Practical exercise 2: TRANSIT PHOTOMETRY
In this practical a ctivity the students will explore one of the most successful indirect methods of detecting exoplanets, namely the transit photometry. This will be facilitated by replacing the host star by a spherical light bulb and the exoplanet by a small, few centimeter sized b all. Orbital motion of the exoplanet will be simulated by suspending the ball on a piece of string from a stand to the level of the bulb, so that giving the ball a nudge will make it revolve around the bulb in circles. The students will record the intensit y of the coming from the light bulb and produce the corresponding light curve. After completing this activity, they should acquire a better understanding of how the method of transit photometry works and on what parameters the light curve of a star with tr ansiting exoplanet depends.
Materials needed
Light bulb (preferably spherical with diffusing globe), light detector (ideally equipped with a data logger or connectable to a PC with suitable software for analyzing measured data; alternatively can use a sto pwatch), laboratory stand with a clamp, cork/polystyrene balls of diverse sizes, piece of string.
Warm up questions
Q: Why is it hard to observe exoplanets directly?nbsp;
A: Their angular separations from their host star are very small and they are much dimmer t han the host star, meaning that their light is “drowned out” by the light coming from the star.
Q: What do astronomers mean by a light curve
A: Light curve is the observed dependence of brightness of an object (typically a star) on time.
Q: What effect may an exoplanet exert on the light which we observe from its host star?nbsp;
A: Provided that the line of sight to the star lies close to the exoplanet’s orbital plane, the exoplanet will periodically obscure its host star’s disk (as observed from the Earth). This will be observed as periodical dips in the host star’s light curve.
Q: Describe the shape of a light curve for a star with transiting exoplanet. On what parameters does it depend?nbsp;
A: See the theoretical introduction for details. The separation of the dips will be given by the orbital period. The depth of the dips depends on the size ratio of the star and the exoplanets.
Q: Does the depth of the dip in the light curve depend on the d istance of the exoplanet from its host star?nbsp;
A: No, it does not, as the orbital radius is very small compared to the distance of the host star from the Earth, so the ratio of the angular sizes of the exoplanet’s and host star’s disk (which determine the po rtion of light blocked by the exoplanet and thereby the depth of the dip) are given directly by the ratio of their physical (linear) sizes.
Q: What are the typical size ratios of exoplanets and their host stars?nbsp;
A: For Earth sized exoplanets orbiting Sun like stars we obtain ratios roughly 1/100 while for Jupiter sized planets we obtain 1/10.
Q: Calculate the percentage of host star’s light blocked by an exoplanet which is 100x smaller than the star. A: The percentage of the light block is given by the ra tio of that is (1/100)^2 = 0.01
Q: How is the light curve of an exoplanet transit affected by the position of its orbital plane relative to the line of sight?nbsp;
A: The main effect is on the length of the observed transit: the longest possible duration of transit is achieved for the edge on geometry, while for large enough inclinations the transits may disappear altogether. The geometry does not generally affect the depth of the dip (neglecting limb darkening and borderline cases where the exoplanet disk merely grazes its host star’s disk), nor does it affect the separation of the individual dips, which is always equal to the orbital period.
Q: Name an example of a space based observatory which is devoted to detecting exoplanets using transit photometry. Use the internet to look for examples of confirmed exoplanets detected by this observatory.
A: For instance the Kepler space telescope which, as of August 2018, detected over 2000 con firmed exoplanets. To name one example the exoplanet designated as Keple r 442b orbits its host star (distance from Earth 1120 ly, mass and luminosity 0.61 and 0.11 those of the Sun, respectively) at a distance 0.4 au with period 112 days. It has mass and radius 2.3 and 1.3 multiples of the mass and radius of the Earth. It lies within the habitable zone of its host star.
Instructions
The students will model an exoplanetary system containing a single exoplanet by light bulb with spherical diffusing globe (size 20 30 cm) and a polystyrene ball (size 2 3 cm; the students may use a number of balls of various sizes). They should attach one end of a piece of string to the ball. The other end should be tied to a clamp affixed to a laboratory stand. The height should be adjusted so that the ball can move in the level of the light sourc e. Orbital motion of the exoplanet can then represented by circular motion of the ball suspended from the stand around the light bulb. The length of the string should be sufficiently large (> 1 m) compared to the radius of the balls motion so that the peri od can be regarded as independent of the radius. Using a light intensity detector placed within the plane in which the ball revolves around the light bulb and, ideally, connected to a PC equipped with suitable software which enable recording and analyzing the data from the detector, the students will measure the light curve of the light bulb. In the case that the data from the light detector cannot be analyzed by a computer, the students will have to use a stopwatch to time the transits while noting down re adings from the detector. The students will then experiment with putting the exoplanets on different orbits, moving the detector slightly off the orbital plane, or, using balls of different sizes. They should note what effects these adjustments have on the recorded light curves. Note that it will probably prove very hard to achieve realistic up to scale size proportions. As a result, the students may observe dependence of the depth of the dips in their light curves on the distance in which the ball revolves the light bulb. The students should be reminded of the fact that this effects disappears in the realistic systems where the orbital radius is much smaller than the distance of the system from the observer (detector).
The detailed instructions for the students are as follows:
1. Using the materials provided and instructions from your teacher, construct a model of an exoplanet orbiting its host star.
2. Place a light detector in the orbital plane of the ball around the
3. Measure the light curve of the light bulb with the orbiting ball over several periods.
4. Experiment with putting the ball on different orbits. Record the corresponding light curves for each type of orbit.
5. Measure the light curve while slightly lifting the detector above the orbital plane (so that the transits still occur).
6. Analyze the light curves you obtained. For each light curve which you measured determine:
- the orbital period of the ball (in seconds)
- the depth of the dip (as a percentage of the light blocked by the ball)
- the width of the dip (in seconds)
The following questions should be answered by the students after completing the practical activity
Q: How does the measured period depend on the size of the ball’s orbit? Does this remain true in the systems where a real exoplanet orbits a real star?nbsp;
A: If the suspension length is large compared to the orbital radius, the students should find that the period. This property of physical pendulum (for small amplitudes of oscillation) is, of course, not shared by gravitational systems where the period as a function of orbital radius follows the 3 rd Kepler’s law.
Q: How does the measured depth of the dip in the light curve change when using balls of different sizes
A: The students should observe that using larger balls will result in deeper minima.
Q: How does the measured width of the dip in the light curve change when increasing the orbital radius?nbsp;
A: Increasing the size of the orbit increases the speed at which the ball moves (because the period should be roughly constant). The students should observe that this will result in narrower minima.
Q: What is the effect of moving the detector somewhat away from the orbital plane?nbsp;
A: Provided that the transits still occur, the students should observe that the width of the minima decreases because when the detector does not lie exactly in the orbital plane, its point of view is that the ball takes shorter path to traverse the bulb’s disk than if it was crossing it radially.