Burton W. Jones

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Burton W. Jones

• Distinguished
• Teacher
• and
• MAA Member

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• Gauss
• and
• Gauss Again

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These famous names

• Newton Legendre
• Euler Gauss Lagrange
• Laplace Bessel
• might all be found on a list of

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Great Astronomers!!!
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Why Astronomy?

• Celestial (orbital) mechanics the mathematics
  of things in orbit

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My introduction to orbital mechanics

• From the good Dr. Gauss himself, in a most
  unexpected way
• The Method of Least Squares

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From Edwards and Penneys Calculus

• The great German mathematician Carl Friedrich
  Gauss (1777-1855) invented the method of least
  squares when he was 18 years old. A short time
  later, he used it to determine the orbit of the
  first-discovered asteroid Ceres, initially
  observed on the first day of the nineteenth
  century, but lost from sight a few weeks later.

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From Larson and Hostetlers Calculus

• The method of least squares was introduced by
  the French mathematician Adrien-Marie Legendre
  (1752-1833).

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Whats a curious fellow to do?

• Summarische Übersicht der zur Bestimmung der
  Bahnen der beiden neuen Hauptplaneten angewandten
  Methoden
• Summary Survey of the Methods for Determining the
  Orbits of the Two New Major Planets

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What its like to read Gauss(Quotes from
Summarische Übersicht)
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What its like to read Gauss(Quotes from
Summarische Übersicht)

• One can easily convince oneself
• From this it is easy to conclude
• From this the following three equations follow
  easily
• It is clear
• Further, one easily recognizes
• I allow myself this easily understood expression
  for the sake of brevity

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Learning orbital mechanics from Gausss
Summarische Übersicht
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Learning orbital mechanics from Gausss
Summarische Übersicht

• Method of successive humiliations
• What I didnt learn (least squares)
• What I later learned (least squares)
• Where it led me (a whole new direction in my
  research, with some interesting impacts on my
  teaching!)

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Todays TalkOrbital mechanics applications in
the standard undergraduate mathematics curriculum
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Orbit basics
E (central angle)
satellite
?
t t0
f (polar angle)
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I. Calculus I
Analysis of Keplers Equation
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Analysis of Keplers Equation

• Problem 1. Given e0.5, M2.5, solve k(E)0.
• (I say, Blah blah blah Newtons method.
  Students reach for their calculators. )

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Analysis of Keplers Equation

• Problem 2. Are there other solutions?
• (Three really clever students put their
  calculators in graphing mode.)

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Analysis of Keplers Equation

• Problem 3. Same questions
• for e 1.5, M 0.2.

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Keplers Equation E-0.5 sin(E)-2.50
E-1.5 sin(E)-0.20
k(E)
k(E)
E
E
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Consider Keplers Equation E - e sin(E)
M 0, 0

Must there be a solution?

If so, must the solution be unique?

Why NASA cares about these questions

(Students look in vain for the right button on
their calculator.)

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k(E) E - e sin(E) M 0

Observe

k is continuous

k(0) 0, k(M1)0 (there is a root)

k '(E) 1 - e cos(E) 0 (root is unique)

Calculus books are jammed with functions, but
rarely look at families of functions

A valuable point of view in real world
applications

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II. Linear Algebra

• Textbook problem
• Consider the bases
• B(2,1,1), (2,-1,1), (1,2,1) and
• B(3,1,-5), (1,1,-3), (-1, 0,2).
• Find the transition matrix from B to B.
• My students always ask

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• Why in the world would I ever want to express
  vectors in any basis other than
• B(1,0,0), (0,1,0), (0,0,1)?

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• Why in the world would I ever want to express
  vectors in any basis other than
• B(1,0,0), (0,1,0), (0,0,1)?
• Answer Things that are very complicated in one
  coordinate system may be simple in another!

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• We can choose our basis to make the Earths
  rotation simple to describe

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• or we can choose our basis to make the
  satellites motion easy to describe

z'
y'
x'
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• Or we could choose the basis so that the
  elliptical orbit has an easy polar
  representation

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• But we really need all three! So
• Express vectors in the most convenient basis,
  then use transition matrices to convert as
  necessary between the various bases!

z
z'
y'
y''
z''
?
x''
y
x'
x
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• First transition matrix
• basis to
• basis

z'
y'
y''
z''
x''
Wp
x'
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• Next transition matrix

z
z'
y'
i
y
x'
x
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• Final transition matrix

z
y
RA
x
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This completes the conversion from
coordinates to coordinates!
z
y
x
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Now we can easily work with satellite positions
in one coordinate system, then change them to
another!
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Vector Time (GMT) 2005/024/133200.000
Vector Time (MET) N/A Weight (LBS)
404236.9 M50 Cartesian
M50 Keplerian
-----------------------------------
-------------------------------- X
4205964.20 a
6731367.37 meter Y -1135248.32 meter
e .0009492 Z
5124138.27 i
51.32425 XDOT 448.028837
Wp 48.90129 YDOT 7575.227356 meter/sec
RA 274.58915 deg ZDOT 1315.109936
f 28.49040

M 28.43854
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III. Differential Equations

• Space Shuttle Mission to Hubble Telescope March
  1, 2002

Initial Space Shuttle Orbit
Transfer Orbit
Hubble Orbit
?v
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• Vector Time (GMT) 2002/060/113232.000
• Vector Time (MET) 000/001030.000
• Weight (LBS) 254068.0
• M50 Cartesian
  M50 Keplerian
• -----------------------------------
  --------------------------------
• X 429037.29 a
  6700526.03 meter
• Y -5724011.02 meter e
  .0395379
• Z 3066426.36 i
  28.47537
• XDOT 7923.139316 Wp
  53.88105
• YDOT 93.941694 meter/sec RA
  174.32026 deg
• ZDOT -476.018849 f
  44.89957

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• SHUTTLE TRAJECTORY DATA
• Lift off time (UTC) 2002/060/112202.000
• Maneuvers contained within the current ephemeris
  are as follows
• IMPULSIVE TIG (GMT) M50 DVx(FPS) LVLH
  DVx(FPS)
• IMPULSIVE TIG (MET) M50 DVy(FPS) LVLH
  DVy(FPS)
• DT M50
  DVz(FPS) LVLH DVz(FPS)
• -----------------------------------------------
  -------------------------
• 060/120713.533 -99.2
  133.3
• 000/004511.533 80.6
  -0.1
• 000/000128.577 -38.0
  -0.0

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• Vector Time (GMT) 2002/060/121157.821
• Vector Time (MET) 000/004955.821
• Weight (LBS) 250747.5
• M50 Cartesian M50
  Keplerian
• -----------------------------
  --------------------------------
• X 2861810.80 a
  6769217.50 meter
• Y 5503917.12 meter e
  .0277775
• Z -3128507.70 i
  28.48130
• XDOT -6827.375539 Wp
  52.94227
• YDOT 817.354008 meter/sec RA
  174.16154 deg
• ZDOT -1143.785242 f
  197.83838

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Apply Eulers Method to a System of Differential
Equations
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IV. Calculus III The Mother Lode
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Derivation of Keplers Laws
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Vector Time (GMT) 2005/024/133200.000
Vector Time (MET) N/A Weight (LBS)
404236.9 M50 Cartesian
M50 Keplerian
-----------------------------------
-------------------------------- X
4205964.20 a
6731367.37 meter Y -1135248.32 meter
e .0009492 Z
5124138.27 i
51.32425 XDOT 448.028837
Wp 48.90129 YDOT 7575.227356 meter/sec
RA 274.58915 deg ZDOT 1315.109936
f 28.49040

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z
h
v
i
r
y
x
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Vector Time (GMT) 2005/024/133200.000
Vector Time (MET) N/A Weight (LBS)
404236.9 M50 Cartesian
M50 Keplerian
-----------------------------------
-------------------------------- X
4205964.20 a
6731367.37 meter Y -1135248.32 meter
e .0009492 Z
5124138.27 i
51.32425 XDOT 448.028837
Wp 48.90129 YDOT 7575.227356 meter/sec
RA 274.58915 deg ZDOT 1315.109936
f 28.49040

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V. My all time favorite student project

• Given the position and velocity YDOT, ZDOT of the International Space Station at
  time t0
• Determine where (azimuth and altitude) and when
  to look for the space station as it flies over
  your location.

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• Solution uses trigonometry, vector calculus,
  linear algebra, and just a dash of computer
  programming.
• Students see the results of the project in an
  extraordinary way they go outside and look in
  the calculated direction at the calculated time,
  and watch the space station fly by!

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(No Transcript)
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Related Project Real-time International Space
Station Tracker
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Research Opportunities Outside the Classroom
Context
History
Astronomy
Mathematics
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Which brings us toGauss Again!(A Current
Project)

• Problem Calculate the date of Easter in a given
  year.
• Certainly of mathematical interest (number
  theory?)
• Certainly of historical interest
• Has its roots in astronomy (full moon, vernal
  equinox, etc.)

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But what does it have to do with Gauss?

• Berechnung des Osterfestes
• Calculation of the Easter Date
• published August 1800

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Gausss Easter Algorithm

• then Easter falls on the 22 d eth of March or
  the d e 9th of April. (For 2000-2099, M24,
  N5)

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Example

• a 2005 mod 19 10
• b 2005 mod 4 1
• c 2005 mod 7 3
• d 19aM mod 30 4
• e 2b4c6dN mod 7 1
• Then Easter falls on the 22 d eth of March,
  or March 27th.

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Gausss explanation of the various steps of the
algorithm
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Gausss explanation of the various steps of the
algorithm

• The analysisdoes not allow itself to be shown
  here in its complete simplicity
• A further development of this circumstance would
  be too long and drawn out here.
• which one can easily convince oneself
• From this it is clear

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A parting gem from Dr. Gauss

• Handschriftliche Bemerkung at the end of his
  article
• From Christmas to Easter is on average,
• however, weekdays.
• (Absolutely no explanation whatever!)

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773 ???? 950

• My own observations
• A complete (Gregorian) calendar cycle consists of
  400 years, or 146,097 days

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773 ???? 950

• My own observations
• A complete (Gregorian) calendar cycle consists of
  400 years, or 146,097 days
• Prime factorization of 146,097 337773
• Coincidence? Seems unlikely, but

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I have no idea!