ASTR 101 Kepler’s Laws · (using a telescope, of course, so that we can see the moons). In real...

ASTR 101 Kepler’s Law | pg. 1

ASTR 101 Kepler’s Laws In this exercise, you will replicate the astronomical investigations of two historically famous

astronomers: Johannes Kepler, and Galilei Galileo. Rather than using real telescopic

observations, you will

• use the Starry Night software package to make a number of hypothetical observations of

the planet Jupiter on a series of successive nights, as though carrying out an actual

observing campaign using a small telescope

• notice several bright objects associated with Jupiter and, like Galileo, realize that they are

travelling with Jupiter as it moves across the distribution of background stars – in short,

that they are moons of that planet

• with simple observations, determine the periods of the four bright moons (that is, work

out how long in days it takes each of them to orbit completely around Jupiter)

• similarly, determine their relative distances from Jupiter itself

• confirm that there is a simple relationship between the period of a moon and its average

distance from the planet (with the more remote moons taking longer to orbit Jupiter)

• test whether this is another manifestation of Kepler’s third law (which dictates how the

planets orbit around the Sun)

In the perfect world, we would develop this understanding simply by watching Jupiter all the time

(using a telescope, of course, so that we can see the moons). In real life, this is unfortunately not

possible! Because of the rotation of the Earth, Jupiter rises in the East and sets in the West every

day, so it is hidden from view (‘below the horizon’) for about half the time. Moreover, because

of the different orbits of the Earth and Jupiter around the Sun, whole months may go by when we

can’t see it at all in the night sky: at such times, Jupiter is ‘behind the Sun,’ on the far side of its

orbit as seen from the Earth.

To do the actual practical exercise, using a real telescope, we would therefore first have to pick the

right set of dates (when Jupiter is ‘up’) to make our observations, and then, like Galileo himself, do

the equivalent of going out every 24 hours – say, precisely at midnight – to see how things have

changed since the previous day. We would also have to hope for clear weather! As we will see,

with the ‘magic’ of the Starry Night software, we will be able to improve on this in some remarkable

ways! Let’s start, however, by considering some practical realities. During the first few weeks

of March 2016, Jupiter was very nicely placed, brightly visible in the midnight sky (and indeed

could still be seen in the southwestern sky during the late evening hours in May). Knowing that,

we will pretend to be Galileo, going out night after night for a number of weeks to watch the

changing positions of the moons of Jupiter. Instead of staying up late and getting cold, however,

we will use the Starry Night program to simulate what we would see through a small telescope.

We begin by turning on the Starry Night software and choosing an appropriate starting date for

our opening sequence of observations. Set the date to Feb 24th 2016, and the time to 11:00 PM.

You should see something like image shown on the next page (although you may have chosen a

different panorama when you first set up your Starry Night software, and of course you may be

in a location at a different latitude, which will change the aspect to some extent). Please note

that I have chosen to represent this image as a negative, with black stars and planets on a white

background: such images are easier to look at on the printed page. (I used the ‘white sky’ option

under the ‘View’ command button.)


In this image, you can see the recognizable constellation Orion in the Southwest (look above the

“SW” marker), with the bright star Sirius somewhat to the lower left (east) of Orion. Farther East,

we can see the nearly full Moon (it was in fact full on Feb 22nd), and to its upper right the bright

point of light that is Jupiter.

Under the ‘Label’ drop-down control button on the task bar, you can ask for labels to be added to

various objects. Here is what happens if you do so for the stars and planets:

Now go to the Search tool at the upper right to find and center on Jupiter. (If you don’t know how,

this is an opportune moment to review the various controls and commands that the Starry night

software provides you with. Take the time to do so by working through the tutorial material and

on-line instruction booklet right now.) When Jupiter is centered in the frame, find the + - buttons

at the lower left of the screen and toggle the ‘+’ key to zoom in on Jupiter until you can readily see

all four of the Galilean moons: Io, Europa, Ganymede, and Callisto. (Leaving the labels turned on

at this stage will help you decide.) Here is what you should see:


Without the labels, you see this:

Note that it is very difficult to distinguish between the stars (the dots of light that are remote

background objects) and the bright Galilean moons (which also appear as dots: they are too small

to show up as discs in the way that the large planet Jupiter itself does). That’s why Galileo first

thought that every dot of light in his field of view was a background star, until he discovered that

four of them were moving along with Jupiter as it drifted in its orbit. Of course, better modern

telescopes do indeed ‘resolve’ these moons, an effect that you can simulate by zooming in even

farther with the Starry Night software, but Galileo himself never really saw them as other than

points of light.


Even when you know that four of these objects are moons, it is difficult to distinguish one from

another – they are not dramatically different in colour or brightness. So if you look at the

arrangement tonight, and then again a couple of weeks from now, after all four of the moons have

moved around, you will be hard-pressed to know which is which! That is a problem that Galileo

had to deal with, but it’s easier for us because (as we will describe) we can monitor the motions

continuously in ways that he could not, and keep careful track. (Moreover, if you are ever

confused, you can turn on the Labels!)

To start with, however, move your cursor to the Date & Time representation on the top left of

your screen. The time showing now should be about 11:00 PM on Feb 24 th (that’s how we first

set it up); try advancing it one hour at a time to see what happens, stopping at (say) 7:00 AM on

Feb 25th. Here is what you will see at the end:

There are two things to notice. One is that Io has moved perceptibly closer to Jupiter, even over

the span of just a few hours! (It is the innermost moon, and moves fastest in its orbit.) The

second change is a bit puzzling, however: the whole field of view seems to have rotated! (Look

back at the picture on the previous page, showing that Ganymede is to the upper right of Jupiter at

11:00 PM; by 7:00 AM, it is to the lower right.) What is going on?

The reason for this is that we are on the rotating Earth, watching Jupiter and its system of moons

rise in the East (at 11:00 PM) with Ganymede leading the way; eight hours later, they are setting

in the West, with Ganymede still in the lead, diving towards the horizon. This behaviour is a

distracting nuisance for our little experiment, but we can eliminate it by choosing a slightly

different way of representing the images.

Go back to Feb 24th at 11:00 PM, with Jupiter and its moons centered in the field of view. First,

turn the Labels off, so that you see only the objects in the sky. Now find the Options tool in the

task bar, open the drop-down list, look under “Orientation” and select “Ecliptic.” As you will

notice, this makes the system of Jupiter and its moons appear horizontal, and they will retain that

orientation as time passes. (Advance the time by an hour or two to confirm the correctness of that

statement.)


We have two more problems to contend with. The first is that as the Earth rotates, Jupiter and its

moons will periodically become invisible to us, setting in the West and not reappearing until half

a day later again in the East. Of course Galileo had no control over this (or the vagaries of

weather!) but we can do something magical: let’s make the whole Earth transparent, as though we

were living on a ball of ultra-clear glass through which we can see objects even when they are

below the horizon. To do that, open the “View” tab in the toolbar, and select “Hide Horizon,”

very near the bottom of the drop-down list. (You might then want to zoom back out to confirm

that your default landscape is indeed completely gone. Toggle it on and off to see the effect.)

The second problem is that we will also lose sight of Jupiter and its moons during daytime hours.

If the Sun rises while Jupiter is still fairly high above the horizon, we are out of luck: the Earth’s

atmosphere scatters so much sunlight that the brightly-lit sky makes it impossible to see any stars

and planets. Even picking out the Earth’s moon, which can be seen in the daytime when it is in

certain phases, can be difficult. (By the way, here’s a photograph to show that this is possible.

Not everyone is aware of that!)

If you use the Starry Night option of looking at black stars and planets on a white sky (as in the

figures I presented above), the problem is already resolved: as you step forward through the hours

and days, the sky always looks the same. But if you prefer a more realistic representation of

bright stars on a dark sky, the stars will vanish in the daytime. You can resolve this issue by

turning off the sun, in a sense – or rather, by simulating what would happen if there was no

atmosphere surrounding the Earth, since the real problem is caused by the scattered light from the

very molecules of air that we breathe. To do so, look under the “View” tab on the Starry Night

toolbar and select the option “Hide Daylight” near the bottom of the drop-down list. (If you are

working in “white sky” mode, the “hide daylight” option will be grayed out.) Let’s review

quickly:

• we have made the Earth transparent, so that we never lose sight of Jupiter and its moons;

• we have eliminated the effects of scattered sunlight (either by working with a white sky

or by explicitly hiding the daylight);

• we have chosen to view Jupiter and its moons in the ‘ecliptic’ projection so they line up

horizontally; and


• we have turned off all the labels so that we can watch the motions ‘as Galileo did’

We are now ready to make some real observations and measurements. Before we do so, however,

let’s develop a sense of what we will see. Go back to our starting point and time (11:00 PM, Feb

24th 2016, with Jupiter and its moons in the centre of the frame). Once there, put the cursor on

the hour button, then press and hold down the ‘up’ arrow on your keyboard to allow the time to

advance, hour by hour, until some number of weeks have passed on the simulation. Watch the

steady movement on the screen. Here is what you will notice:

• Jupiter appears to drift across the background field of stars (or rather, since we are

centred on Jupiter, we see a steady progression of the background stars parading on past)

• meanwhile, the four bright dots that are lined up with Jupiter are clearly associated and

moving with it, looping back and forth from one side to the other as they do so;

• moreover, the innermost moon (Io!) moves quickly, and changes from one side of Jupiter

to the other every day or two; the outer moons move more slowly.

It is this behaviour that we are now going to study and quantify.

The Actual (Simulated) Observations

We want to work out two things:

• what is the orbital period, in days and hours, for each of the four moons? (Io, Europa,

Ganymede, Callisto)

• how far are the moons from Jupiter? – not in units like miles or kilometers, but simply

relative to one another (for example, we might discover that Ganymede is four times, say,

as far from Jupiter as Io is)

There are complications in determining each of these, so we will now consider them in turn.

The Orbital Periods

Let us start with Callisto; ignore the others for now. (Turn the labels on briefly to identify it.)

Using the hour entry on the time control, step forward or back until you can identify a moment at

which the moon seems to be as far away from Jupiter on the left side of the screen as it ever gets.

Make a careful note of the date (in the format month-day-hour: for example, 02-26-21:00: Feb

26th 9:00 PM).

From this new starting point, advance the time by one hour at a time until Callisto has returned to

its original extreme position on the same side of Jupiter. At this stage, record the new date and

time, and work out exactly how many days and hours have passed since you started the exercise:

this is of course the measured orbital period of Callisto in its orbit around Jupiter. Express your

result in days, including a decimal part (for example, 10 days 6 hours = 10.25 days).


Having done that for Callisto, now do the same for the other three. By the way, when you are

studying the inner moons (especially Io and Europa), you should feel free to expand the scale –

that is, zoom in some more – so that their behaviour is more easily followed.

Here is a small table (Table 1) for you to fill in:

Moon Starting time (target

moon at extreme left

of its orbit)

Finishing time (target

moon again at extreme

left)

Days passed (= orbital period

of the target moon)

Io

Europa

Ganymede

Callisto

Distances from Jupiter

One complication here is that the moons may not be moving in perfectly circular orbits, so a moon

may go farther from Jupiter out to the left than it does to the right. But we will assume for

simplicity that the orbits are perfect circles, which is pretty close to reality. This means that when

you see one of the moons off to one side as far from the planet as it ever gets, that is a reasonable

representation of its true distance.

The second thing to know is that we need to know how far the individual moons are from the

centre of Jupiter, not the edge of the planet. (This is something we discuss in the course in

connection with Newton’s law of gravity.) So you must measure their distances as shown here:

Finally, please remember that we will be interested in comparing the distances of the various moons from the planet, so it is critically important that you do not change the scale (the zooming) of the simulation when you make the following four separate measurements. It has to be the same throughout this part of the exercise.

Here is what to do:

• look back at Table 1 (above) for the entries opposite Callisto, to identify a date on which

you estimated it was it at its extreme distance from the planet. Adjust the date of

observation to move the simulated observations back or forward to that day, and then

adjust the zoom until Callisto is clearly visible at the edge of the screen.


• using a tape measure or a ruler, measure how far the image of Callisto lies from the centre of Jupiter on the screen at that time

• repeat this exercise for each of the other three moons in turn without changing the scale

(don’t zoom in or out!). In other words, look back at Table 1 in each case, and adjust the

time of observation to the moment at which the moon of interest was at its extreme

distance from Jupiter itself; then measure how far that moon was from the centre of

Jupiter at that moment.

Here is another small table (Table 2) to fill in:

Moon Time of extreme position

(from Table 1, above)

Distance of moon from

centre of Jupiter (measured on the image)

Io

Europa

Ganymede

Callisto

We are now going to simplify matters a little. In the Solar System, we adopt a system of units in

which the Earth is said to be one astronomical unit (1 AU) from the Sun. (This is an average

value, in fact, because the Earth’s orbit is not a perfect circle. We are sometimes a little closer,

sometimes a little farther away.) One AU is about 150,000,000 km, which is a big and awkward

number, so we prefer the use of Astronomical Units to make calculations easier. For instance,

Jupiter itself is about 5.2 AU from the Sun – that is, a bit more than 5 times the Earth’s average

distance, something that is easy to picture.

We will do something similar here. We will introduce (for this exercise only) a new unit of

distance called the Ionian Unit (= I.U.). The moon Io is then, by definition, 1.0 Ionian Unit from

Jupiter. The others are farther, but to work out their distances in these new units we simply have

to divide the numbers in the table above. (For instance, if we measured Europa to be 4.5 cm from

the center of Jupiter, but found that Io is 1.5 cm from the planet, then Europa must be 4.5 / 1.5 =

3.0 I.U. from Jupiter.)

Here’s another table (Table 3) that allows us to summarize that, and to include the orbital periods:

Moon Distance from Jupiter

in Ionian Units (I.U.)

Orbital period of Moon

in days (from Table 1)

Io 1.0

Europa

Ganymede

Callisto


As you can see (provided you have not made some silly mistake) the more remote moons take

longer to go around Jupiter than those that are in closer. We have noticed, in short, that there is a

correlation. But it’s interesting to ask whether there is some straightforward mathematical law

that relates these -- or is it just a general relationship of no particular reliability?

(Imagine a variety of people strolling through and around a big park on different paths. Some

people would move quickly, some more slowly. Their various rates of progress might in turn

depend in a very subtle way on the layout of the park: the most interesting flower beds, where

people tend to walk slowly and linger, might be nearest the centre, for example, or distributed in

some completely random way. So any observed relationship might be complex.)

To address that question, we will do as any scientist would: use a piece of graph paper (or the

plotting tools in an Excel spreadsheet, if you know how to do that) to make a plot of the data

above. If the data points appear to lie along some smooth relationship, rather than being

higgledypiggledy, we can conclude that there is probably some underlying basic physical law!

Here (on the next page) is some squared paper to use. By the way, I know that some of you have limited science or math background, so if you are not familiar with making plots, or lack confidence in doing so, please look back at the course home page. Near where you found the link to this assignment, you will see another link, one that leads you to a simple description of why making such graphs is so informative -- and how to do so.

Incidentally, if you make a plot from numbers that you have put into an Excel spreadsheet, it is particularly easy to get things quite wrong! The simple guide we have provided on "How to Make and Interpret Graphs" will make this all very clear.


You should find that there is indeed a very clear indication of a simple relationship here – the

moons with larger orbits have longer periods in a way that is clearly smoothly predictable, and if

a fifth moon were to be added somewhere in between you can well imagine that we would be able

to work out what its orbital period would be. Of course, noting the existence of a relationship

does not immediately lead to an understanding of the cause. (Remember that Newton’s

introduction of gravity, and his consequent explanation of Kepler’s laws, came fully seventy

years after Kepler’s work!) But this is a good example of how science works.

We can do one last thing, which is actually to test whether the relationship we have found is

consistent with the same basic physics that describes the orbits of the planets around the Sun.

Kepler’s third law (which you do not have to memorize!) states that the “square of the orbital

period of a planet around the sun is proportional to the cube of its distance from the sun.” Let

us now see if the same relationship applies within the system of Jovian moons.


To do so, we need a final data table (Table 4) – one in which we simply add two more columns of

data to Table 3. For each moon, figure out “distance cubed” by multiplying the moon’s distance

by itself twice (that is, distance x distance x distance) and entering that value into the fourth

column. Then figure out “period squared” by multiplying the orbital period by itself once

(period x period) and entering that value into the fifth column.

Moon Distance from

Jupiter in I.U.

(from Table 3)

Orbital period

(from Table 3)

Distance cubed Period squared

Io 1.0 1.0

Europa

Ganymede

Callisto

Having done that, make one last plot, using the data in the last two (rightmost) columns of the

table you have just created. (There is some squared paper on the next page, or again you can use

an Excel spreadsheet.) If you have done this all correctly, you should see what appears to be

pretty nearly a straight-line relationship. This allows you to safely conclude that the same

physical law that governs the orbiting planets in the Solar System is at work in the satellite system

of Jupiter. This is in fact a manifestation of Newton’s Universal Law of Gravity.


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ASTR 101 Kepler’s Laws · (using a telescope, of course, so that we can see the moons). In real...

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