# Changes to day length at solstices vs. equinoxes

by AstroDoug
Tags: equinoxes, length, solstices
 P: 6 Probably dumb question. I am trying to understand why the change in day length from day to day is greatest at the equinoxes and least at the solstices. This effect is greater at higher latitudes (N or S). I'm assuming it has to do with the tilt of the earth's axis relative to the incident sun rays, which is greatest at the solstices (when the sun "stands still"), but I'm too stoopid to work out the geometry to my satisfaction. I know that the change in day length can be described by a sine wave curve, which demonstrates this effect, but I'm trying to understand why that is the case. For example, the contribution to the equation of time from the earth's orbital speed variation can also be shown as a sine wave curve, but the explanation is given by Kepler's First and Second Laws, Newtonian dynamics, etc. You can go beyond just saying "it's a sine wave curve" and look at the underlying geometry/physics. I've looked at a lot of animations explaining the mechanism of seasonal change in day length, but I haven't found anything that particularly focuses on this aspect (why the rate of change is greatest at the equinoxes and least at the solstices). Does anyone have a reference to a relatively simple explanation (trig OK, calc gonna be a challenge), animation preferred if possible?
 Homework Sci Advisor HW Helper Thanks P: 12,873 Welcome to PF; Just to be clear - "change in day length" from what? Anyway, such effects are due to the combination of the Earth orientation at different times of the year and the rotation of the Earth. You should be able to see it by sketing out the positions at the specified times.
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Changes to day length at solstices vs. equinoxes

 Quote by Simon Bridge Welcome to PF; Just to be clear - "change in day length" from what? Anyway, such effects are due to the combination of the Earth orientation at different times of the year and the rotation of the Earth. You should be able to see it by sketing out the positions at the specified times.
Thanks for your reply. I was referring to the change in the length of daylight from day to day at the equinoxes (maximum change) and the solstices (minimal change). I think I need to improve my sketching tools/ability!
 Sci Advisor P: 6,068 A simple way to see it is to look at it as a math problem. In elementary calculus you learn that the derivative of a function = 0 at local min or max. The solstices are local min and max, so the rate of change (derivative) = 0.
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P: 12,873
 Quote by AstroDoug Thanks for your reply. I was referring to the change in the length of daylight from day to day at the equinoxes (maximum change) and the solstices (minimal change).
In that case... what mathman said.
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Sorry for the delay in responding. Thanks for your input, I will delve into the resources you've provided. Also thanks for the reality check on the responses I got on AstronomyForum.
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 Quote by mathman A simple way to see it is to look at it as a math problem. In elementary calculus you learn that the derivative of a function = 0 at local min or max. The solstices are local min and max, so the rate of change (derivative) = 0.
Thanks for your reply. The problem I'm having is to understand the underlying physical phenomena that allow us to describe change in daylight with this kind of function. For example, the equation of time has a component that is due to the elliptical orbit of the earth, and that component can be plotted separately as a sine wave. The shape of the curve is explained by Kepler's Laws and ultimately by the theory of gravity. This is the kind of explanation (different laws/theories, I guess) I am trying to get to with the issue of the variation in change in daylight. Now, I admit to only a very dim recollection of college calculus, so if your response is, "Yeah, like I said in my original post," I'll take your word for it ...
 Sci Advisor P: 6,068 The daylight variation is the result of the earth's axis change in orientation with respect to the sun, since it is fixed with respect to the stars.
 P: 6 Sorry, I'm getting a little punchy. I'm referring to the rate of change in length of daylight, in other words, why the change in length of daylight from day to day is greater at the equinoxes and less at the solstices. I know that the change is described by a sine wave curve, and this is the nature of that curve, but I am trying to figure out what the underlying physics/geometry is to explain that. This is what I was referring to in the comment on the contribution of the elliptical orbit to the equation of time. This can also be expressed as a sine wave, but there is an underlying physical reason (theory of gravity) to explain why the earth speeds up in its orbit at perihelion and slows down at aphelion. I'm looking for a similar explanation to the different rate of change in the length of daylight during the year.
 P: 46 I worked through this a few weeks back and this is how I did it (not saying it's the easiest way, but it seemed to give correct results): It helped a lot to consider the Earth fixed in space at the origin and its rotational axis as fixed vertically, then relatively the sun will follow a circle tilted slightly from the horizontal plane making one revolution per year. For half the year the sun will be below the horizontal plane, the other half above. You can then split the Earth exactly in half with a plane that is facing the sun, this tells you which half is in daylight and which is not. You can then also draw a circle of constant latitude around the Earth, the proportion of that circle that is in daylight will give you the length of your day. Using some maths you can figure out the intersection points of the plane facing the sun and the circle of constant latitude, with those you can calculate the amount of daylight. If there are no intersection points then you're in constant daylight or darkness. Addition: You can then plot the curve of daylight against time of year and you'll see the sine-curve-like line (it's not exactly a sine curve though) - this is why the rate of change of daylight is not constant.
 Sci Advisor PF Gold P: 1,382 The sun runs "fast" and "slow" during large parts so the the year. See the attached analemma image. And there are times when the slow/fast sun delta times change quickly. This in combination with your location on Earth's surface changes the times of sunrise and sunset. Add to that the political nature of time zone boundaries and it becomes confusing figuring out apparent - not actual - day length. We live by a political clock. The actual duration of light is as others have described - from and astronomical point of view. How we perceive in or day to day lives may be somewhat different. Attached Thumbnails
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 Quote by sgb27 I worked through this a few weeks back and this is how I did it (not saying it's the easiest way, but it seemed to give correct results): It helped a lot to consider the Earth fixed in space at the origin and its rotational axis as fixed vertically, then relatively the sun will follow a circle tilted slightly from the horizontal plane making one revolution per year. For half the year the sun will be below the horizontal plane, the other half above. You can then split the Earth exactly in half with a plane that is facing the sun, this tells you which half is in daylight and which is not. You can then also draw a circle of constant latitude around the Earth, the proportion of that circle that is in daylight will give you the length of your day. Using some maths you can figure out the intersection points of the plane facing the sun and the circle of constant latitude, with those you can calculate the amount of daylight. If there are no intersection points then you're in constant daylight or darkness. Addition: You can then plot the curve of daylight against time of year and you'll see the sine-curve-like line (it's not exactly a sine curve though) - this is why the rate of change of daylight is not constant.
Thanks! This is more or less what I've been trying to do, so I may be on the right track, but I think my sketching powers/powers of visualization are failing me, not to mention maths. I am going to buy a compass and protractor, really set this up, and maybe the math will be clearer. I foresee that I will be the eccentric old guy in the nursing home with his pads and pencils, trying to explain this to the other denizens.

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