This activity is related to a Teachable Moment from May 6, 2016. See "A Teachable Moment You Can See! The Transit of Mercury."

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### Overview

Activity Notes

A transit happens when a planet crosses in front of a star. From our perspective on Earth, we only ever see two planets transit the sun: Mercury and Venus. Both are pretty rare events. Transits of Mercury happen only about 13 times per century and the next transit of Venus won't happen until 2117! While rare, transits have been used for centuries to track the movements and properties of planets in our own solar system. And more recently, scientists have started using transits of distant stars to find exoplanets – planets orbiting other stars.

This activity features real-world applications of math concepts related to transits and gives students practice calculating the movements of planets in our solar system and other star systems.

### Management

• Students can work individually or in pairs to solve the problems on the student worksheet.
• Calculators are optional but can simplify the calculations.

### Background

In the early 1600s, Johannes Kepler discovered that both Mercury and Venus would transit the sun in 1631. It was fortunate timing: The telescope had been invented just 23 years earlier and the transits wouldn’t happen in the same year again until 13425. Kepler didn’t survive to see the transits, but French astronomer Pierre Gassendi became the first person to see the transit of Mercury (the transit of Venus wasn’t visible from Europe). It was soon understood that transits could be used as an opportunity to measure the apparent diameter – how large a planet appears from Earth – with great accuracy.

In 1677, Edmond Halley observed the transit of Mercury and realized that the parallax shift of the planet – the variation in Mercury’s apparent position against the disk of the sun as seen by observers at distant points on Earth – could be used to accurately measure the distance between the sun and Earth, which wasn’t known at the time.

Today, radar is used to measure the distance between Earth and the sun with greater precision than can be found using transit observations, but the transit of Mercury still provides scientists with opportunities for scientific investigation in two important areas: exospheres and exoplanets.

Exosphere Science

Some objects, like the moon and Mercury, were originally thought to have no atmosphere. But scientists have discovered that these bodies are actually surrounded in an ultra-thin atmosphere of gases called an exosphere. Scientists want to better understand the composition and density of the gases that make up Mercury’s exosphere and transits make that possible.

“When Mercury is in front of the sun, we can study the exosphere close to the planet,” said NASA scientist Rosemary Killen. “Sodium in the exosphere absorbs and re-emits a yellow-orange color from sunlight, and by measuring that absorption, we can learn about the density of gas there.”

This image of Mercury passing in front of the sun was captured on Nov. 8, 2006 by the Solar Optical Telescope, one of three primary instruments on the Hinode spacecraft. Image credit: Hinode JAXA/NASA/PPARC

Exoplanet Discoveries

When Mercury transits the sun, it causes a slight dip in the sun’s brightness as it blocks a tiny portion of the sun's light. Scientists discovered they could use that phenomenon to search for planets orbiting distant stars, called exoplanets, that are otherwise obscured from view by the light of the star. When measuring the brightness of far-off stars, a slight recurring dip in the light curve (a graph of light intensity) could indicate an exoplanet orbiting and transiting its star. NASA’s Kepler mission has found more than 1,000 exoplanets by looking for this telltale drop in brightness.

This diagram shows how scientists use the light curves of distant stars to search for planets outside our solar system. A drop in the light curve, as illustrated here, is a good indication that there's a planet orbiting the observed star. Credit: NASA Ames

The Kepler mission measures the brightness of stars. The data will look like an EKG showing the heart beat. Whenever a planet passes in front of its parent star as viewed from the spacecraft, a tiny pulse or beat is produced. From the repeated beats, we can detect and verify the existence of Earth-size planets and learn about the orbit and size of the planet. Video credit: NASA Ames and Dana Berry

Additionally, scientists have begun exploring the exospheres of exoplanets. By observing the spectra of the light that passes through an exosphere – similar to how we study Mercury’s exosphere – scientists are beginning to understand the evolution of exoplanet atmospheres as well as the influence of stellar wind and magnetic fields.

### Procedures

1. Explain how transits have been used throughout history and today to study the movements and properties of planets in our solar system and beyond.

2. Introduce students to Kepler's Third Law:

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit (or the average distance to the sun).

3. Explain how Kepler's Third Law can be used to calculate the movements of planets:

What Kepler's Third Law means is that for our solar system and planets around stars with the same mass as our sun, R3 = T2, where R is a planet's distance from the sun in astronomical units (AU) and T is the planet's orbital period in years.

Because the distance between Earth and the sun (1 AU) is 149,600,000 km and one Earth year is 365 days, the distance and orbital period of other planets can be calculated when only one variable is known.

4. Have students use the provided worksheet and formula to calculate the orbital periods and distances from the sun for Mercury and Venus.
 Planet Orbital Period (years) Orbital Period (days) Distance from Sun (AU) Distance from Sun (km) Mercury 0.387 AU Venus 224.7 days Earth 1 year 1 AU
5. Explain how students can use light curves and Kepler's Third Law to calculate the properties of exoplanets, or planets around other stars:

The Kepler mission studies the brightness of stars and finds exoplanets by measuring the light curves of thousands of stars. A light curve is a graph of light intensity over time. Stable stars generally put out a mostly even amount of light. When the brightness drops, there is a dip in the light curve. There are different causes for dips in brightness, but when a dip occurs at a regular, predictable interval, it could be caused by an exoplanet transiting that star.

To calculate the properties of planets around other stars (exoplanets), we must modify our formula to account for the variation in the star’s mass as compared with our sun. So we use R = ∛(T2 · Ms) where Ms = is the star’s mass in relation to our sun's mass.

To find the orbital period of an exoplanet using a light curve, determine the length of time between each dip in the light curve, represented by a line that drops below the normal light intensity.

6. Have students study the light curves provided on the worksheet to determine the orbital period and other properties for Kepler-5b, 6b, 7b and 8b. Then, have them calculate the missing information for Kepler-452b – the most Earth-like exoplanet yet discovered – using the data provided.
 Planet Mass of Parent Star (relative to sun) Orbital Period (days) Distance from Parent Star (AU) Distance from Parent Star (km) Kepler-5b 0.05064 AU Kepler-6b 1.21 Ms Kepler-7b 9,350,000 km Kepler-8b 1.21 Ms Kepler-452b 1.04 Ms 156,500,000 km

### Discussion

• Kepler-5b, 6b, 7b and 8b with their low Kepler- numbers have relatively short orbital periods compared with Kepler-452b. Why do you think that is?

Stars observed by the Kepler mission with confirmed exoplanets are named Kepler-1, Kepler-2, Kepler-3, etc. The exoplanets around these stars are given a letter that corresponds to the order in which they were discovered, starting with b. So, Kepler-5b is the first exoplanet discovered around the fifth star found to have a planet.

Kepler-5b, 6b, 7b, and 8b were all discovered early in the mission (as indicated by their low Kepler- number). Kepler discoveries require multiple drops in the light curve to be confirmed. This can be accomplished in several weeks with orbital periods like those of Kepler-5b, 6b, 7b, and 8b. Kepler-452b, with its long orbital period, took several years to cause multiple dips in the light curve. As a result, it took much longer to be discovered, and many other exoplanets were discovered first, hence the high number.

### Assessment

• Individuals or pairs of students should compare work and discuss how their results agree or disagree. For disagreements, students should talk through how they arrived at their answers.

• Check students' work against the answer key:  Planet Orbital Period (years) Orbital Period (days) Distance from Sun (AU) Distance from Sun (km) Mercury 0.24 years 88.0 days 0.387 AU 57,900,000 km Venus 0.62 years 224.7 days 0.72 AU 108,200,000 km Earth 1 year 365.2 days 1 AU 149,600,000 km
 Planet Mass of Parent Star (relative to sun) Orbital Period (days) Distance from Parent Star (AU) Distance from Parent Star (km) Kepler-5b 1.37 Ms 3.55 days 0.05064 AU 7,580,000 km Kepler-6b 1.21 Ms 3.23 days 0.04559 AU 6,820,000 km Kepler-7b 1.36 Ms 4.89 days 0.06250 AU 9,350,000 km Kepler-8b 1.21 Ms 3.52 days 0.04828 AU 7,220,000 km Kepler-452b 1.04 Ms 384.84 days 1.046 AU 156,500,000 km
• Student answers may vary slightly from each other, and from the answer key. This is the result of variations in students' interpretations of drops in the light curve and from the actual measurements. Data provided was also rounded from more precise mission data. Student answers should generally agree with the answers provided, but shouldn't be marked as incorrect for not matching exactly.

### Extensions

Kepler Exoplanet Discoveries - Click on each planet referred to in the student worksheet to view an interactive of the exoplanet system, the light curve that led to its discovery, and a chart pinpointing its location in the sky.

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