperbolic Orbit Lecture 3: The Two Body Problem 22 (Vallado, 2013)

Hyperbolic Orbit Lecture 3: The Two Body Problem 23

Hyperbolic Orbit Interplanetary transfers use hyperbolic orbits everywhere –Launch –Gravity assists –Arrivals –Probes Lecture 3: The Two Body Problem 24 (Vallado, 2013)

Properties of Conic Sections Lecture 3: The Two Body Problem 25

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 0at periapsis and apoapsis Always 0 for circular orbits Only elliptic orbits (h=r a v a =r p v p ) Lecture 3: The Two Body Problem 26

Flight Path Angle Another useful relationship is: Lecture 3: The Two Body Problem 27

Specific Energy Recall the energy equation: Note at periapse h=r p v p r p =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 2 nd and 3 rd Laws Expression of Kepler’s 3 rd Law Proves 2 nd Law Lecture 3: The Two Body Problem 32

Proving Kepler’s 2 nd and 3 rd Laws Expression of Kepler’s 3 rd Law Proves 2 nd Law Lecture 3: The Two Body Problem 33

Proving Kepler’s 2 nd and 3 rd Laws Expression of Kepler’s 3 rd Law Proves 2 nd Law Lecture 3: The Two Body Problem 34

Proving Kepler’s 2 nd and 3 rd Laws Expression of Kepler’s 3 rd Law Proves 2 nd Law Lecture 3: The Two Body Problem 35

Proving Kepler’s 2 nd and 3 rd Laws Mean angular rate of change of the object in orbit Shuttle (300km)90 min Earth Obs (800 km)101 min GPS (20,000 km) ~12 hrs GEO (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