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
2/13
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
Juliana Peña Ocampo 000033-049
3/13
Questions
(Extracted from the IB Mathematics HL Internal Assessment Teacher Support Material)
Juliana Peña Ocampo 000033-049
4/13
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
1000
2000
3000
4000
5000
6000
7000
8000
0 2 4 6 8 10 12
Orb
ital
rad
ius
/ m
illio
ns
of
km
Position from Sun
Graph 1: Orbital radii of Planets
Quadratic
Cubic
Exponential
Juliana Peña Ocampo 000033-049
5/13
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)
Juliana Peña Ocampo 000033-049
6/13
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
1000
2000
3000
4000
5000
6000
7000
0 2 4 6 8 10 12
Orb
ital
rad
ius
/ th
ou
san
ds
of
km
Order from Jupiter
Graph 2: Orbital radii of moons of Jupiter
Quadratic
Cubic
Exponential
Juliana Peña Ocampo 000033-049
7/13
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)
Juliana Peña Ocampo 000033-049
8/13
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
1000
2000
3000
4000
5000
6000
7000
8000
0 2 4 6 8 10 12
Orb
ital
rad
ius
/ m
illio
ns
of
km
Position from Sun
Graph 3: Orbital radii of Planets (with Pluto)
Exponential
y = 31.211e0.5454x
0
1000
2000
3000
4000
5000
6000
7000
8000
0 2 4 6 8 10 12
Orb
ital
rad
ius
/ m
illio
ns
of
km
Position from Sun
Graph 4: Orbital radii of Planets (without Pluto)
Exponential
Juliana Peña Ocampo 000033-049
9/13
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.
Juliana Peña Ocampo 000033-049
10/13
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
Juliana Peña Ocampo 000033-049
11/13
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.
The function for this relationship is
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
8
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
Juliana Peña Ocampo 000033-049
12/13
The function for this relationship is
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
8
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)
Juliana Peña Ocampo 000033-049
13/13
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.