- 3044 - EXOPLANET - Pegasi and Doppler shift astronomy? The planet “51 Pegasi b” was discovered in 1995. It was the first planet discovered orbiting a normal star, like our Sun. When watching the star astronomers were able to detect a rhythmic wobble using the “Doppler Shift Technique.”
--------------- 3044 - EXOPLANET - Pegasi and Doppler shift astronomy?
- The best way to understand this astronomy is to apply it to our own Solar System. Our Sun too has a wobble because our Sun and the planet Jupiter orbit each other around a common center of mass that is just outside the radius of the Sun. We know this because the Sun traces out this very small circle every 12 years which is the exact period of Jupiter’s orbit.
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- In fact, there is another wobble that we can detect where the Sun orbit’s the mutual center of mass with the planet Saturn every 29.5 years. The total graph of the Sun’s wobbles have been measured since 1960 to 2025 projections.
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- All of these many loops and ellipsis about the Solar System’s center of mass can be defined for our Sun. Now, let’s assume we are 30 lightyears away looking face-on, or down on the Solar System wobbles, could we detect them?
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- The radius of the Sun would be 0.0005 arc seconds. The entire range of wobbles radii over this period of 65 years would only be 0.015 arc seconds. This is 100 times smaller than the angular resolution of the Hubble Space Telescope. So, we could not detect the Sun’s wobbles looking down on it from 30 lightyears distant.
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- However, looking at our Solar System edge-on from 30 lightyears distant we could detect the Doppler Shift as the Sun was moving toward us during ½ of its orbit around the center of mass, the Blueshift, and when it was moving away from us during the second ½ of its wobble, the Redshift.
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- Astronomers did this for the star that had the planet “51 Pegasi b” orbiting it. The wobble had a pattern of repeating every 4 days. Pefasus is 158 lightyears away, not just 30 lightyears. The Blueshift peaked at 57 meters per second coming towards us and a Redshift of 57 meters per second going away from us. This is a wobble difference of about 256 miles per hour .
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- The period of orbit of only 4 days, corresponding to the planet’s year, meant that the planet was orbiting very close to its star. Its surface temperature would be about 1,000 Kelvin. Its orbit being about 1/8th of Mercury’s orbital distance from the Sun.
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- Its star is 6% more massive than our Sun. It is 18% larger in diameter. Its orbit is 4.5% of an astronomical unit, ie: Earths orbit.
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- The Doppler Shift for Pegasi repeats itself every 4 days, so we need Kepler’s and Newton’s formula that calculates the period as a function of radius. The period^2 = the radius of orbit^3, p^2 = a^3
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------------------- p^2 = 4*pi^2 * a^3 / G * ( M1 + m2)
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------------------- p = Pegasi orbital period = 4.23 days
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------------------- M1 = mass of the star = 1.06 Solar Mass
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------------------- m2 = mass of the planet
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------------------- a = radius of the planet’s orbit
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------------------ G = the force of gravity = 6.67*10^-11 m^3/kg*sec^2
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------------------ Solar Mass = 2*10^30 kilograms
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------------------ 1.06 Solar Mass = M1 = 2.12*10^30 kilograms
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------------------ p = 4.23*24*3600 = 3.65*10^5 seconds
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------------------- (3.65*10^5)^2 = 4*pi^2*(radius)^3 / 6,67*10^-11 * 2.12*10^30
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------------------- radius = 7.81*10^9 meters
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------------------ 1 AU = 1.5*10^11 meters
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------------------ radius = 0.052 AU.
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Therefore, Pegasi is orbiting about 5% of the Earth-Sun distance which is very close to its star. For comparison the orbit of Mercury is 39% of an AU.
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- To find the mass of Pegasi we use the information of its orbital velocity about the star, which is 57 meters per second. Let’s consider the Pegasi is a single planet. The planet and the star are orbiting a center of mass just like the Sun and Jupiter ( if we ignore the other planets for a moment.)
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- There is a conservation of momentum and momentum is equal to mass times velocity. The orbiting momentum of the star must be the same as the orbiting momentum of the planet in opposite directions.
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------------------- Ms*Vs = Mp * Vp
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------------------- Vp = the velocity of the planet can e calculated from the radiud calculated above and the period of orbit that we measured.
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------------------ Vp = 2*pi* radius / period
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------------------ Vp = 2*pi*7.81*10^9 meters / 3.65*10^5 seconds
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------------------ Ms = mass of the star = 2.12*10^30 kilograms
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------------------ Vs = velocity of the star = 57 meters / second
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------------------- 2.12*10^30 * 57 = Mp * 2*pi*7.891*10^9 / 3.65*10^5
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----------------- Mp = mass of eh planet = 9*10^26 kilograms
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----------------- Mass of Jupiter = 1.9*10^27 kilograms
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----------------- Mass of Pegasi = 0.47 Jupiter Mass.
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- This is a minimum mass for Pegasi because the orbiting planet and the Doppler Shift measurements are not likely edge-on. The actual mass is likely 2 times the measured mass. So, Pegasi is about the mass of Jupiter.
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- The velocity measured depends on the incline angle of the orbit we are observing, but can’t see. Only on a direct edge-on measurement is the measure velocity, and therefore the measured mass equal to the true velocity and true mass.
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- The actual mass is the minimum mass divided by the cosine of the orbital inclination (which normally we do not know). If the angle is between 60 and 90 degrees the actual is probably double the minimum.
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- By random chance this applies to about 33% of the stars. If the angle is between 84.3 and 90 degrees the actual mass is 10 times the minimum. This randomly occurs in about 6% of the cases.
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So, in conclusion Pegasi is thought to be about the size of Jupiter orbiting very close to its star having a surface temperature of 1,000 Kelvin. Probably not a good place for searching for life. It is so close to its star its atmosphere is being blasted away. But, wouldn’t you like to learn more?
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February 12, 2021 EXOPLANET - Pegasi Doppler shift ? 840 3044
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