Astronomical Question

GA

A bit embarassing but I think that's it

At least your not like my wife who one day couldn't figure out how to attach a bottle sprayer to a garden hose because she had the wrong end.

That is still my fault even though she unhooked the hose from the house then switched it 180 end to end then tried to put the spray bottle on.

Wanna hear how you turn on a vacuum cleaner or put your vehicle in gear? We have covered those as well.

In the next week or two I am expecting I will be working with her on demystifying the secrets to toilet flushing and how to get through closed doors.

Count your blessings. At least she's not trying to teach you the finer points in child birth! With her foot as the baby!

keywords "unintentional autosuggestion psychosis"
? so you can feel helpful(/required) at home ((((muhahaa))))

we've a saying that repetition is the mother of wisdom e.g. you repeat some s..t you start believing in it - is what the "keywords" about . . . there are a typical situation where when mosquito bites your S in a open air concert you bunch the nose the one's right behind(/next) to you . . . just don't [s]beep[s] complicate your life because you had nothing else to do

I didn't help her with the hose. That was the whole start of her melt down. Same with the putting her car in gear and the vacuum cleaner.

She gets herself so uptight about irrelevant things that she can't focus on whats right in front of her. Then from there proceeds to take it out on me for not taking it serious when she appears to not be able to handle a common action most 5 year old could rationalize and reason out in a few seconds.

That's a good partial answer - but a major factor is the angle that the Ecliptic (the Sun's path in the sky) makes with the horizon at sunrise and sunset. Due to the 23.5 degree tilt of the Earth's axis, this angle changes with respect to the horizon between sunrise and sunset. In the two diagrams below, notice how different the Ecliptic angle is between sunrise and sunset on the same day (today) in the northern hemisphere. (Notice the equator maintains the same angle.) This is what causes the difference in the change of sunrise vs the change in sunset.

So just to be a bit clearer:

At night the Ecliptic is 23.5 degrees above the Equator; at dawn it is 23.5 degrees below the Equator. As the Sun moves along the Ecliptic, it sets quickly at night and the setting time doesn't vary much. At dawn it grazes the horizon and 'daybreak' moves along the horizon quite a bit from day to day, changing the time of daybreak a lot.

By the way, the times change fastest near the equinoxes.

Thank you for your reply. I'm trying to visualize your explanation. When I do, I will let you know.

I don't know about you, but I need to study a 3D model of the Solar system to see the various relationships, especially when it comes to the moon...

A heliocentric 3D model would tell you something else: the Moon is not strictly a satellite of Earth. The Earth/Moon system is actually a binary planet (like binary stars). How to tell? The Moon's orbit about the Sun is everywhere convex.

I still don't get it. On the two diagrams that you provided (day and night) it looks to me as though the angle between the ecliptic plane and the celestial equatorial plane (horizontal "the horizon") is constant. Unless you mean that the instantaneous angle (points on the same celestial meridian) between ecliptic and equator are constantly varying as both are great circles? But if that were true, the path of the sun would show as a curve on the diagrams you provided.

Can you see what I am missing?

They probably are curves on those diagrams, but between the map scale and magnitutde of curvature (quite small given that the Earth's orbit is a better circle than you can draw with a compass), the curvature as shown would be negligible at best.

-See "Yahoo Weather" ;It is very unusual!!

The morning Sun rising is difference -is shorter than the Dawn difference

(Western Hemisphere-these days...), it is contradicting your statement!!

Or they have quite a big mistake...

Not only is he a good mechanic he's an astrophysicist too. And also trying to housebreak a female at the same time as well, what a guy!Duke.

The earth's orbit is not a circle but an ellipse with the sun at one of the foci. The earth is actually closer to the sun when it is winter in the northern hemisphere. When it is closer to the sun, not only is it moving faster but proportionally through a greater angle in its orbit (because the sun is closer), lengthening the day. Thus the length of the daylight varies not only due to the tilt of the earth but due to the distance from the sun.

A closely related topic is the "analemma" - look here:
http://apod.nasa.gov/apod/ap020709.html

Thanks for the suggestion. Rixter (reply 9) makes the same suggestion. Still, I don't think that the change in declination or of Greenwich Hour Angle at noon UCT explains the difference in rate at which sunrise gets earlier and sunset gets later.

Look at it this way: Imagine you are on the equator and are timing sunrises and sunsets. This takes the earth's inclination out of the equation. If the earth's orbit were perfectly circular, these would occur at 12 hour intervals throughout the year.

The earth's orbit is actually elliptical with the closest point or perihelion occurring Jan 4 this year. At perihelion, the time between the sunrise/sunset events is greater that 12 hours because the earth is moving faster in its orbit and is closer to the sun. At aphelion (in July) the interval is less than 12 hours. The time of these events occurs earlier and earlier in the day between January and July.

The rate of change is not linear. It starts out slowly in January, speeds up, then slows down by July when it starts decreasing. Its this rate of change that causes the sunrise and sunset times to not be symmetrical.

How much does the eccentricity of Earth's orbit contribute to the advance/delay (depending on the season) per day. What is the max deviation over the course of a (sidereal?) year? Let's stick to points on the equator as Earth's equatorial plane intersects the eccliptic plane (where, if Earth's orbit were perfectly circular, days and nights would each be exactly 12 [solar] hours).

There's another factor which I neglected. Sunrise and sunset are measured when the top edge of the sun clears the horizon, thus making the length of daylight on average longer than 12 hours. In addition, the atmosphere acts like a prism, bending the light and increasing the daylight still further. These effects are greater at the solstices than at equinox because the sun strikes the horizon vertically at equinox and at an angle at the solstices due to the earth's declination. There are two sinusoidal effects which add together to generate the "equation of time" mentioned earlier which relates sun time with clock time. The ellipticity of the earths orbit causes a variation of a period of 1 year. The effect of the declination causes a 6 month sinusoidal variation.

This article provides a pretty good explanation with some numbers:

http://en.wikipedia.org/wiki/Equation_of_time

Why don't we just have the Indiana Legislature set the time of sunrise and sunst? They successfully passed a bill setting pi equal to 3.0 because those other numbers were too hard to remember.

What bill was that I live here in Indiana and have not heard of that one.

In 1897 Representative T.I. Record of Posen county introduced House Bill #246 in the Indiana House of Representatives. The bill, based on the work of a physician and amateur mathematician named Edward J. Goodwin

I can scarcely imagine the sort of legislation he would have introduced had the dear doctor introduced him to Euler's identity:

e^iπ + 1 = 0

----

(that's pi in the exponent. Doesn't look much like pi in this editor's font)

Calculate eΠ - Π. If you don't get 20, something is wrong with your calculator.

But my calculator computes in base-pi. It was made in Indiana. By decree.

Mine doesn't look like pi either, and it doesn't stay up in the exponent... GRRR

Perhaps, as with my calculator, you can mitigate the problem somewhat by shoveling more coal into the stoker.

The "Equation of Time" shows the difference between clock time and sun time.

The plot of error in minutes versus time of year (black) is a combination of a 1 year period waveform (green) and a 6 month period waveform (red).

The green waveform is a result of the earth's orbit being elliptical and not circular. At perihelion, the earth is closest to the sun and moving faster in its orbit and sun time falls back. At aphelion, the earth is moving slower and is farther from the sun and the sun moves ahead.

The red curve is a little harder to explain. Sun time is determined by the direction of the sun from the earth in a direction perpendicular to the earth's axis. Imagine the earth's orbit as seen from the star Polaris, which is in the direction of the north polar axis. Since the axis is inclined at 23 degrees, a circular orbit would appear elliptical with the sun at the center.

In the same amount of time, at the solstices (short axis of ellipse) the earth moves through a larger angle (as seen from Polaris) than it does at the equinoxes (at the ends of the ellipse along the long axis). This causes the 6 month period red curve.

Thank you for your persistence. I was in the process of plotting the black curve using the nautical almanac. I was aware that it reflected Newton's equal areas proposition and hence the green curve is understandable. I did not know (and still cannot quite see) that the equation of time incorporates the red curve. I think I am having trouble visualizing sunrise and sunset from the perspective (as you suggested) of the celestial pole. I will study this tomorrow.

Thanks for your effort.

Here's another way to look at it. If you drive a stake into the ground to use as a sundial, at high noon, the shadow will point either directly north, directly south, or there will be no shadow, depending on your latitude and time of year. The north/south position of the sun doesn't matter, only the direction perpendicular to the axis. This is what you would see far above the north pole.

We've been talking about on the equator so far. I think that at higher latitudes, the effect of changing length of days has a bigger effect than the equation of time does on the asymmetry of sunrise/sunset times.

I'm a very visually-oriented person; I can look at graphs and drawings and very quickly see concepts that seldom if ever become clear from studying equations. So I took the time to make some drawings that have clarified the answer to the OP's question in my mind. The drawings are believed to be accurate for my latitude, but the concepts are the same for all latitudes. Hopefully they will be of use to others as well.

All of these drawings represent the Earth as seen with the Sun off to the left and either viewed looking along the ecliptic plane or normal (3D perpendicular) to it. The first pair show the Earth at the Northern hemisphere Summer Solstice:

Obviously the night half of the Earth is shown in grey. The North pole is tilted 23.5° toward the sun, and the daytime line (yellow) is much longer than the nighttime line (blue). The sunset local meridian is in red.

Since this is taking some time, I'll complete this in separate posts.

Continuing, here is the same set of drawings at an equinox.

The North pole is tilted away from the viewer in the Ecliptic view. The day and night lines are of equal length:

At the Northern Winter Solstice, the N pole is tilted away from the Sun, and the Day line is much shorter than the night line:

I'll finish in a third post.

Finally, here's the pair of drawings one month after the Summer Solstice:

Here the N pole is tilted only about 21° toward the Sun and roughly 11° away from the viewer in the Ecliptic view. Clearly the length of the day line is shorter than in the summer solstice view, and the night line is longer. The graphical key to my understanding is to observe that the sunset point is moving along a less curved portion of the day-night ellipse, while the sunrise point is moving along a more tightly curved portion of the same ellipse. These different curvatures lead to different rates of change. I leave it to others to do the math...

FWIW, I originally tried to submit these drawings as jpegs, but even as 300dpi jpegs, the quality was poor. The ones actually used are png images. I still wish CR4 would figure out a way to allow PDFs!

The magic formula for calculating the amount of daylight is:cos (θ) = tan(δ) x tan(ø)

θ (theta) is the angle between between solar noon and the sunset or sunrise line

δ (d) is the declination angle (angle of sun above or below equator)

ø (phi) is the latitude

The diagram above shows where this formula comes from.The earth is shown with the equator horizontal and the sun above the equatorial plane at angle (d). We assume the earth radius = 1. The observer is at the latitude (phi). Latitude phi is a circle of distance sin(phi) above the equator with radius cos(phi). The terminator from this view is at an angle d from the vertical. The distance X in the figure is then sin(phi) x tan(d). On the latitude circle, X = cos(phi) (which is the radius of the circle) x cos(theta). Dividing both sides by cos(phi) gives the formula.

Thanks. It took me 5 or 6 times back and forth between the text and the sketch to truly follow it, but I do agree, at least at the solstices.

On the other hand, to me it is an incomplete solution, because it is not solved for θ, and certainly not for time. Of course θ= arccos(tan(δ)*tan(Φ)), and T=12*θ/π.

Now on the third hand, it can't be a completely precise solution, because it implies that the time from sunrise to solar noon is exactly the same as the time from solar noon to sunset, and I'm pretty sure that is not usually true (which is why the original question was asked).

There are two solutions for Θ, one positive and the other negative which correspond to sunset and sunrise. The trick is that δ, the sun's declination, changes during the year changing a little bit between each sunrise/sunset.

If the earth's orbit were perfectly circular, the earth would move through the same number of degrees per day and δ would be a sine curve with an amplitude of 23.45 degrees and a period of 1 year. Since the earth's orbit is an ellipse, there are various calculations that need to be made to determine where the earth is in its orbit at each sunset/sunrise and to calculate δ. These messy calculations are shown in:

http://en.wikipedia.org/wiki/Sunrise_equation

Is it legitimate to substitute sunrise/sunset with the terminator?

Good point. There are two additional factors here... Sunrise and sunset are defined as when the top limb of the sun touches the horizon, so daylight at both ends will be extended by the radius of the sun (about 1/4 degree) divided by cos(δ). (δ is the declination, between -23.45 and +23.45 degrees). The second factor is atmospheric refraction which extends daylight by about .8 degrees divided by cos(δ) on both ends. (When you see the sun touch the horizon, it is already below the horizon.) So both of these corrections added together make the actual sunrise earlier and sunset later.

Well I believe that the Rixter/Warner Hypothesis has resolved this matter. Thanks and GA to both of you for your effort. I had a shot at assigning values but quickly developed a serious nosebleed. Still, I may have advanced the science of geometrics with what I think of as the "Mercator-ish Projection.

I've been trying to more fully understand the apparent motions of the Sun and Moon myself. What you've shown is a good start, but unless I'm mistaken, it is only correct for points on the equator, and does not account for the latitude of the viewer's location on Earth. After all, locations near the poles have no sunrises for part of the year.

I have to agree with you that our analysis so far only applies at the equator. At higher (and lower) latitudes, as the length of daylight is changing over time, the rate of change over a day would skew the sunrise/sunset times to be asymmetrical.

You can go to the following site and print a calendar for latitude=0 and then latitude=60. At the higher latitude there is much more asymmetry in the sunrise/sunset times. I would have to say that this is likely the main factor in the asymmetry at most latitudes.

http://aa.usno.navy.mil/data/docs/RS_OneYear.php

This site also has some good information:

http://en.wikipedia.org/wiki/Sunrise_equation