orbital radii of planets and moons of jupiter

IB Mathematics HL Type II Task Orbital radii of planets and moons of Jupiter Colegio Colombo Británico School Number: 000033 Juliana Peña Ocampo Candidate Number: 000033-049 May 2008

Juliana Peña Ocampo 000033-049

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Contents

Questions ..........................................................................................................................................3

Answers.............................................................................................................................................4

Answer 1.a ................................................................................................................................................ 4

Answer 1.b ................................................................................................................................................ 5

Answer 2.a ................................................................................................................................................ 6

Answer 2.b ................................................................................................................................................ 7

Answer 2.c ................................................................................................................................................. 8

Answer 3 ................................................................................................................................................. 10

Bibliography .................................................................................................................................... 13


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Questions

(Extracted from the IB Mathematics HL Internal Assessment Teacher Support Material)


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Answers

Answer 1.a In order to find the relationship between the position and the orbital radius, the data was graphed with

Microsoft Excel and three types of best fit – quadratic, cubic and exponential – were used to find the

function relating these two variables. The result is shown in Graph 1.

From this, it is possible to conclude that the exponential regression is the appropriate function relation

the position with the orbital radius because it closely approaches all points. Additionally, unlike the

quadratic and cubic functions, the exponential function is always growing (its rate of growth is positive

always), whilst the other functions have changes in their rate of growth – sometimes they are

increasing, sometimes they are decreasing. This is particularly notable in the quadratic function, but also

occurs in the cubic one.

Using Excel, the formula for the exponential function was found to be

-1000

0

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7000

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0 2 4 6 8 10 12

Orb

ital

rad

ius

/ m

illio

ns

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km

Position from Sun

Graph 1: Orbital radii of Planets

Quadratic

Cubic

Exponential


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Answer 1.b The orbital radius of an asteroid in position 5 can be calculated by replacing by 5 in [1].

Asteroids in the main asteroid belt are located 2 to 4 AU (approximately 300 to 600 million km) from the

Sun1. is almost in the middle of this range, which shows that the function [1] does have applicability

and will very possibly find asteroids.

Regarding specifically the dwarf planet Ceres, which lies in the asteroid belt, its orbital radius was

discovered by Carl Friedrich Gauss by using a different method than this one, by measuring its orbital

period. Its orbital radius was found to be 2.7 AU, approximately 403 million km.2 This model, therefore,

is not able to predict the position of Ceres accurately.

1 (1)

2 (2)


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Answer 2.a Again, Excel was used to find the relationship between the position of the moons and their orbital radii,

with three regression types, quadratic, cubic and exponential. The results are in Graph 2.

It is quite obvious that the exponential function is the best fit. Although all lines fit the four points, only

the exponential function gives an accurate trend line of greater or lesser positions than the data

graphed.

This function is

0

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0 2 4 6 8 10 12

Orb

ital

rad

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Order from Jupiter

Graph 2: Orbital radii of moons of Jupiter

Quadratic

Cubic

Exponential


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Answer 2.b Using [3], was replaced by the values 3, 4, 9 and 10 to predict the orbital radius of the moons in these

positions. The results are in the table bellow.

x y

3 153.29

4 251.47

9 2987.87

10 4901.59

The actual scientific data3 is show in the table below.

Name Order from Jupiter

Orbital radius / thousands of km

Amalthea 3 181

Thebe 4 212

S/2000 9 7435

Leda 10 11094

From this information, the formula seems to prove relatively accurate for moons 3 and 4, with only

about 30 thousand km in difference from the actual results, but very inaccurate for moons 9 and 10,

with a margin of error in the range of millions of km. This could be due to the very small sizes of the

farther moons (4 km radius for S/2000 and 8 km radius for Leda4), which would make the gravitational

attraction between them and Jupiter lesser and allowed them to drift further away.

This model is therefore applicable only moons of Jupiter as far as Callisto.

3 (3)

4 (3)


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Answer 2.c The model for predicting orbital radii of planets is very accurate up to Saturn. From Uranus onwards, it is

somewhat less accurate. Also, from Graph 1, it can be seen that the quadratic and cubic models are

actually more accurate for Uranus, Neptune and Pluto than the exponential model.

A reason for this inaccuracy could be due to Pluto's status as a planet. Pluto is no longer considered a

planet, but rather a dwarf planet. Not considering Pluto as part of this data actually transforms the

function into a slightly more accurate one, as seen in graphs 3 and 4.

y = 32.473e0.5341x

0

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Orb

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Position from Sun

Graph 3: Orbital radii of Planets (with Pluto)

Exponential

y = 31.211e0.5454x

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Orb

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Position from Sun

Graph 4: Orbital radii of Planets (without Pluto)

Exponential


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As for the model for moons of Jupiter, all Galilean moons accurately fit the model, as seen in graph 2.

Since there were only four data points, a highly accurate model was easily found to predict the orbital

radius of a Galilean moon.


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Answer 3

Scaled model for inner planets

Planet Order from Sun Scaled orbital radius

Mercury 1 1.0000

Venus 2 1.8687

Earth 3 2.5838

Mars 4 3.9361

The scaling factor for the inner planets was the orbital radius of Mercury, 57.9 million km, in order for

the scaled orbital radius of Mercury to be equal to 1. Graphing and applying exponential regression

yields graph 5.

The function for this relationship is

y = 0.689e0.4435x

02468

1012141618

0 1 2 3 4 5 6 7 8

Scal

ed

orb

ital

rad

ius

Order from Sun

Graph 5: Scaled orbital radius of the inner planets


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Scaled model for Galilean moons

Moon Order from Jupiter Scaled orbital radius

Io 5 1.0000

Europa 6 1.5179

Ganymede 7 2.4208

Callisto 8 4.2602

The scaling factor for the Galilean moons was the orbital radius of Io, 442 thousand km, in order for the

scaled orbital radius of Io to be equal to 1. Once again, these results were graphed and an exponential

regression was used, as shown in graph 6.


However, it is not appropriate to compare the two models, for the scaled orbital radii of inner planets

and Galilean moons, because the Galilean moons start at position 5. Taking this into account, graph 6

was made again, but this time having Io at position 1, Europa at position 2, and so on. This is show in

graph 7.

y = 0.087e0.4815x

0

2

4

6

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10

12

0 2 4 6 8 10 12

Scal

ed

orb

ital

rad

ius

Order from Jupiter

Graph 6: Scaled orbital radius of the Galilean moons


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And so we have two comparable functions, [4] and [6], for the orbital radii of the inner planets and the

Galilean moons:

Planets Galilean moons

It is quite interesting how similar these two functions are. First of all, they are both exponential

functions in the form . The values of the constants and are also very similar. The difference

between the constants of each model is less than 0.1, and the difference between the constants of

each model is less than . This shows the close similarity these two models have, and is a clear

example of how two similar physical phenomena, the orbits of planets and the orbits of the moons of

Jupiter, are interrelated by mathematical models.

y = 0.5969e0.4815x

0

2

4

6

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10

12

0 1 2 3 4 5 6 7

Scal

ed

orb

ital

rad

ius

Order from Jupiter

Graph 7: Scaled orbital radius of the Galilean moons (with Io starting at position 1)


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Bibliography

1. Arnett, Bill. Asteroids. Nine Planets. [Online] February 26, 2006. [Cited: February 13, 2008.]

http://www.nineplanets.org/asteroids.html.

2. 1 Ceres. JPL Small-Body Database Browser. [Online] August 2003. [Cited: February 13, 2008.]

http://ssd.jpl.nasa.gov/sbdb.cgi?sstr=Ceres;orb=1;cov=0;log=0#elem.

3. Hamilton, Calvin J. Jupiter. Views of the Solar System. [Online] March 6, 2008. [Cited: March 10,

2008.] http://www.solarviews.com/eng/jupiter.htm.

orbital radii of planets and moons of jupiter

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