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2119 - Math Discovers Exoplanets. If technology continues to improve and
astronomers can capture some reflected light from the exoplanet its
spectroscopy can detect the elements that are in the planet’s atmosphere. Detecting
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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-
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------------------------------ International Space Station crossing the Sun
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------------------------------------ 2119 - 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: The function of a wave in
relation to time = Amplitude* sin(n*2pi* frequency * time). “n” is integer multiples of the fundamental
frequency. These frequencies are called
harmonics.
-
- ( 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. 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). Ok, what does Fourier
Analysis have to do with this?
-
- 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. 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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- October 13, 2018 was Review 710
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--------------------- Thursday, October 11, 2018 -------------------------
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