Lab 8 – Planetary Orbits.

Kepler’s Laws describe the motion of one object in orbit around a much more massive object.  This is the case for the planets that orbit the Sun whose mass is 1000 times greater than the most massive planet Jupiter.  It is also true for the moons that go around the planets in our Solar System. It is not true for Pluto whose moon Charon is only 1/6 its mass. But it is true for all satellites that have been launched from Earth.  Kepler’s Laws can be derived from Newton’s laws of motion, which apply to all non-relativistic motion.  The are the following:

1. The orbits of the planets are ellipses with the Sun at one focus.
2. The speed of the planet as it goes about its orbit increases or decreases so that the area between it and the Sun over a given time stays the same.
3. The amount of time it takes for a planet to complete its orbit, called its period, squared is equal to the distance of the planet from the Sun, actually the semi-major axis of the ellipse, cubed.

Laboratory Tools

In this lab we will make use of the , developed by the University of Nebraska-Lincoln.  This simulator demonstrates Kepler’s three laws of planetary motion. The simulator has a box for Orbit Settings which allows you to select a planet or choose a distance (semi-major axis) and eccentricity (how squished is the ellipse) of your choice. The Animation Controls box starts and stops the animation and controls the speed, while the Visualization Options box will allow you to add the orbits of the planets for comparison.   There are four tabs under the diagram that can be selected. Three correspond to the different laws of Kepler and the fourth is Newtonian Features which we will not use.

Assignment

First use the Kepler’s 1st Law tab. Check all of the boxes so that all features are shown.  Now vary the eccentricity of the orbit with the slider in the Orbit Settings box. Describe how the semi-major and semi-minor axes change with eccentricity and how the focus and center change too,

Next use the Kepler’s 2nd Law tab.  Click on the sweep continuously box and then start sweeping. Change the parameters to Venus and then Earth in the Orbit Settings box. What do you notice about the areas swept out by the planets motion? Then try Mercury and Pluto, how do the colored regions change? Play with the eccentricity slider in the Orbit Settings box. How do the swept out regions change with eccentricity?

Now use the Kepler’s 3rd Law tab.  The plot shows period versus semi-major axis.  According to Kepler’s 3rd Law all orbits will fall on a single curve in this plot. If you set the orbit to Jupiter’s in the Orbit Setting box you will see that the period is 11.9 years. What semi-major axis would give a period of exactly 10 years?  What semi-major axis gives a period of exactly 100 years?

Questions

1. If you discovered a new comet, what would you need to know to determine its period from Kepler’s Laws?
2. What does Kepler’s second law tells us about the speed of a comet when it is far from the Sun?
3. What property of an orbit determines the distance between the Sun and the center of the orbit?

Lab 9 – The Copernican Model

We now all grow up knowing that the Sun is the center of the Solar System, so it can be hard to understand what a momentous effect it had on scientific thought in the 17 century.  The most interesting thing about changing from an Earth centered to a Sun centered model was that the data didn’t really change.  The Ptolemaic system did a perfectly good job of predicting the positions of planets. Copernicus Sun-centered model doesn’t predict the positions of the planets any better, it just seems simpler. In fact, the thing that we actually see, the motion of the Sun and planets compared to Zodiac constellations looks very similar in both models.

In this lab we will explore the Ptolemaic Sun-centered model and the Earth-centered model of Copernicus. In particular we will look at how the same motion as seen from the Earth can be caused by very different motions.

Laboratory Tools

In this lab we will be making use of two simulators developed by the astronomy department at the University of Nebraska-Lincoln. The simulators are:

• The, which shows the motion of a planet in the Ptolemaic Earth centered system. The simulator shows an Orbit View from above the plane of motion and Zodiac Strip that shows how the planet and Sun move relative to the Zodiac constellations.  The Planetary Parameters box allows you to change the planet or change the parameters to a planet that doesn’t exist. The Control and Settings box lets you start the animation and control what is shown.
• The, which shows the motion of a planet in the Copernican Sun centered system. This simulator shows a top down view of two planets orbiting the Sun in the Diagram box and again how the planet and Sun move relative to the Zodiac constellations in the Zodiac Strip.  You can choose which planets are used in the Orbit Sizes box and the animation in the Animations Control   The Timeline box gives the names of certain astronomical events, you can ignore this box.

Assignment

First open the Ptolemaic System Simulator.  Start the animation and look at the Zodiac Strip.  Describe how the planet moves relative to the Zodiac constellations. Try all four preset planets and describe what is similar in their motion and any differences.  Now look at the Orbit View.  What in the Orbit View causes the motions you have noted in the Zodiac Strip view?  You can adjust the planets parameters using the sliders. Which parameter is most important in changing the motion of the planet?

Now let’s open the Planetary Configuration Simulator. Set the observer’s planet as Earth and the target planet as which every planet you were using last with the previous simulator.  Looking at the Zodiac Strip is the motion the same or different?  Now look at the Diagram box.  Using this view explain the what causes the motions you see in the Zodiac Strip. Is it easier to explain these motions from this view or using the Ptolemaic model?  The Planetary Configuration Simulator also allows one to change the planet you are observing from. Change the observer’s planet to Jupiter and let the target planet be the Earth.  Describe the motion in the Zodiac Strip under these conditions. How about if the target is Mercury, Venus or Mars?

Questions

1. Would aliens living on Jupiter be more or less likely to have come up with a Sun centered model for the Solar System?
2. For what purpose is the Ptolemaic model actually a  “better” model of the Solar System?
3. For what purpose is it absolutely necessary that one uses the Sun-centered model of the Solar System?

Lab 10 – Cosmic Distance Ladder

Measuring distances in astronomy is very challenging because we can not travel the distance we want to measure. On Earth we primarily measure distance either by laying down a ruler between two points or by traveling between two points and measuring the time it takes. If we know our velocity then we can calculate the distance by distance = velocity x time. We don’t even have to travel, instead we can send electromagnetic radiation which we know travels at the speed of light or 300,000 km/s and just measure the time it takes the radiation to go the distance (and usually back).

In astronomy two other basic approaches are used to determine distance. The first is triangulation or parallax in astronomy speak. In this method one measures the angle of an object from one location and then moves to a different location and measures the angle again.  This creates a triangle and allows one to calculate the distance of the object.  Since the Earth moves 2AU every 6 months this is the natural way to get two measurements from two different locations. However, the farther away the object the smaller the change in angle which limits this approach.

The second method used in astronomy to measure distance is to measure the apparent brightness of an object and compare it to the objects known luminosity called absolute magnitude in astronomy speak. Since we know brightness decreases as distance squared this allows us to measure the distance. The only problem with this approach is having a way to determine the luminosity of objects. Luckily there are pulsating variables, the HR diagram and supernova which all give us ways to estimate the luminosity of things without knowing distances.  If the apparent magnitude, m, and absolute magnitude, M, of an object are known then the logarithm of the distance, d, can be determined from

m – M = -5 + 5 log₁₀ d

Laboratory Tools

We will make use of four simulators in this lab all produced at the University of Nebraska-Lincoln. The four simulators are:

• –  The left panel shows the Map, you can move the observer (the red X) along the road. The top right panel shows the Observer’s View, which depends on the observers position as determined by the red X.  The bottom right panel has the Controls.  Here you can take a measurement, clear your measurement set the error and show a ruler. You can only make changes for Preset A.  For Presets B and C you can not change the error or see the boat on the Map.
• –  The top left panel shows the Absorption Line Intensities  for different temperature stars. The red line can be dragged change a stars temperature and spectral class. The panel below this, Simulated Spectrum, shows how the spectrum of that star would look.  The bottom panel on the left, Distance Modulus Calculation, determines the distance to the star based its apparent magnitude and location in the HR diagram.   The distance is the number in blue.  The top right panel shows the HR Diagram. while the lower right panel shows the Star’s Attributes. Here you can change its apparent brightness and luminosity class.
• – The left panel shows the HR Diagram. The y-axis on the left side (red) shows the absolute magnitude M; the right side (blue) shows the apparent magnitude, m.   Click-dragging on the plot will allow you to move the stars up or down.  The upper right panel, Cluster Selection, allows you to choose the star cluster plotted in the HR Diagram.  The middle right panel, Diagram Options, just allows you to add a horizontal bar which is useful for reading off what the apparent magnitude is that corresponds to an absolute magnitude. The bottom right panel, Distance Modulus Calculator, will calculate the distance for a given apparent and absolute magnitudes.
• – The top panel Light Curve Plot shows a model supernova light curve in red and allows you to select actual supernova data from a dropdown menu. You can also click on a horizontal bar to more clearly see how the apparent and absolute magnitudes compare. The bottom panel is a Distance Module Calculator, enter an apparent magnitude, m, and an absolute magnitude, M, and it will display the distance after the log.

Assignment

Starting with the parallax explorer first move the observer around and notice how the Observer’s view changes.  Then take two measurements far apart, show the ruler and measure the distance to where the lines cross. Now clear the measurements and set the error to 3.0. The boat is in the diamond region outlined by the two cones. Take a third measurement in between your first two and measure the distance range within all three cones.  Now switch to Preset B, take to measurements and determine the range of distances. Note there is no boat shown in this case.  Finally use Preset C. Take two measurements, you are only able to move the observer to 2 preset points. What is the distance range in this case? Include a table like the one below in your report

Now turn to the Spectroscopic Parallax Explorer.  The default should start with a star that  has a temperature of 5840K and an apparent magnitude of 1.0.  The luminosity class should be V. Record the distance.  Make changes to those quantities as shown in the table below and record the distances.

Now turn to the HR Diagram Star Fitting Explorer. Here we will try to fit various star clusters to the main sequence. Some are much easier to fit than others you may want to start from the bottom of the list and work your way up. From the fit we can read off an apparent and absolute magnitude, you can show the horizontal bar to get two values. Then enter them in the Distance Module Calculator panel to get the distance. Record your values in a table like this

Finally let’s take a look at the Supernova Light Curve Fitting Explorer.  Here we can try and match data from supernova to a typical light curve, starting from the last choice may be easier. Note you can move the points in both the x and y directions. Use the horizontal bar to read off the apparent and absolute magnitudes and then calculate the distance. Record your values in a table like this.

Questions

1. At what temperatures does the luminosity class have a strong effect on the distance determined and when does it have a small effect?
2. Which method finds the smallest distances? Which method finds the largest distances?
3. Which method do you think is the most precise?