Tim Derr
Astronomy
Newton’s Generalization of Kepler’s Third Law
Introduction
In the early 1600’s, an astronomer by the name of Johannes Kepler sought to find an
answer as to why the planets move around the Sun and what was the force sweeping them about
it. He developed three laws he used to describe the concept of planetary motion; including his
third law suggesting there existed a force that moved planets around the Sun and that this force
weakens as it reaches out to farther planets. This force that Kepler was referring to would later
be defined as gravity by Sir Isaac Newton. Towards the end of the 17th century, Newton began
to apply his theory of gravity to planetary motion and Kepler’s laws. In this paper, a study will
be conducted in an attempt to validate Newton’s generalization of Kepler’s third law.
Kepler’s first law stated that each planet’s orbit around the Sun took the shape of an
ellipse, with the Sun at one focus (or foci) of the ellipse, but with the other vacant. An ellipse is
a type of oval in which every point is the same total distance from two fixed points called the
foci. Newton was able to prove this law using calculus and showing that a planet’s orbit will be
elliptic if the centripetal force (the force directed toward the center of a curve along which an
object is moving) varies as
from the Sun (located at a foci).
Kepler’s second law states that a planet traveling along an elliptical path moves faster
when it is closer to the Sun and slower when its farther away. It also suggests that a line from
the planet to the Sun sweeps out equal areas in equal intervals of time due to the change in its
speed. Newton was able to prove this law using the assumption that as a planet approaches the
Sun, gravity is pulling the planet forward and sideways at the same time, causing the planet to
increase its speed. As the planet is moving away from the Sun, the gravitational force is pulling
both backwards and sideways resulting in a decrease in speed as well as curving it around back
toward the Sun.
Kepler’s third law was used to compare the speed of a planet to another planet in a
different orbit by generating an equation that made the assertion that “the ratio of the cube of the
semimajor axis of a planet’s orbit to the square of its orbital period around the Sun is the same
for each planet” (Koupelis, 2011, p.59). Newton modified this law using his theory regarding
the laws of motion and universal gravity to formulate an equation that can be used for a binary
system. A binary system is a system in which two objects are orbiting each other due to their
mutual gravitational attraction and Newton’s generalization of this law was developed to see if
one could estimate the mass of an object orbiting something other than the Sun. This
generalization of Kepler’s third and final law is what this study will attempt to validate by
estimating the mass of Jupiter by observing the Galilean moons of Jupiter.
Methods
Once again, the four Galilean moons of Jupiter will serve as the focus of this study and
will be closely observed to test whether Newton’s generalization of Kepler’s third law is indeed
valid. By observing the orbits of these four moons (Fig. 1), one can calculate and analyze the
furthest distance the moon travels from Jupiter and the time it takes for each moon to travel
around Jupiter. This information is essential for analyzing Kepler’s third law, which; therefore,
is necessary to interpret Newton’s generalized version. Kepler’s third law regarding planets
motion around the Sun is symbolized as
= C.
In this equation, “a” is equivalent to the semimajor axis of the planet’s elliptical orbit (in
other words the radius of its orbit, which is approximately its average distance from the Sun.
Finding the “a” or radius of an object’s orbit, we must assume that its orbital path is nearly
circular, which Kepler found is characterized by most planets in our solar system. “P” represents
the planet’s orbital period, which is the time it takes a planet to complete a full orbit around the
Sun, also known as the sidereal period. “C” is a constant and as Kepler had calculated, this value
should equal about 1 for every planet if “a” is computed in AU (astronomical units, or the
average distance from the Earth to the Sun) and “P” is computed in Earth years.
Fig. 1: Contemporary Laboratory Experiences in Astronomy’s (CLEA) program illustrating the revolution of
Jupiter’s moons. This is the program that will be used for the observations of this study.
But the formula used in Kepler’s third law only applies to planet’s orbiting the Sun and is
just the first part in Newton’s modification of the equation, which is symbolized as
=(
)
(m1 + m2). The letters “a” and “P” represent the same figures defined previously, but in
Newton’s equation they represent objects traveling about another object other than the Sun. “G”
represents the gravitational constant and “m1” and “m2” are the masses of the two objects being
analyzed. This formula can be simplified if “a” is measured using AU and “P” is measured using
years to:
=
. In this case, “m1” will be the mass of one of Jupiter’s moons and “m2”
as well as “MJupiter” is the mass of the Jupiter. With this equation, since the masses of the
moons are extremely less than the mass of Jupiter, we can ignore “m1” and therefore conclude
that
=
. So in regards to Newton’s application of gravitational force to Kepler’s
equation, we can predict that for all objects orbiting Jupiter,
= Jupiter’s mass.
In summary, this study will use the broadest form of Newton’s modification of Kepler’s
equation depicted at the top of this page (where “m1” will represent the mass of one of Jupiter’s
moons and “m2” will represent the mass of Jupiter) in order to test whether each of the Galilean
moons produces a value approximately equivalent to the mass of Jupiter relative to the Sun;
therein validating Newton’s generalization of Kepler’s third law.
Results
To begin solving for the variable “a”, or the semimajor axis, we need to determine what
this value is in pixels. Since this study is using a computer program to observe the motion of
Jupiter’s moons, the x and y coordinates are in units of pixels rather than units of kilometers
(km) or astronomical units (AU). The CLEA program has four different magnifications that you
can observe with (shown at the bottom of Fig. 1) and this study observed all moons at the
“100X” magnification setting. To measure the semimajor axis, you must record the x-coordinate
for the center of Jupiter and the farthest x-coordinate its moon reaches from the center of the
planet. The center of Jupiter was determined to have an x equal to 315 and Fig. 2 shows a table
listing the x coordinates and semimajor axes in pixels for the moons of Jupiter.
Moon
1st Furthest
x-coordinate
2nd Furthest xcoordinate
373x
1st
Semimajor
Axis
58pix
255x
2nd
Semimajor
Axis
60pix
Average
Semimajor
Axis
59pix
Io
Europa
406x
91pix
217x
98pix
94.5pix
Ganymede
463x
148pix
163x
152pix
150pix
Callisto
575x
260pix
48x
267pix
263.5pix
Fig. 2: The “1st” calculations are from the orbit on the right side of Jupiter and the “2 nd” calculations or from the
orbit on the left side of Jupiter (when looking at the software).
From the values illustrated in Fig. 2, the most important information is the average
semimajor axis because this will be used to represent “a” in Newton’s equation. The next step is
to change the units from pixels (pix) to kilometers (km) and eventually astronomical units (AU).
To do this, we first need to determine how many kilometers 1 pixel represents. One can
determine this by taking Jupiter’s diameter in kilometers and dividing it by the planet’s diameter
in pixels. Using the computer software, it has been determined that Jupiter stretches from 303x
to 324x, or 21 pixels. From work done by previous astronomers we know that Jupiter is
142,984km in diameter and then can solve that:
⁄
. Now we can
take the average semimajor axis in pixels for each of Jupiter’s moons and multiply it by
6,808.76km to get “a” in kilometers. From here, we can then convert the distance in kilometers
to astronomical units with this equation:
(
)
. Below, Fig. 3 displays these
numbers for the four Galilean moons of Jupiter.
Moon
Semimajor Axis (pix)
Semimajor Axis (km)
Semimajor Axis (AU)
Io
59pix
401,716.84km
.0027 AU
Europa
94.5pix
643,427.82km
.0043 AU
Ganymede
150pix
1,021,314km
.0068 AU
Callisto
263.5pix
1,794,108.26km
.0120 AU
Fig. 3: Shows the semimajor axis (a) for all units. In this study, the semimajor axis in AU will be used to determine
Jupiter’s mass.
For this study’s purposes, we will use the semimajor axis in astronomical units (AU) to
serve as “a” when we determine Jupiter’s mass. Next, we must find “P”, or the orbital period of
Jupiter’s moons. To find this in the computer program, you must mark where a moon is along its
orbital plane and then you observe how many hours it takes for the moon to complete a full
revolution back to that same spot. Fig. 4 displays the findings for “P” for the four moons of
Jupiter in units of hours, days, and years. To convert from hours to days, one uses the equation:
(
)
. To convert from days to years, one uses the equation:
(
)
Moon
Orbital Period (hours)
Orbital Period (days)
Orbital Period (years)
Io
42 hrs
1.75 days
.0048 yrs
Europa
85 hrs
3.54 days
.0097 yrs
Ganymede
172 hrs
7.17 days
.0196 yrs
Callisto
408 hrs
17 days
.0466 yrs
.
Fig. 4: Shows orbital period (P) in all units. For this study, the orbital period in years will be used to calculate
Jupiter’s mass.
For this study’s purpose, we will be using the orbital period (P) in years to calculate
Jupiter’s mass. Now that we have the values for “a” and “P” for each of Jupiter’s moons, we can
calculate Jupiter’s solar mass by using the equation
. For example, to calculate Jupiter’s solar
mass using Io’s values for “a” and “P”, the equation would be:
solar units.
Each moon’s predictions for Jupiter’s mass are illustrated in Fig. 5 along with a comparison to
Jupiter’s actual mass in solar units.
Calculated Mass of Jupiter (solar units)
0.001
0.000954
0.00095
0.0009
0.000854
0.00085
0.000845
0.000818
Solar Mass of Jupiter
0.000796
0.0008
0.00075
0.0007
Io
Europa
Ganymede
Callisto
Jupiter's Real
Mass
Fig. 5: Calculations using Newton’s generalization of Kepler’s 3rd law for each of Jupiter’s moons to predict the
mass of Jupiter. The last column is Jupiter’s actual solar mass used to compare the accuracy of the predictions.
By calculating the solar mass of Jupiter four times with each of its moons, it helps
validate the research by making more accurate assumptions as to what the true mass of Jupiter is.
Taking into account the four masses that were calculated, the average mass of Jupiter results in
.000828 solar mass. This is a .000126 difference in solar mass when compared to Jupiter’s real
solar mass of .000954. The likely reason as to why the estimated calculations are less than the
actual mass of Jupiter is probably due to the CLEA software used to observe the moons. It is
unlikely one would have any values that are greater than the actual mass because the moons
never go past their maximum semimajor axis in the software. With that said, when I was
measuring the “a” and the “P” in pixels, I most likely didn’t reach the maximum or actual
distances/sidereal periods for each moon. And when ‘a’ is cubed, it magnifies this discrepancy
and has a slight effect on the data.
When determining whether this .000126 difference in solar mass is significant in terms of
validating Newton’s generalization, we must analyze the data in regards to how many standard
deviations it is from the mean. The purpose of finding the standard deviation is to determine
how “spread out” all of the calculations are, or, in other words, if the data is close enough to the
mean (the average of all the numbers) to make the assumption that your estimations are valid. If
the data falls within 1 standard deviation from the mean, then we can state that our estimated
calculations for Jupiter’s mass are acceptable with a high confidence level. If it falls within 2 or
3 standard deviations, then the data is still acceptable, but with a lower confidence level since
this suggests that our data is a bit more spread out and less conclusive. If the data I calculated
does not fall within 3 standard deviations from the mean, then my use of Newton’s
generalization of Kepler’s third law is not valid or unacceptable in terms of calculating Jupiter’s
mass.
To determine this number, you measure how far each value is from the mean and square
it. Then those values are added together and divided by the population size (in this case 5 – my 4
estimations and the actual mass of Jupiter). Lastly, you take the square root of this number and
see if it falls within 1, 2, or 3 standard deviations from the mean. With my data, I calculated a
value of .0000543 which is within 1 standard deviation from the mean of .0008543 solar units.
This means that my estimations will be accepted by the science community as accurate, thus
validating Newton’s generalization of Kepler’s third law.
Conclusion
In conclusion, I was able to accurately estimate the orbital period and semimajor axis for
each of the Galilean moons of Jupiter. These measurements were initially calculated in pixels
for the semimajor axis and hours for the sidereal period, but were then converted into
astronomical units and years, respectively. These figures were then used to calculate the mass of
Jupiter using Newton’s generalization of Kepler’s third law which can be simplified to the
equation,
= Jupiter’s mass in solar units. This process was repeated for each of the four
Galilean moons and each calculated value was compared to the actual mass of Jupiter. In order
to determine whether these estimations were seen as rational by the science community, the data
was analyzed to see whether it fell within 3 standard deviations from the mean mass of Jupiter in
solar units. The results yielded a value within 1 standard deviation which meant I was able to
conclude that Newton’s generalization of Kepler’s third law is indeed valid.
References
Koupelis, T. (2011). In Quest of the Universe. Sudbury, MA: Jones and Bartlett Publishers.