That was really fascinating! Your analysis was like watching a Sudoku puzzle unfolding - very engaging! You have to give it to Al-Burini though for coming up with such an elegant first order solution in the 11th century. He probably didn’t know about the about the variation of refractive index with altitude and definitely knew nothing about plate tectonics or that the Earth is an oblate spheroid, He hiked up a mountain in Pakistan in the 11th Century; had an instrument that could measure inclination/declination with a precision of less than 5 seconds of arc, and came up with a solution that was within +/- 15% of the modern value. That’s stunning! Actually, I think it’d be great if you could research how people calculated trigonometric functions a 1000 years ago - there were no calculators or Kaye & Laby tables back then. Keep up the good work that you are doing.
Thanks! I totally agree with you. Al-Buruni was a great scientist, and doesn’t get enough credit. His method compared to Eratosthenes is simpler, easier to do, and you get the added bonus of knowing the distance to the horizon. Yet we all learn about Eratosthenes, but Al Biruni was never mentioned in any of my physics courses.
There is a formula used in geodesy to calculate the curvature of light from atmospheric parameters that can be measured: c = [7.90e-5 K/mbar] P / T² (0,0343 K/m + dT/dh) where c = curvature of light rays calculated from atmospheric parameters pressure P in mbar, absolute temperature T in Kelvin and most importantly the temperature gradient dT/dh in °C/m. Actually in geodesy they calculate the so called refraction coefficient k, which can be calculated using the radius of the earth R from c as follows: k = c R Note: refraction (c) depends mainly on the temperature gradient dT/dh. Pressure and absolute Temperature have a minor influence and the wavelength of light, humidity and CO2 concentration can be neglected in practice. When we know the curvature of light, which is the inverse of the radius of curvature, we can use the following formula to calculate the radius of the earth by taking refraction into account: R = (h cos(α) - 0.5 c h²) / (c h + 1 - cos(α)) where R = measured radius of the earth, h = observer altitude above the ground (sea level or plain in case of Al-Biruni), α = dip angle from horizontal to apparent refracted horizon and c = curvature of light. There is a simpler approximation of this formula: R = 1 / (0.5 α²/ h + c) I derived all this formulas myself and checked them against literature and own experiments and calculations. The formula for c was derived using Fermat‘s principle and calculus. You can find this derivation on my website, just google Deriving Equations for Atmospheric Refraction bislins
9:46 Where does this equation for K come from? Is there any resources I can use to find this, I can't find anything resembling it and I'm not sure how it's derived.
One way to get it is to take a total derivative of the left side, and plug in the Euler Lagrange equations, you will find that if L has no explicit phi dependence you get 0. Therefore it must be a constant. I’m surprised you couldnt find this derivation. For standard lagrangian mechanics (where the independent variable is time, instead of phi) this constant is the energy.
@@physicsalmanac I figured the left side of the equation went to zero because there is no dependency on phi, that much was clear, but I'm confused on how the equation with the K came about, I'm not sure if I'm missing something(which is entirely possible, I'm not exactly well versed in Lagrangian mechanics) but if the partial of the Lagrangian with respect to r is zero, how does this result in the left side of the K equation? Everything before and after that made sense, I'm just struggling with where the left side of the K equation is coming from.
@@arthursgarage6550 The left side of the equation and the lagrangian do depend on phi, because r and r' depend on phi... but there is no explicit phi dependence in this case. Meaning there is no explicit term with phi in the lagrangian. L in general = L( r', r, phi). So the total derivative of L is: dL/dp = DL/Dr' * dr'/dp + DL/Dr * dr/dp + DL/Dp. Where p=phi, d = total derivative, D=partial derivative. and recall r'=dr/dp. Now apply this total derivative to the left side of the equation, then substitute in the Euler-Lagrange equation, and you will see the the only term that remains is DL/Dp, which is 0 if there is no explicit dependence on phi.
18:40 This really seems to undermine the whole point of this calculation. You introduce an approximation in a calculation to estimate the error in an approximation. The fact that you then go on to say that the error is independent of the height of the mountain is just mind boggling to me; yeah, it's irrelevant because you just approximated it away. If it really doesn't matter, then the whole premise of both this error analysis AND Al-Biruni's calculation is called into question. Since if the mountain isn't high enough to see enough of the curvature of the Earth, then the calculation would never work, but if the mountain is high enough and you're able to see far enough, then as your original premise stated it calls into question how much of that is distorted by the atmosphere, which does depend on the height. Also I don't think your substitution of R = h / [sec(phi) - 1] is valid, since it assumes phi is the correct angle, and NOT the observed one distorted by atmosphere. So A for effort dude but you really dropped the ball at the end there at the end.
The additional corrections you’re referring to, that I’ve dropped are higher order corrections. I’m only including 1st order corrections here. a*R is small, and phi is small. So sec(phi) is very close to 1 (1+ a small number). So a*R*sec(phi) = a small number plus a really small number. I’m approximating by ignoring the really small number. But you’re right that this dropped term would depend on height, since phi depends on height. You can always forgo the very last step and keep the sec(phi) if you want better precision. This would be a miner correction. Really miner it turns out. Even if you took your mountain to be Mt. Everest (the best you could do), then the error would still be about 15%. Also the horizon is a result of the curvature of the earth. Even if our eyes can’t “see” the curve. It would not exist if the earth wasn’t curved. The substitution of R is valid, as phi is NOT the measured angle. Alpha is what we measure. Phi is the correct geometric angle, but not the angle being measured. So it's unknown. Similar to R. R is the true radius of the earth, but not the one being measured. This is also why we take sec(phi) be 1. We don't know exactly what it is, but we do know its barely bigger than 1. So we make an approximation, to estimate are error in R measured, due to alpha differing from phi. Good comment though. I like that your thinking about if the analysis makes sense, instead of blindly following.
Al Birunri could have never measured the dipangle accurately enough for this to be relevant anyway It is said he arrived at a radius of 10 miles less than reality. If he did,that was by coincidence. He could never have measered the angle at that accuracy. Not even remotely .
Great video, sorry you had to bach the guy although 11 centuries earlier and made the calculation without the calculus jargon that you had to go through, your explanation was intricate and somewhat burdensome for laymen like me, I was just waiting for you at the end to say that Elbiruni was a genius instead you said he got lucky, maybe he knew things that he couldn't explain to his peers back then.
Thanks. I meant no disrespect to Al Biruni. Even if he was aware of atmospheric refraction (it’s not entirely impossible as snells law was discovered around the time he was born), the math needed to calculate it would not be invented til 600 years later. So he did the best he could have done at the time.
AL Biruni’s correctly measured the radius of earth. your calculations further proves that AL Biruni is perfectly correct. thank you for the video
You’re welcome. Thank you for watching!
That was really fascinating! Your analysis was like watching a Sudoku puzzle unfolding - very engaging!
You have to give it to Al-Burini though for coming up with such an elegant first order solution in the 11th century. He probably didn’t know about the about the variation of refractive index with altitude and definitely knew nothing about plate tectonics or that the Earth is an oblate spheroid,
He hiked up a mountain in Pakistan in the 11th Century; had an instrument that could measure inclination/declination with a precision of less than 5 seconds of arc, and came up with a solution that was within +/- 15% of the modern value. That’s stunning!
Actually, I think it’d be great if you could research how people calculated trigonometric functions a 1000 years ago - there were no calculators or Kaye & Laby tables back then.
Keep up the good work that you are doing.
Thanks! I totally agree with you. Al-Buruni was a great scientist, and doesn’t get enough credit. His method compared to Eratosthenes is simpler, easier to do, and you get the added bonus of knowing the distance to the horizon. Yet we all learn about Eratosthenes, but Al Biruni was never mentioned in any of my physics courses.
There is a formula used in geodesy to calculate the curvature of light from atmospheric parameters that can be measured:
c = [7.90e-5 K/mbar] P / T² (0,0343 K/m + dT/dh)
where c = curvature of light rays calculated from atmospheric parameters pressure P in mbar, absolute temperature T in Kelvin and most importantly the temperature gradient dT/dh in °C/m.
Actually in geodesy they calculate the so called refraction coefficient k, which can be calculated using the radius of the earth R from c as follows:
k = c R
Note: refraction (c) depends mainly on the temperature gradient dT/dh. Pressure and absolute Temperature have a minor influence and the wavelength of light, humidity and CO2 concentration can be neglected in practice.
When we know the curvature of light, which is the inverse of the radius of curvature, we can use the following formula to calculate the radius of the earth by taking refraction into account:
R = (h cos(α) - 0.5 c h²) / (c h + 1 - cos(α))
where R = measured radius of the earth, h = observer altitude above the ground (sea level or plain in case of Al-Biruni), α = dip angle from horizontal to apparent refracted horizon and c = curvature of light.
There is a simpler approximation of this formula:
R = 1 / (0.5 α²/ h + c)
I derived all this formulas myself and checked them against literature and own experiments and calculations.
The formula for c was derived using Fermat‘s principle and calculus. You can find this derivation on my website, just google
Deriving Equations for Atmospheric Refraction bislins
Very nice website. Thanks for the info.
This was recommended by Bob The Science Guy. Subscribed.
Oh yeah? Well thanks Bob the Science Guy, and thank you for subscribing!
9:46 Where does this equation for K come from? Is there any resources I can use to find this, I can't find anything resembling it and I'm not sure how it's derived.
One way to get it is to take a total derivative of the left side, and plug in the Euler Lagrange equations, you will find that if L has no explicit phi dependence you get 0. Therefore it must be a constant.
I’m surprised you couldnt find this derivation. For standard lagrangian mechanics (where the independent variable is time, instead of phi) this constant is the energy.
@@physicsalmanac I figured the left side of the equation went to zero because there is no dependency on phi, that much was clear, but I'm confused on how the equation with the K came about, I'm not sure if I'm missing something(which is entirely possible, I'm not exactly well versed in Lagrangian mechanics) but if the partial of the Lagrangian with respect to r is zero, how does this result in the left side of the K equation? Everything before and after that made sense, I'm just struggling with where the left side of the K equation is coming from.
@@physicsalmanac If you have any resources I could read up on the derivation of this formula it would be greatly appreciated!
@@physicsalmanac Update, is the equation with the K the Legendre transform?
@@arthursgarage6550 The left side of the equation and the lagrangian do depend on phi, because r and r' depend on phi... but there is no explicit phi dependence in this case. Meaning there is no explicit term with phi in the lagrangian. L in general = L( r', r, phi). So the total derivative of L is:
dL/dp = DL/Dr' * dr'/dp + DL/Dr * dr/dp + DL/Dp. Where p=phi, d = total derivative, D=partial derivative. and recall r'=dr/dp. Now apply this total derivative to the left side of the equation, then substitute in the Euler-Lagrange equation, and you will see the the only term that remains is DL/Dp, which is 0 if there is no explicit dependence on phi.
Very informative presentation, thank you!
You are most welcome! Thanks for watching!
Jinkies, this is a-lot to keep track of😅
Amazing vid tho!
18:40 This really seems to undermine the whole point of this calculation. You introduce an approximation in a calculation to estimate the error in an approximation. The fact that you then go on to say that the error is independent of the height of the mountain is just mind boggling to me; yeah, it's irrelevant because you just approximated it away. If it really doesn't matter, then the whole premise of both this error analysis AND Al-Biruni's calculation is called into question. Since if the mountain isn't high enough to see enough of the curvature of the Earth, then the calculation would never work, but if the mountain is high enough and you're able to see far enough, then as your original premise stated it calls into question how much of that is distorted by the atmosphere, which does depend on the height.
Also I don't think your substitution of R = h / [sec(phi) - 1] is valid, since it assumes phi is the correct angle, and NOT the observed one distorted by atmosphere.
So A for effort dude but you really dropped the ball at the end there at the end.
The additional corrections you’re referring to, that I’ve dropped are higher order corrections. I’m only including 1st order corrections here. a*R is small, and phi is small. So sec(phi) is very close to 1 (1+ a small number). So a*R*sec(phi) = a small number plus a really small number. I’m approximating by ignoring the really small number. But you’re right that this dropped term would depend on height, since phi depends on height. You can always forgo the very last step and keep the sec(phi) if you want better precision. This would be a miner correction. Really miner it turns out. Even if you took your mountain to be Mt. Everest (the best you could do), then the error would still be about 15%.
Also the horizon is a result of the curvature of the earth. Even if our eyes can’t “see” the curve. It would not exist if the earth wasn’t curved.
The substitution of R is valid, as phi is NOT the measured angle. Alpha is what we measure. Phi is the correct geometric angle, but not the angle being measured. So it's unknown. Similar to R. R is the true radius of the earth, but not the one being measured. This is also why we take sec(phi) be 1. We don't know exactly what it is, but we do know its barely bigger than 1. So we make an approximation, to estimate are error in R measured, due to alpha differing from phi.
Good comment though. I like that your thinking about if the analysis makes sense, instead of blindly following.
Excellent work
Why thank you for saying so! 🙂
engaging but font size will be better. subscribed.
Thanks for the feedback! And for the sub of course.
Al Birunri could have never measured the dipangle accurately enough for this to be relevant anyway
It is said he arrived at a radius of 10 miles less than reality. If he did,that was by coincidence. He could never have measered the angle at that accuracy. Not even remotely
.
Great video, sorry you had to bach the guy although 11 centuries earlier and made the calculation without the calculus jargon that you had to go through, your explanation was intricate and somewhat burdensome for laymen like me, I was just waiting for you at the end to say that Elbiruni was a genius instead you said he got lucky, maybe he knew things that he couldn't explain to his peers back then.
Thanks. I meant no disrespect to Al Biruni. Even if he was aware of atmospheric refraction (it’s not entirely impossible as snells law was discovered around the time he was born), the math needed to calculate it would not be invented til 600 years later. So he did the best he could have done at the time.