ASTRO 801
Planets, Stars, Galaxies, and the Universe

Lab 1, Part 1

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Background

One of the observations that Galileo is famous for making is the discovery of four Moons of Jupiter, which these days we refer to as the "Galilean Moons."

This was one of the observations that contributed to the revolution in our understanding of the true nature of the Solar System. What Galileo personally observed was what he thought were stars near Jupiter, and night after night, he witnessed their positions change with respect to Jupiter.

Test this with Starry Night!

I have created a Starry Night save file (.snf) to let you jump to see this directly (a copy is posted in Canvas). If you would like to set it up yourself, you can do the following:

  • set the date and time to June 28, 2009 at 2:00am;
  • turn on the Planet / Moon labels so you can see which objects are visible;
  • either by mousing around or using the Find feature, locate Jupiter and right-click on it to center it;
  • zoom in so the field of view is approximately 40 arcminutes wide;
  • set the time flow rate to 1 day;
  • click on the forward one step button to watch the positions of the objects change as each day goes by.

You should witness exactly what Galileo did—as you click on the forward button, each night the arrangement of the four Galilean Moons (Io, Europa, Ganymede, and Callisto) changes with respect to Jupiter.

Now, let's do something that Galileo could not. Let's look at Jupiter from above its North Pole so that we can see physically what is going on.

Test this with Starry Night!

I have created a Starry Night save file (.snf) to let you jump to see this directly (a copy is posted in Canvas), but if you would like to set it up yourself you can do the following:

  • set the viewing location to the surface of Jupiter, and set the latitude and longitude to 90 degrees North, 0 degrees West;
  • turn on the Planet / Moon labels so you can see which objects are visible;
  • use the up / down arrows to move about 0.01 AU or so above Jupiter's North Pole;
  • set the time flow rate to 1 hour;
  • click on the forward one step button to watch the positions of the objects change as each hour goes by.

What you can see in this latter view is the orbit of the moons, but what you see in the former view is what appears to be a side to side change in position of the moons. In this lab, we are going to measure that side to side motion and use that data to calculate the mass of Jupiter using Newton's version of Kepler's Third Law.

Let's talk about how the side view and top view compare. Below is an image that shows the top view (that is, as seen from Jupiter's North Pole) of the orbit of a moon:

Graphic illustration to show the overhead view (above Jupiter's N. pole) of the orbit of a Galilean moon.
Image of a hypothetical moon orbiting a hypothetical planet with the radius labeled and the apparent separation as seen by a distant observer labeled
Credit: Penn State Astronomy & Astrophysics

If you study the image, you will note that when the moon is in front of Jupiter or behind it, we can describe its projected side to side distance from the planet as zero (in any units). When the moon is at a right angle from the Earth/Jupiter line, it will be seen at its maximum separation from the planet. If you consider how it will appear from Earth as it orbits and moves between these two extremes, it will appear to trace out a sine curve from maximum separation, to zero, to maximum separation, to zero, and back again. Sine curves have a few basic properties:

  • They are repetitive. Their pattern repeats over time.
  • The maximum and minimum values of a sine curve are the same every time (i.e., the peak value is the same for every peak, which is also equal to the trough value). For the moons, this value is equal to its distance from Jupiter.
  • The distance between successive peaks is equal to the period of the sine wave. This is the length (of time, in this case) over which the pattern repeats. For the moons, this is equal to the amount of time it takes for them to orbit once around Jupiter.

Below is a sample plot for a fictitious moon of Jupiter. The x-axis is labeled Julian Date, which is an easier way of sequentially marking days than relying on our calendar, which is difficult to work with on a plot because the number of days per month varies. Each tick mark on the x-axis is one day. The y-axis shows projected separation from Jupiter in units of Jupiter diameters.

Plot of a sine curve to show how to find the orbital period needed to perform the calculation for Kepler's 3rd law.
Two-dimensional plot of the apparent separation between a fictitious moon and its host planet as a function of time
Credit: Penn State Astronomy & Astrophysics

If you again refer to the first image on this page, you will notice that when the moon goes from rightmost maximum separation to zero to leftmost maximum separation to zero to rightmost maximum separation again, that is the period of one orbit around Jupiter. On the curve above, the time from maximum to maximum peak is the same as the time from rightmost maximum separation to rightmost maximum separation. So, you can, therefore, estimate the period of a moon's orbit directly from one of these curves.

The amplitude of the curve illustrated above is the distance in Jupiter diameters when the moon is at either its left or rightmost maximum separation from the planet. That is a direct measurement of the semi-major axis of its orbit. Therefore, by plotting one of these curves, you can measure both P and a for a moon, which are the two quantities you need for a Kepler's third law calculation.

Recall that: P 2  = (4π   2  x a 3 ) / G(m 1  + m 2 ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

So, therefore, if you have P and a measured, you get the sum of ( m 1  +  m 2 ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. , which is in this case the sum of ( m Jupiter  +  m moon ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. .

On the next page are instructions for using a simulated observatory to take data on Jupiter's moons for the purpose of measuring the mass of Jupiter.