New Mass of Jupiter Lab

New Mass of Jupiter Lab

The Relationship between Revolution and Period

Astronomers cannot directly measure many of the things they study, such as the masses and distances of the planets and their moons. Nevertheless, we can deduce some properties of celestial bodies from their motions despite the fact that we cannot directly measure them. In 1543 Nicolaus Copernicus hypothesized that the planets revolve in circular orbits around the sun. Tycho Brahe (1546-1601) carefully observed the locations of the planets and 777 stars over a period of 20 years using a sextant and compass. These observations were used by Johannes Kepler, a student of Brahe’s, to deduce three empirical mathematical laws governing the orbit of one object around another. Kepler’s Third Law is the one that applies to this lab. For a moon orbiting a much more massive parent body, it states the following…

page1image1766208

where

M is the mass of the parent body in units of the mass of the sun.
a is the length of the semi-major axis in units of the mean Earth-Sun distance, 1 A.U. (astronomical unit). If the orbit is circular (as we assume in this lab), the semi-major axis is equal to the radius of the orbit.

T is the period of the orbit in Earth years. The period is the amount of time required for the moon to orbit the parent body once.

In 1609, the telescope was invented, allowing the observation of objects not visible to the naked eye. Galileo used a telescope to discover that Jupiter had four moons orbiting it and made exhaustive studies of this system, which was especially remarkable because the Jupiter system is a miniature version of the solar system. Studying such a system could open a to understand the motions of the solar system as a whole. Indeed, the Jupiter system provided clear evidence that Copernicus’ heliocentric model of the solar system was physically possible. Unfortunately for Galileo, the inquisition took issue with his findings; he was tried and forced to recant.

Here are two different versions of the simulation address.

https://phet.colorado.edu/en/simulation/gravity-and-orbits
https://phet.colorado.edu/sims/html/gravity-and-orbits/latest/gravity-and-orbits_en.html

Open the simulation by clicking the “play” button in the picture of the simulation. On the bottom, select the “To Scale” page. We will use this simulation to verify Kepler’s Law by collecting data on a planet orbiting a star of various masses.

Procedure:

**If you reset the simulation at any point to change values you will need to reset all of the initial parameters. Please always make sure you have the correct star size and distance.

Diagram

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  1. On the right hand side, click the boxes for ‘Measuring Tape’, ‘Grid’, and ‘Path’.
  2. Set the Star Mass to 0.5
  3. Put your pointer on the word planet and it will allow you to move the planet to only 1 grid box away from the Star (it starts at 2 boxes away) – go towards the star to the next intersection on the same horizontal line.
  4. Press the play button and watch the planet begin to orbit the star. As it approaches 1 complete cycle, pause it, and use the forward button to step the planet to what looks like one complete orbit.
  5. Measure the Major Axis by using the Measuring Tape. Place your pointer on the measuring tape, and move it, adjusting the rightmost red cross hairs to measure the major axis of the orbit. Note: it gives a value in THOUSANDS of miles. So whatever value it says, add three zeros at the end. Ex: 52,100 thousand miles=52,100,000 miles. Record this value under the Major Axis column.
  6. Find the Period where it gives a value of ‘Earth Days’ above the Clear button. Record this under the Period column.
  7. Calculate the Amplitude (the semi-major axis) in miles by dividing your Major Axis by 2. Record.
  8. Calculate the Amplitude in AU by multiplying your amplitude in miles by 1.0758 x 10-8. Record.
  9. Calculate the Period in years by dividing your period in Earth Days by 365. Record.
  10. Calculate a3 by multiplying your Amplitude in AU by itself three times. Record.
  11. Calculate T2 by multiplying you Period in years by itself twice. Record.
  12. Calculate the Mass of the Planet by dividing the Amplitude cubed by the Period squared. Record.
  13. Calculate the Percent Difference by doing the following calculation:
page2image1755456

14. For example: If you calculated 0.6 for M on the first trial, where the given Star Mass is 0.5, you would do the following…

Percent of Error:

  1. Repeat the procedure, adjusting the star mass and starting grid distance as given on the table.
Table

Description automatically generated

Only complete “1 (our Sun). Note to use two grid boxes.

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