The Kepler Mission
In this series of exercises, participants will
- Identify characteristics that affect transit detections.
- Recognize transit-like events and measure the period in time series brightness data.
- Distinguish between testable and un-testable hypothesis for a given dataset.
- List planetary properties that can be determined from transit data.
- Explain properties affecting the width and location of the habitable zone.
- Create a graph of Kepler data using Excel.
- Apply physical models to determine planetary properties of previously identified planets.
Engage: Discuss extrasolar planets & model transit
Start your lesson with a class or small group discussion about the possibility of planets around other stars. Introduce the vocabulary term extrasolar planet to describe the planets discovered around stars other than the Sun.
Create a model transit e.g. from the GEMS lesson "Detecting Planet Transits." An example model would be a small light bulb representing the star with a bead on a dowel representing an extrasolar planet. In the discussion, elicit the concept that the brightness of the star is dimmed due to a planetary transit.
Have participants make some predictions: Which type of system make it easier to find planets using this technique. If it doesn't matter, write EQUAL CHANCE. Submit the predictions on a separate sheet of paper with your name on it.
- Less massive stars or more massive stars.
- Planets with orbits that are closer to circular or highly elliptical orbits.
- Face-on orbits or edge-on orbits.
- Small diameter planets or large diameter planets.
- Small mass planets or large mass planets.
- Planets close to star or planets far from star.
Have participants navigate to the transit simulator: http://astro.unl.edu/naap/esp/animations/transitSimulator.html and allow them to experiment freely for a few minutes.
Discuss the AU, Rjup, Mjup units. Discuss meaning of semimajor axis and eccentricity. Ask the participants to brainstorm the meaning of the graph shown. Ask if the entirely 3.56 day orbit of the Option A preset is shown in the graph. Be sure that participants recognize that:
- the horizontal axis is time but that the entire orbit is not shown and the horizontal scale rescales itself so it is essential to read the values.
- the vertical axis represents observed star brightness in some normalized units where the highest is always '1' and that the transit of the planet causes the observed brightness to diminish. This scale also rescales automatically.
Explore: Test factors affecting visibility of transit
Ask participants to use simulation (show theoretical curves) to answer 'What factors make it easier to detect a transit?' (or ask them to test their predictions made earlier)
After participants have tested their predictions, discuss what 'easier to detect' means in the context of a transit. Should elicit both length of eclipse and depth of eclipse.
Have participants work in small groups to formulate a hypothesis about the ease of detecting extrasolar planets. Have participants share their hypotheses.
Many first draft hypotheses will relate to the number of planets found (e.g. If it is easier to detect transits around low mass stars, then I expect to find more planets if I search low mass stars.) Discuss the TESTABILITY of their hypothesis using this simulation since the simulation will not tell you about the number of stars found. Have participants focus on the simulation features that are quantitative such as the length of eclipse as function of total orbit and/or depth of eclipse and create a new TESTABLE WITH SIMULATION hypothesis in the 'if .... then...' format. A mnemonic to assist with hypothesis generation is the 'if mix then dry' where the 'mix' refers to the independent variable being on the 'x' axis and the 'dry' refers to the dependent variable being on the 'y' axis.
Participants should then test their hypothesis and plot a graph of the dependent variable vs. the independent variable they selected. The leader may wish to guide the later discussion by encouraging a broad range of hypotheses be tested.
The discussion of testable hypothesis might include an extension to discussing a 'good science fair question' which is testable with available tools and knowlege versus a 'good question' which may not be testable with tools or information at hand.
Have participants reflect back on their predictions. What prior knowledge were they activating? Were they just guessing? Is it useful to make predictions during an activity if there is no prior knowledge? Also discuss the need to check back and correct mistaken thinking. Was there a misconception that led to a prediction that was off base? Many participants may not have heard much about transit searches but may have heard about radial velocity searches so the prior knowledge activated was not entirely relevant.
Explain: Factors affecting ideal transit detectability
Summarize the participants' findings about common factors affecting transit detectability including observer-system geometry, stellar mass/radius to planetary radius ratios, and semi-major axis, percentage of orbit time in eclipse and period of planet.
Elaborate: Noise and Detectability
So far, participants have only examined the theoretical curves which have perfect transit signatures. Have participants enable the 'show simulated measurements' feature to simulate actual measurements. (Turning off the theoretical curve makes it more of a real challenge!) Break participants up into groups to determine the maximum noise level they could have in their instrument or detections in order to be able to detect one to three of the detected systems which are pre-loaded into the system. Both the scroll bar and typing into the box are valid ways to change the noise to signal value. The systems available to test are: OGLE-TR-113 b, TReS-1, XO-1 b, HD 209458 b, OGLE-TR-111 b, OGLE-TR-10 b, HD 189733 b, HD 149026 b, OGLE-TR-132 b. Groups should report the value and the value as a percentage of the eclipse depth. Have more than one group determine the minimum noise level so that a discussion about how scientists compare their answers and verify the results of others is part of the scientific process.
Ask participants to find the pre-set most like Earth. Here are two sources for planetary data.
Engage: Habitable Zone
Have participants brainstorm a list of factors which affect the ability of Earth to host life. Have them extend their list to factors which may affect the probability of life on extrasolar planets. Depending on the knowledge level of the participants, they may use the word brightness of star instead of luminosity/mass of star. Use prompts such as 'What if the star doesn't shine for very long?' to help elicit stellar lifetime. Depending on the knowledge level of the participants, you may also want to include the luminosity change of the star as a function of time and galactic factors such as nearby stellar density and metallicity.
An extension is for participants to experiment with the Circumstellar Habitable Zone simulator to test their predictions. In the figure below, the habitable zone is shown in green.
Image Credit: Kepler Mission.
Explore: Taking Simulated Transit Data
Have participants make a list of planetary properties they think they could determine from measuring the % flux change and transit length for a star of known mass/luminosity/radius. They will check their predictions using a simulation.
In groups, have the participants work through the following flash simulation which allows the simulation of taking transit data.
Divide group up so they all do different stars. [A5, M0, K5, F0, K0, F5, G5, M5, G2, G0]. Participants must manually record the results and the formulas as there is not the functionality for saving results nor for navigating through the simulation after it is complete.
Share the data for each as a 'key' for participants and check that groups that were assigned the same stellar type got similar results. Reinforce that scientists share and verify data.
Compare the list of planetary characteristics determined using the simulation to their original (probably short) list of characteristics they thought they would be able to determine. Encourage them to be amazed and emphasize that the use of MODELS (some might call these theories) that describe how the universe works help us learn more about the universe than just what the observations can directly tell us.
Explain: Where do the formulas come from?
Depending on the participant level and interest, the leader may decide it is important to reinforce the origin of the equations used in the simulation which include Kepler's 3rd Law for non-solar systems, the no-atmosphere approximation of the planetary temperature given the distance from the star, the fraction of radiation intercepted by the planet from a perfect blackbody (Stefan-Boltzmann) emitter.
Elaborate: Finding planetary properties of real planets detected by the transit technique.
Participants will now download Kepler data, determine the transit depth and period and will determine the planetary properties. The leader must provide information about the stellar type of the parent star (including surface temperature in Kelvin, stellar mass in solar masses, and stellar radius in solar radii). This information is in Table 2 below.
As of August 31, 2010, the first 5 planets discovered by Kepler have their transit data published at http://archive.stsci.edu/prepds/kepler_hlsp/. Divide the participants into groups and assign each group one object to examine (Kepler 4b, Kepler 5b, Kepler 6b, Kepler 7b, Kepler 8b). Each group will graph the data to determine the period and eclipse depth.
- Right click on the relevant Kepler Time Series file (e.g. hlsp_exo_kepler_phot_KPLR10874614_kep_v1.0_dtr.txt) and choose Save Link As to save the file to the local computer.
- Open Excel.
- Choose File:Open and select the file you downloaded.
- The text import wizard opens (in Excel 2003). Select Fixed Width.
- Scroll down through the preview file window to identify on which row the actual time series data starts (e.g. row 19) and modify the Start import at Row number to be the same as the first row of data. The data is two columns versus the header which looks like fractured sentences that begin with a '#.'
- Click Next to go to step 2 of the Text Import Wizard.
- The vertical lines indicate where Excel has decided to put a column break. It is probably not between the two columns of data you have. Click in the gap between the two columns of data (e.g. at 15 characters in from the left) to create a column break between the two columns of data.
- Click Finish to import your two columns of data.
- Create a scatter graph of the data by highlighting the two columns of data and selecting Insert: Chart.
- Select Chart type: XY Scatter.
- If everything goes well, you should have a graph with large values (>2 million) on the horizontal axis and values which range from approximately 0.95 to 1 on the vertical axis.
- The vertical axis is in the same units as the vertical axis in the first simulator; normalized flux, so that a 0.99 indicates that 99% of the total stellar flux is detected or 1% is blocked by the planet. This would be a 1% drop in brightness.
- The horizontal axis is in fractions of Julian date which is a universal astronomical timekeeping method. (You can convert to calendar date using the US Naval Observatory Calculator; zero Julian Date is in 4713 BCE.)
- [advanced - make horizontal axis easier to read by subtracting a constant from each date or change the display of the axis to be in scientific notation or show more decimals]
- Determine the period and eclipse depth, or percent drop in brightness.
- Easy version: Go back into simulator and enter in values for your star and planet and have the simulator do the math for you to get the planetary distance, temperature, and radius.
- High School version: Convert the period in days to period in years and use the stellar mass to compute the distance the planet orbits from its star. (P^2 * Mstar = D^3) Compute the exoplanet surface temperature using the stellar temperature, stellar radius, and planetary distance. (Tstar * sqrt(Rstar/(214 * 2 * Distance)) where some of the factors account for units conversion and then the fraction of light absorbed from Stefan-Boltzmann). Compute the exoplanet radius using the percentage brightness drop and the relative cross sectional areas of the star and planet. sqrt(109 * Rstar^2 * %brightnessdrop /100). This will be in units of Earth's radius.
Evaluate: Compare to previously published results and discuss finer points
Compare period determinations by different groups. Discuss multiple ways period could be determined from this data. Is one method better than another? (yes, averaging over multiple periods is superior to using only one pair of data) How does determining period in this type of data compare to determining period of a wave?
What, if any differences were found? Note the difference in units for planetary mass and radius.
Can also discuss two complexities - variable stars and giant stars. Giant stars might have same surface temperature - but are unlikely to have a habitable planet due to lifetime and prior violent activity issues. Variable star profiles are discussed in detail on the Kepler Science Website.
These are the stellar and planetary parameters as published in Borucki, W. J. et al., Kepler Planet-Detection Mission: Introduction and First Results. Science 19 February 2010:
Vol. 327. no. 5968, pp. 977 - 980
Table 1 Properties of the exoplanets detected by Kepler. The state of the current observations is insufficient to support claims of nonzero eccentricity. Therefore, parameter estimates are based on the assumption of a circular orbit. Calculations of the equilibrium temperatures assume a Bond albedo = 0.1 and efficient transport of heat to the night side. Epoch = HJD-2454900.0. RJ is the Jupiter equatorial radius. MJ is the mass of Jupiter. Errors are ± 1 and represent formal errors only.
Table 2 Characteristics of the stars hosting Kepler planets. Kp is the stellar magnitude calculated for the Kepler band pass. The values are similar to those produced by an R filter (7) for most star types. Right ascension (RA) and declination (dec) refer to the J2000.0 equinox. For three of the stars (Kepler-4, -5, and -7), the model fits give two peaks in the distributions of the mass and radius. The values listed here are thought to be the best estimate. Teff , effective temperature. M* and R* are the mass and radius of the host stars, respectively. MSun and RSun are the mass and radius of the Sun, respectively.
Exercise created by Zo Webster and implemented at 2010 GEARS workshop.
GEARS is Funded by NASA Office of Education Grant NNX09AH83A and supported by the Georgia Department of Education, Columbus State University, and Georgia Southern University.
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