Sunrise on Mercury Christina Crow, Emily Tarvin & Kevin Bowman - - PowerPoint PPT Presentation

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Sunrise on Mercury Christina Crow, Emily Tarvin & Kevin Bowman July 5, 2012 Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 1 / 30 Table of Contents Introduction 1 Keplers Laws 2 Useful Formulas

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SLIDE 1

Sunrise on Mercury

Christina Crow, Emily Tarvin & Kevin Bowman July 5, 2012

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 1 / 30

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SLIDE 2

Table of Contents

1

Introduction

2

Kepler’s Laws

3

Useful Formulas for Ellipses

4

Derivation of Semi-Major Axis, Semi-Minor Axis, and Distance from Center to Focus

5

Using Area to Obtain a Function of Time

6

Using Vectors to Track the Sun’s Position

7

The Unique Phenomenon

8

Conclusion

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 2 / 30

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SLIDE 3

Introduction

Introduction

Mercury is the closest planet to the Sun.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 3 / 30

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SLIDE 4

Introduction

Introduction

Mercury is the closest planet to the Sun. Its aphelion measures 69,816,900 km.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 3 / 30

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SLIDE 5

Introduction

Introduction

Mercury is the closest planet to the Sun. Its aphelion measures 69,816,900 km. Its perihelion measures 46,001,200 km.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 3 / 30

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SLIDE 6

Introduction

Introduction

Mercury is the closest planet to the Sun. Its aphelion measures 69,816,900 km. Its perihelion measures 46,001,200 km. Mercury is the fastest planet in our solar system, at an average speed

  • f 48 km/s.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 3 / 30

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SLIDE 7

Introduction

Introduction

Mercury is the closest planet to the Sun. Its aphelion measures 69,816,900 km. Its perihelion measures 46,001,200 km. Mercury is the fastest planet in our solar system, at an average speed

  • f 48 km/s.

This great speed causes Mercury to have a very short year; only 87.9 Earth days.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 3 / 30

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SLIDE 8

Introduction

Introduction

Mercury is the closest planet to the Sun. Its aphelion measures 69,816,900 km. Its perihelion measures 46,001,200 km. Mercury is the fastest planet in our solar system, at an average speed

  • f 48 km/s.

This great speed causes Mercury to have a very short year; only 87.9 Earth days. A solar day on Mercury lasts 2 Mercurian years, or 176 Earth days, and a sidereal day lasts 58.6 Earth days, or 2

3 of a Mercurian year.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 3 / 30

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SLIDE 9

Introduction

Introduction

Mercury has a 3:2 spin:orbit resonance, and because its orbital speed is much greater than its rotational speed, an interesting occurance happens during sunrise and sunset on Mercury.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 4 / 30

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SLIDE 10

Introduction

Introduction

Mercury has a 3:2 spin:orbit resonance, and because its orbital speed is much greater than its rotational speed, an interesting occurance happens during sunrise and sunset on Mercury. The purpose of this project is to explore, explain, and illustrate this unique phenomenon.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 4 / 30

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SLIDE 11

Kepler’s Laws

Kepler’s Laws

Kepler’s First Law states that planets travel along elliptical orbits with the Sun as a focus.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 5 / 30

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SLIDE 12

Kepler’s Laws

Kepler’s Laws

Kepler’s First Law states that planets travel along elliptical orbits with the Sun as a focus. Kepler’s Second Law states that a line joining a planet and the Sun sweeps out equal areas in equal intervals of time, therefore a planet travels fastest at perihelion and slowest at aphelion.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 5 / 30

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SLIDE 13

Kepler’s Laws

Kepler’s Laws

Kepler’s First Law states that planets travel along elliptical orbits with the Sun as a focus. Kepler’s Second Law states that a line joining a planet and the Sun sweeps out equal areas in equal intervals of time, therefore a planet travels fastest at perihelion and slowest at aphelion. Kepler’s Third Law states that a planet’s sidereal period (or year) is proportional to the square root of its semimajor axis cubed.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 5 / 30

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SLIDE 14

Kepler’s Laws

Kepler’s Laws

Kepler’s First Law states that planets travel along elliptical orbits with the Sun as a focus. Kepler’s Second Law states that a line joining a planet and the Sun sweeps out equal areas in equal intervals of time, therefore a planet travels fastest at perihelion and slowest at aphelion. Kepler’s Third Law states that a planet’s sidereal period (or year) is proportional to the square root of its semimajor axis cubed. All three of these laws were considered throughout this project.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 5 / 30

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SLIDE 15

Useful Formulas for Ellipses

Useful Formulas For Ellipses

We denote the semi-major axis of the ellipse as a, the semi-minor axis, b, and the distance from the center of the ellipse to a focus, c.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 6 / 30

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SLIDE 16

Useful Formulas for Ellipses

Useful Formulas For Ellipses

We denote the semi-major axis of the ellipse as a, the semi-minor axis, b, and the distance from the center of the ellipse to a focus, c. An ellipse can be described by the equation r1 + r2 = 2a where r1 and r2 are the distances from both foci to a corresponding point on the curve.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 6 / 30

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SLIDE 17

Useful Formulas for Ellipses

Useful Formulas For Ellipses

r1 r2 F1 F2 C major axis minor axis 2a 2c 2b

This graphic was obtained from http://mathworld.wolfram.com/Ellipse.html

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 7 / 30

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SLIDE 18

Useful Formulas for Ellipses

Useful Formulas For Ellipses

Another more familiar formula for an ellipse is x2 a2 + y2 b2 = 1 . Since we are at the origin, x0 and y0 are 0.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 8 / 30

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SLIDE 19

Useful Formulas for Ellipses

Useful Formulas For Ellipses

Another more familiar formula for an ellipse is x2 a2 + y2 b2 = 1 . Since we are at the origin, x0 and y0 are 0. The ellipse can be expressed in polar coordinates x = a cos(φ) and y = b sin(φ) for some parameter, φ.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 8 / 30

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SLIDE 20

Useful Formulas for Ellipses

Useful Formulas For Ellipses

The eccentricity of an ellipse is the ratio c

a.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 9 / 30

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SLIDE 21

Useful Formulas for Ellipses

Useful Formulas For Ellipses

The eccentricity of an ellipse is the ratio c

a.

(Note: A circle has eccentricity 0, and a parabola has eccentricity 1. Mercury’s orbit has the greatest eccentricity of all the planets in our solar system. Its eccentricity is 0.205.)

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 9 / 30

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SLIDE 22

Useful Formulas for Ellipses

Useful Formulas For Ellipses

The eccentricity of an ellipse is the ratio c

a.

(Note: A circle has eccentricity 0, and a parabola has eccentricity 1. Mercury’s orbit has the greatest eccentricity of all the planets in our solar system. Its eccentricity is 0.205.) The area of an ellipse can be expressed as A = πab.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 9 / 30

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SLIDE 23

Derivation of Semi-Major Axis, Semi-Minor Axis, and Distance from Center to Focus

Derivation of Semi-Major Axis, Semi-Minor Axis, and Distance from Center to Focus

In order to find the values

  • f a, b, and c unique to

Mercury’s elliptical orbit, we derived the equations based on aphelion and perihelion. We observe that the semi-major axis is described by a = aphelion + perihelion 2

b r a c a

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 10 / 30

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SLIDE 24

Derivation of Semi-Major Axis, Semi-Minor Axis, and Distance from Center to Focus

Derivation of Semi-Major Axis, Semi-Minor Axis, and Distance from Center to Focus

It is also apparent that the distance from the center to a focus is described by c = a − perihelion

b r a c a

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 11 / 30

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SLIDE 25

Derivation of Semi-Major Axis, Semi-Minor Axis, and Distance from Center to Focus

Derivation of Semi-Major Axis, Semi-Minor Axis, and Distance from Center to Focus

To find the length of the semi-minor axis, we must use the formula r1 + r2 = 2a and set r1 = r2. Hence, r = a. Using the Pythagorean Theorem, we find that b =

  • a2 − c2

.

b r a c a

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 12 / 30

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SLIDE 26

Derivation of Semi-Major Axis, Semi-Minor Axis, and Distance from Center to Focus

Derivation of Semi-Major Axis, Semi-Minor Axis, and Distance from Center to Focus

We are also able derive a, b, and c in terms of eccentricity.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 13 / 30

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SLIDE 27

Derivation of Semi-Major Axis, Semi-Minor Axis, and Distance from Center to Focus

Derivation of Semi-Major Axis, Semi-Minor Axis, and Distance from Center to Focus

We are also able derive a, b, and c in terms of eccentricity. Since eccentricity is defined as c

a and c =

√ a2 − b2, we rewrite eccentricity as

√ a2−b2 a

and solve for b to obtain b = a

  • 1 − eccentricity2

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 13 / 30

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SLIDE 28

Using Area to Obtain a Function of Time

Using Area to Obtain a Function of Time

Using Kepler’s Second Law, we derived an equation that represents time in terms of area.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 14 / 30

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SLIDE 29

Using Area to Obtain a Function of Time

Using Area to Obtain a Function of Time

Using Kepler’s Second Law, we derived an equation that represents time in terms of area. We can use area as a measure of time since they are directly proportional.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 14 / 30

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Using Area to Obtain a Function of Time

Using Area to Obtain a Function of Time

The function of position that gives time is demonstrated by the equation A(φ) = 1 2(cb sin(φ) + abφ)

a cosΦ, b sinΦ Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 15 / 30

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Using Area to Obtain a Function of Time

Using Area to Obtain a Function of Time

To obtain the equation in terms of one Mercury year, we multiply A(φ) by

1 abπ.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 16 / 30

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Using Area to Obtain a Function of Time

Using Area to Obtain a Function of Time

To obtain the equation in terms of one Mercury year, we multiply A(φ) by

1 abπ.

The new equation is A(φ) = 1 2abπ(cb sin(φ) + abφ)

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 16 / 30

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SLIDE 33

Using Area to Obtain a Function of Time

Using Area to Obtain a Function of Time

Now that we have a function of the area that gives time, we needed to invert this equation to obtain a function of time that gives position.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 17 / 30

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Using Area to Obtain a Function of Time

Using Area to Obtain a Function of Time

Now that we have a function of the area that gives time, we needed to invert this equation to obtain a function of time that gives position. We let Mathematica compute this for us. We created an animation that uses the inverted function to show Mercury orbiting the Sun and demonstrates the increased speed of Mercury at perihelion as well as the decreased speed at aphelion.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 17 / 30

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Using Vectors to Track the Sun’s Position

Using Vectors to Track the Sun’s Position

The Sun is at (−c, 0) and Mercury’s position is at (a cos(θ[t]), b sin(θ[t]))

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 18 / 30

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SLIDE 36

Using Vectors to Track the Sun’s Position

Using Vectors to Track the Sun’s Position

The Sun is at (−c, 0) and Mercury’s position is at (a cos(θ[t]), b sin(θ[t])) Therefore, the vector from Mercury to the Sun can be denoted by (−c − a cos(θ[t]), −b sin(θ[t]))

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 18 / 30

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SLIDE 37

Using Vectors to Track the Sun’s Position

Using Vectors to Track the Sun’s Position

The Sun is at (−c, 0) and Mercury’s position is at (a cos(θ[t]), b sin(θ[t])) Therefore, the vector from Mercury to the Sun can be denoted by (−c − a cos(θ[t]), −b sin(θ[t]))

M2S a cosΘt,b sinΘt c,0

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 18 / 30

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Using Vectors to Track the Sun’s Position

Using Vectors to Track the Sun’s Position

Suppose γ is the angle of Mercury’s rotation. Since Mercury orbits twice for every three rotations, the angle-to-area ratio is 2π

2 3 = 3π.

This implies that in time t, Mercury will have rotated γ = 3πt + γ0 where γ0 is the initial angle. Hence, the vector for Mercury’s horizon line is (cos(3πt + γ0), sin(3πt + γ0)) .

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 19 / 30

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Using Vectors to Track the Sun’s Position

Using Vectors to Track the Sun’s Position

M2S Horizon Line

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 20 / 30

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Using Vectors to Track the Sun’s Position

Using Vectors to Track the Sun’s Position

In order to find the angle between the vector representing the horizon line and the vector from Mercury to the Sun, we used the definition

  • f the dot product and cross

product. We normalized both of these vectors so that their magnitude is 1. cos(β) =

MercurytoSun·HorizonLine ||MercurytoSun||||HorizonLine||

sin(β) =

MercurytoSun×HorizonLine ||MercurytoSun||||HorizonLine||

Β Horizon Line M2S

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 21 / 30

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SLIDE 41

The Unique Phenomenon

The Unique Phenomenon

The graph represented by the sine function is

0.5 1.0 1.5 2.0 1.0 0.5 0.5 1.0

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 22 / 30

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SLIDE 42

The Unique Phenomenon

The Unique Phenomenon

Zoomed in from 0.4 to 0.6:

0.45 0.50 0.55 0.60 0.015 0.010 0.005 0.005 0.010 0.015

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 23 / 30

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SLIDE 43

The Unique Phenomenon

The Unique Phenomenon

The graph represented by the cosine function is

0.5 1.0 1.5 2.0 1.0 0.5 0.5 1.0

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 24 / 30

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SLIDE 44

The Unique Phenomenon

The Unique Phenomenon

As you can see from the sine and cosine graphs, there is a strange

  • ccurrence at t = 0.5 and t = 1.5.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 25 / 30

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SLIDE 45

The Unique Phenomenon

The Unique Phenomenon

As you can see from the sine and cosine graphs, there is a strange

  • ccurrence at t = 0.5 and t = 1.5.

Both of the functions ”dip” at these values for t.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 25 / 30

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SLIDE 46

The Unique Phenomenon

The Unique Phenomenon

As you can see from the sine and cosine graphs, there is a strange

  • ccurrence at t = 0.5 and t = 1.5.

Both of the functions ”dip” at these values for t. How does all of this apply to Mercury’s sunrise and sunset?

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 25 / 30

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SLIDE 47

The Unique Phenomenon

The Unique Phenomenon

0.45 0.50 0.55 0.60 0.015 0.010 0.005 0.005 0.010 0.015

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 26 / 30

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SLIDE 48

The Unique Phenomenon

Miscellaneous

A change in the Sun’s size can be easily detected on Mercury. To understand the perspective of the Sun from Mercury, we can find the angle, α: α = tan−1 RadiusoftheSun DistancefromMercurytotheSun

Α Sun's Radius

  • Dist. to Sun

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 27 / 30

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SLIDE 49

Conclusion

Conclusion

This unique phenomenon occurs because when Mercury is at perihelion, its orbital speed is so much faster than its rotational speed.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 28 / 30

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Conclusion

Conclusion

This unique phenomenon occurs because when Mercury is at perihelion, its orbital speed is so much faster than its rotational speed. As a result, an observer on Mercury could witness a double sunrise during a single perihelion passage, and a double sunset during the next perihelion passage which would occur during the same solar day.

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 28 / 30

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SLIDE 51

Conclusion

Acknowledgements

Richard

  • Dr. Smolinsky

SMILE coordinators

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 29 / 30

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SLIDE 52

Conclusion

References

Exploring Mercury: The Iron Planet, Robert G. Strom http://mathworld.wolfram.com/Ellipse.html http://nssdc.gsfc.nasa.gov/planetary/factsheet/mercuryfact.html http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html http://www.drennon.org/science/kepler.htm http://www.youtube.com/watch?v=nPprOO2u1gk http://dictionary.reference.com/browse/ellipse http://en.wikipedia.org/wiki/Mercury (planet)

Christina Crow, Emily Tarvin & Kevin Bowman () Sunrise on Mercury July 5, 2012 30 / 30