The document discusses Kepler's laws of planetary motion, which describe the orbits of planets, comets, and other celestial bodies as elliptical paths with the sun at one focus. It also highlights the contributions of Johannes Kepler and Carl Friedrich Gauss, particularly Gauss's successful prediction of the orbit of the dwarf planet Ceres using these laws. Kepler's laws have become foundational in the field of astronomy, affecting our understanding of celestial mechanics and orbital dynamics.

1. Math in SPACE
Planetary Orbits
Presented By Gregory Tewksbury
A Mathematics Major from The University of Cincinnati

2. Suddenly I See: Kepler's Laws All Three

3. How do you
determine the orbit?
Any Guesses?
(Pictured is Ceres)

4. But First…

5. Oops, I meant Conic Sections..

6. Why did I mention Conic Sections?

7. Why did I mention Conic Sections?
• Conic sections are used to show the orbits of planets, dwarf planets,
comets, and more!

8. Orbits & Conic Sections
• Orbits can be modeled with an ellipse, BUT most planetary orbits are
almost circles, so it is not apparent that they are actually ellipses.
• This is described in Kepler’s Laws of Planetary Motion

9. Side Note: Drawing an Ellipse

10. Now, let’s dive into Kepler’s Laws
of Planetary Motion!

11. Who Was Kepler?
• Johannes Kepler (December 27, 1571 – November 15, 1630) was a
German mathematician, astronomer, and astrologer.
• He was a sickly child and his parents were poor.
• But his evident intelligence earned him a scholarship to the University
of Tübingen to study for the Lutheran ministry.

12. Kepler’s Firsts
In his book Astronomia Pars Optica, for which he earned the title of founder of
modern optics he was the:
• First to investigate the formation of pictures with a pin hole camera;
• First to explain the process of vision by refraction within the eye;
• First to formulate eyeglass designing for nearsightedness and farsightedness;
• First to explain the use of both eyes for depth perception.
In his book Dioptrice (a term coined by Kepler and still used today) he was the:
• First to describe: real, virtual, upright and inverted images and magnification;
• First to explain the principles of how a telescope works;
• First to discover and describe the properties of total internal reflection.
BUT WAIT, THERE IS MORE!

13. PLUS…
In addition:
• His book Stereometrica Doliorum formed the basis of integral calculus.
• First to explain that the tides are caused by the Moon (Galileo reproved him for this).
• Tried to use stellar parallax caused by the Earth's orbit to measure the distance to the
stars; the same principle as depth perception. Today this branch of research is called
astrometry.
• First to suggest that the Sun rotates about its axis in Astronomia Nova
• First to derive the birth year of Christ, that is now universally accepted.
• First to derive logarithms purely based on mathematics, independent of Napier's tables
published in 1614.
• He coined the word "satellite" in his pamphlet Narratio de Observatis a se quatuor Iovis
sattelitibus erronibus
Drum Roll Please…

14. But his main claim to fame was…

15. But his main claim to fame was…
… being first to correctly explain planetary
motion, thereby, becoming founder of
celestial mechanics and the first "natural
laws" in the modern sense; being universal,
verifiable, precise!

16. Kepler’s Laws
In astronomy, Kepler's laws of planetary motion are three scientific
laws describing the motion of planets around the Sun. Kepler's laws are
now traditionally enumerated in this way:
1. The orbit of a planet is an ellipse with the Sun at one of the two
foci.
2. A line segment joining a planet and the Sun sweeps out equal areas
during equal intervals of time.
3. The square of the orbital period of a planet is proportional to the
cube of the semi-major axis of its orbit.

17. Kepler’s Laws (cont)
(Right) Illustration of Kepler's three laws with
two planetary orbits.
1. The orbits are ellipses, with focal points
푓1 and 푓2 for the first planet and 푓1 and
푓3 for the second planet. The Sun is
placed in focal point 푓1.
2. The two shaded sectors 퐴1 and 퐴2 have
the same surface area and the time for
planet 1 to cover segment 퐴1 is equal to
the time to cover segment 퐴2.
3. The total orbit times for planet 1 and
3
2 : 푎2
planet 2 have a ratio 푎1
3
2 .

18. Kepler’s First Law
The orbit of every planet is an ellipse with the Sun at one of the two foci.
• Mathematically, an ellipse can be represented by the formula:
• 푟 =
푝
1+휀 cos 휃
• Where 푝 is the semi-latus rectum
• 휀 is the eccentricity of the ellipse
• 푟 is the distance between the Sun and the planet
• And 휃 is the angle to the planet’s current position from its closest approach, as seen from the
Sun.
• So as result, the polar coordinates of a planet (with the Sun being 0,0 ) would be
푟, 휃
• If 휀 = 0, the eccentricity is equal to zero and the orbit would be a circle

19. The Law of Orbits
• Kepler’s First Law is sometimes referred to as the Law of Orbits
• The elliptical shape of the
orbit is a result of the
inverse square force of
gravity.
• The eccentricity of the
ellipse is greatly
exaggerated here.

20. Orbit Eccentricity
• The eccentricity of an ellipse can be
defined as the ratio of the distance
between the foci to the major axis
of the ellipse.
• The eccentricity is zero for a circle.
• Of the planetary orbits, only Pluto
has a large eccentricity.

21. Examples of Ellipse Eccentricity
Planet Eccentricity
Mercury 0.206*
*This is NOT a mistake.
This is it’s actual value
Venus 0.0068
Earth 0.0167
Mars 0.0934
Jupiter 0.0485
Saturn 0.0556
Uranus 0.0472
Neptune 0.0086
Pluto*
0.25
* I know Pluto is a dwarf
planet

22. Kepler’s Second Law
A line joining a planet and the
Sun sweeps out equal areas
during equal intervals of time.
• The orbital radius and angular
velocity of the planet in the
elliptical orbit will vary.
• This is shown in the animation
where the planet travels
faster around the sun and
slower away from the sun
• Kepler’s Second Law states
that the blue sector will have
a constant area

23. The Law of Areas
• Kepler’s Second Law is sometimes referred to as The Law of Areas
• This empirical law discovered by
Kepler arises from conservation of
angular momentum.
• When the planet is closer to the sun,
it moves faster, sweeping through a
longer path in a given time.

24. Kepler’s Third Law
• The square of the orbital period of a planet is directly proportional to
the cube of the semi-major axis of its orbit.
• This law describes the relationship between the distance of planets
from the Sun, and their orbital periods.
• Kepler enunciated in 1619 this third law in a laborious attempt to
determine what he viewed as the "music of the spheres" according to
precise laws, and express it in terms of musical notation. So it was
known as the harmonic law.

25. Kepler’s Third Law (cont)
• Mathematically, the law says that
푃2
푎3 has the same value for ALL
planets in the solar system.
• Where 푃 is the time it takes the planet to complete one orbit around the
Sun(also known as the period)
• And 푎 is the semi-major axis (also known as the mean value or average value
between the maximum and the minimum distances between the planet and
the Sun)

26. The Law of Periods
• Kepler’s Third Law is sometimes referred to as the Law of Periods.
• This law arises from the
law of gravitation. Newton
first formulated the law of
gravitation (퐹 = 퐺
푚1푚2
푟2 )
from Kepler's 3rd law.

27. The Law of Periods (cont)
• The equations on the top right
are those used for periods
(amount of time for a planet to
orbit the sun.
• Kepler's Law of Periods in the
form shown on the bottom right,
is an approximation that serves
well for the orbits of the planets
because the Sun's mass is so
dominant. But more precisely
the law should be written

28. Now that we have the
framework…

29. Carl Friedrich Gauss
• Johann Carl Friedrich Gauss
(April 30, 1777 – February
23, 1855) was a German
mathematician, who
contributed significantly to
many fields, including
number theory, algebra,
statistics, analysis,
differential geometry,
geodesy, geophysics,
electrostatics, astronomy,
Matrix theory, and optics.
• He also had a large part in
determining orbits,
especially what he did with
Ceres

30. Ceres
• The image on the right is Ceres, a
dwarf planet in our solar system
• It is located between Mars and
Jupiter (as shown below and on
the next slide)

31. The Solar System

32. Ceres
• Guiseppi Piazzi (1746-1826) discovered Ceres on January 1, 1801.
• Piazzi first thought it was a star.
• Piazzi collected 22 observations over 40 nights
• Ceres then vanished behind the sun’s rays, on February 11, 1801
• Piazzi’s data consisted of triplets (time, right ascension, declination)
giving position of asteroid at different times
• Piazzi’s observations were published in September of that year

33. Determining Ceres Orbit
• Astronomers wanted to recover Ceres’ orbit using the data Piazzi published
• Carl Gauss, who was 24 at the time, tackled the problem.
• Gauss had to solve a system of 17-by-3 linear equations (17 equations in 3
unknowns).
• Sir Isaac Newton said such orbit calculations were “among the toughest
problems in astronomy.”
• Gauss’ work led him to discover least squares approximation and what we
now call Gauss-Jordan elimination.
• Gauss made his computations in only a few weeks. On December 7, 1801,
astronomers found Ceres, exactly where Gauss said it would be!

34. Why This Was A Milestone
• Before Gauss created his method, astronomers would actually plot
the points of their observation of an object and create their orbit
prediction off of that.
• Needless to say, that required multiple observations. But Gauss only
picked 3 observations (January 2, January 22, and February 11, 1801)
• Precision was critical!

35. So Gauss Predicted It, Is It Right?
• The answer is YES! As I said earlier, on December 7, 1801
astronomers found Ceres, exactly where Gauss said it would be!
• Plus, comparing his prediction to currently known info, the percent
error is usually less than 1% (which is great)

36. Did Gauss Use Kepler’s Laws?

37. Did Gauss Use Kepler’s Laws?
• Yes! We used all of the laws!
• We used Kepler’s First Law to model the orbit and we stated it was an
ellipse
• We used Kepler’s Second Law to determine which direction the
major-axis is pointing
• We used Kepler’s Third Law to check our answer (to see if it falls on
the linear equation)

38. (Left) Carl Friedrich Gauss, considered one of the three
greatest mathematicians of all time (along with
Archimedes and Sir Isaac Newton).
(Right) Gauss at 24, when he computed the orbit of
Ceres.
(Left) Gauss’ sketch of the orbit of
Ceres.
(Right) Image of Ceres from the
Hubble telescope.

39. “. . . for it is now clearly shown that the
orbit of a heavenly body may be
determined quite nearly from good
observations embracing only a few days;
and this without any hypothetical
assumption.’’
Carl Friedrich Gauss

40. A Special Thanks to…
• Powers Educational Services for the use of their computer monitor
and letting me have a day off!
• Aaron Eiben for the experience and opportunity
• The University of Cincinnati’s Mathematics Department for all they
have done for me (too much to list)
• The Cincinnati Observatory for allowing me to do this presentation
• AND especially YOU for listening and taking part!