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--------------------- 2625 - FOURIER - Math Discovers Exoplanets
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- Jean Baptista Joseph Baron Fourier, a French mathematician, born 1768, died 1830, became famous for his mathematical treatment of heat flow. Heat flow depends on the temperature difference between two points, the heat conducting properties of the material, the shape of the material, and more.
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- The result of model is a complex set of differential equations that characterize the heat flow from one place to another. Fourier solved these equations using an infinite series of trigonometric functions.
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- His mathematical theorem founded in 1807 is now known as the “Fourier Series“, or “Fourier Analysis“. The series expands a periodic function as an infinite sum of sine waves of varies frequencies and amplitudes. His theorem applies to any periodic wave and has very wide science and engineering application. “Any wave can be approximated by super positioning sinusoidal waves with different frequencies“.
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- The human ear performs a Fourier analysis as sound waves of different frequencies enter the ear and vibrate tiny hairs at those specific frequencies. The listener can then discern the different pitches and harmonics in the sound they hear.
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- The Fourier Theorem:
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- The function of a wave in relation to time = Amplitude* sin(n*2pi* frequency * time).
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- “n” is integer multiples of the fundamental frequency. These frequencies are called harmonics.
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- ( Fourier got a little over obsessed with heat. When he was sick he overheated his house expecting a cure. Wrapped in a blanket, he tripped and fell down the stairs and died.)
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- Our Sun wobbles. Our Sun orbits its spot in the sky in small circles caused by the gravity pull of Jupiter. They are not perfect circles because the Sun is also affected by the lesser amount of gravity pull from Saturn.
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- We know this because if we measure the wobble it has a primary period of 12 years and another wobble super imposed on the primary of 30 years. The orbit of Jupiter about the Sun is 11.862 years. The orbit of Saturn about the Sun is 29.458 years.
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- When the Sun’s wobble is moving towards us it is traveling 20 meters/second faster than when it is moving away from us (+ or- 10 meters/second). If you walk slowly across the room you are traveling 1 meter/second. (2.2 miles per hour). So you can see this is a very small change in velocity that has to be detected.
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- How can astronomers measure such a small change in velocity to detect this slight wobble of a star caused by its orbiting planets?
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- The answer comes from the light spectrum. When light from the Sun, or a star, is passed through a prism it spreads the light into a spectrum, a rainbow of colors from red to blue. A careful study of this rainbow of colors finds fine, dark absorption lines that occur because the light passes through elements that absorb that particular frequency.
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- The particular frequency corresponds to a particular electron orbit, where the electron absorbs the photon and jumps into a higher orbit. Each element has unique electron orbits and therefore unique absorption lines. By measuring absorption lines in a spectrum you can identify each element in the periodic table, 100 different elements such as sodium, carbon, silicon, etc.
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- An even closer look at these absorption lines in the spectrum can identify a shift in their frequency over time. When the wobble is moving towards us the lines shift toward the blue end of the spectrum (higher frequency). When the wobble is moving away from us the lines shift towards the red end of the spectrum (lower frequency).
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- Ok, what does Fourier Analysis have to do with this?
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- This change in frequency in the spectroscopy spectrum is also known as the “Doppler Shift“. It is no different from the Doppler Shift of sound waves that occur when a speeding race car approaches ,then , whizzes past you.
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- Current technology with today’s telescopes and spectrometers can measure Doppler Shifts of +or- 10 meters per second. This has already allowed astronomers to discover over 4,000 planets orbiting other stars in our Milky Way. There are over 400,000,000,000 stars in our galaxy so there are many more out there to be discovered.
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- However, +or-10 meters/second can only detect Jupiter size planets which are 317 times Earth size. In order to measure Earth size, terrestrial planets, orbiting stars we must be able to measure +or- 0.1 meters/second Doppler Shifts. This is measuring wobble velocities equal to the speed of a turtle scampering across the floor.
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- Believe it or not astronomers are figuring out how to do just that in their quest to find life on Earth-like planets like our own. Direct observations are nearly impossible because the brightness of the star totally overwhelms any light reflected off the surface of a nearby planet. It would be like seeing a firefly in the face of a circus search light. However, the Doppler technique may work to allow us to find these small terrestrial planets.
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- If we find enough planets circling stars we are likely to find one orbiting edge-on to the star. In other words, the planet just happens to pass directly in front of the star and directly behind the star.
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- If this happens astronomers can measure in addition to the mass of the planet, the size of the planet because of the stars brightness changes as the planet passes in front and blocks some of the light. Now they can calculate the size and density of the planet.
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- If technology continues to improve and astronomers can capture some reflected light from the planet its spectroscopy can detect the elements that are in the planet’s atmosphere. Knowing this will allow astronomer to predict if life is likely to exist on that planet. They will be able to predict if an iron core and a consequential magnetic field exists to protect life from the star’s cosmic rays and ultraviolet radiation.
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- The breakthrough in this improved technology comes from computing power. First we need to measure the starlight 365 days out of the year for 5 years, i.e. we need a dedicated telescope. And, we must take 175 observations each night, (365 * 5 * 175) = 300,000 observations.
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- With this enormous amount of data and the computing power to detect the slightest sinusoidal wobble in the light spectrum and performing the Fourier transform on the data so the period of the orbiting object will pop out of the data, astronomers will detect Earth-size terrestrial planets orbiting other stars.
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- Fourier would fall down the stairs if he knew what astronomers are doing with his mathematics.
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- February 28, 2020 710 2051 2119 2673
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