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This is an inquiry related to the calculations for solar eclipses. I am pretty good with scientific concepts but not so good with the actual math. Any help is appreciated.
I know the basics of how solar eclipses cause lunar shadows ( umbra and penumbra) to fall on small portions of the Earth. I also know that the distance between the sun, moon, and earth affect these shadows. As a kid I learned that I could create an quasi-eclipse (ie: "blot out the sun") with a simple penny if I held it the right distance between my eye and the sun. This is because of something known as " apparent size" I think.
Assuming (just for the sake of simplifying the discussion) that it is possible to change the size and position of the moon without affecting gravity here is what I would like to know:
1. If I wanted to create a penumbra big enough to cover the whole Earth (well, the sun facing side of course) how big would the moon need to be? What about for an umbra that big?
2. Like the penny, would moving the moon closer or farther from the Earth help make the umbra/penumbra bigger? If so what is the optimal combination of size (smallest diameter) and distance (between the sun and earth) to put the moon so the whole planet is in shadow?
Of course I would love to see the math involved though I make no promises that I will understand it. Thanks again. 66.102.205.169 ( talk) 06:04, 27 September 2009 (UTC
Let's think about perfect circles on a piece of paper. Assume that the sun has radius s, the moon has radius m and that the centre of the sun and the centre of the moon are distance d apart. We can parametrise the perimeter of the sun by S(t) = (s.cos(t), s.sin(t)) and the perimeter of the moon by M(θ) = (d + m.cos(θ), m.sin(θ)). The tangent line to the perimeter of the sun at S(t) is given by cos(t).x + sin(t).y = s. The tangent line to the perimeter of the moon at M(θ) is given by cos(θ).x + sin(θ).y = m + d.cos(θ).
Consider the red lines in this picture. Drawing some pictures will show you that the red lines are tangent to the sun's perimeter and the moons perimeter, and that happens when t = τ and θ = τ, for some τ ∈ [0,2π). Consider the blue lines in this picture. Drawing some pictures will show you that the blue lines are tangent to the sun's perimeter and the moons perimeter, and that happens when t = τ and θ = τ + π, for some τ ∈ [0,2π).
Equating the equations of the tangent lines with the added conditions that θ = t or θ = t + π gives solutions for t. Substituting these solutions into the equations for the tangent lines to the perimeter of the sun give the following: the equations of the blue and red lines are given by
Next you work out the equation of the line passing through the centre of the sun and the intersection of the top blue and red lines. This equation is horrible and I won't write it here. This gives the centre of maximal circles fitting inside the upper-penumbra region; just make the circles bigger until they are tangent to the top blue and red lines. The bottom red line comes into play after a while and so the centres of circles will kink after a while.
The equations get more and more ugly. If you wanted an exact solution then I could give you one, but it would make little sense to a human being. ~~ Dr Dec ( Talk) ~~ 20:04, 28 September 2009 (UTC)
Well folks, the amount of mathematical discussion on what I thought was a simple question has absolutely blown me away. I will probably spend the next several weeks trying to make heads or tails of all this but I **really** appreciate the feedback. 66.102.206.234 ( talk) 03:13, 5 October 2009 (UTC)
I'm reading through a book on complex analysis, and have got stuck on one of the first problems of the text ("Introduction to Complex Analysis" by Nehari). It reads as follows:
The term "regular" may be taken to mean "differentiable". According to my following "proof", however, one has G'(z)=Re[F'(z)], where G=Re[F]. Indeed, let F(z)=u(z)+iv(z), where u and v are real functions; as F is regular on its domain D, for any z in D it must be that u and v are differentiable at z and hence F'(z)=u'(z)+iv'(z). Then one has
the above seems to hold without any need whatsoever for F'(z)=0 (which would then imply that u'(z)=v'(z)=0). Either the book is wrong or I'm missing some fundamental piece of information here; any help on ascertaining which of the two is so would be appreciated. -- Nm420 ( talk) 17:14, 27 September 2009 (UTC)
Mathematics desk | ||
---|---|---|
< September 26 | << Aug | September | Oct >> | September 28 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
This is an inquiry related to the calculations for solar eclipses. I am pretty good with scientific concepts but not so good with the actual math. Any help is appreciated.
I know the basics of how solar eclipses cause lunar shadows ( umbra and penumbra) to fall on small portions of the Earth. I also know that the distance between the sun, moon, and earth affect these shadows. As a kid I learned that I could create an quasi-eclipse (ie: "blot out the sun") with a simple penny if I held it the right distance between my eye and the sun. This is because of something known as " apparent size" I think.
Assuming (just for the sake of simplifying the discussion) that it is possible to change the size and position of the moon without affecting gravity here is what I would like to know:
1. If I wanted to create a penumbra big enough to cover the whole Earth (well, the sun facing side of course) how big would the moon need to be? What about for an umbra that big?
2. Like the penny, would moving the moon closer or farther from the Earth help make the umbra/penumbra bigger? If so what is the optimal combination of size (smallest diameter) and distance (between the sun and earth) to put the moon so the whole planet is in shadow?
Of course I would love to see the math involved though I make no promises that I will understand it. Thanks again. 66.102.205.169 ( talk) 06:04, 27 September 2009 (UTC
Let's think about perfect circles on a piece of paper. Assume that the sun has radius s, the moon has radius m and that the centre of the sun and the centre of the moon are distance d apart. We can parametrise the perimeter of the sun by S(t) = (s.cos(t), s.sin(t)) and the perimeter of the moon by M(θ) = (d + m.cos(θ), m.sin(θ)). The tangent line to the perimeter of the sun at S(t) is given by cos(t).x + sin(t).y = s. The tangent line to the perimeter of the moon at M(θ) is given by cos(θ).x + sin(θ).y = m + d.cos(θ).
Consider the red lines in this picture. Drawing some pictures will show you that the red lines are tangent to the sun's perimeter and the moons perimeter, and that happens when t = τ and θ = τ, for some τ ∈ [0,2π). Consider the blue lines in this picture. Drawing some pictures will show you that the blue lines are tangent to the sun's perimeter and the moons perimeter, and that happens when t = τ and θ = τ + π, for some τ ∈ [0,2π).
Equating the equations of the tangent lines with the added conditions that θ = t or θ = t + π gives solutions for t. Substituting these solutions into the equations for the tangent lines to the perimeter of the sun give the following: the equations of the blue and red lines are given by
Next you work out the equation of the line passing through the centre of the sun and the intersection of the top blue and red lines. This equation is horrible and I won't write it here. This gives the centre of maximal circles fitting inside the upper-penumbra region; just make the circles bigger until they are tangent to the top blue and red lines. The bottom red line comes into play after a while and so the centres of circles will kink after a while.
The equations get more and more ugly. If you wanted an exact solution then I could give you one, but it would make little sense to a human being. ~~ Dr Dec ( Talk) ~~ 20:04, 28 September 2009 (UTC)
Well folks, the amount of mathematical discussion on what I thought was a simple question has absolutely blown me away. I will probably spend the next several weeks trying to make heads or tails of all this but I **really** appreciate the feedback. 66.102.206.234 ( talk) 03:13, 5 October 2009 (UTC)
I'm reading through a book on complex analysis, and have got stuck on one of the first problems of the text ("Introduction to Complex Analysis" by Nehari). It reads as follows:
The term "regular" may be taken to mean "differentiable". According to my following "proof", however, one has G'(z)=Re[F'(z)], where G=Re[F]. Indeed, let F(z)=u(z)+iv(z), where u and v are real functions; as F is regular on its domain D, for any z in D it must be that u and v are differentiable at z and hence F'(z)=u'(z)+iv'(z). Then one has
the above seems to hold without any need whatsoever for F'(z)=0 (which would then imply that u'(z)=v'(z)=0). Either the book is wrong or I'm missing some fundamental piece of information here; any help on ascertaining which of the two is so would be appreciated. -- Nm420 ( talk) 17:14, 27 September 2009 (UTC)