Kelper & Tycho - how did they come to their conclusions?

ps200306

Popoutman wrote: »

When Venus is at greatest elongation from the sun, the Earth is at quadrature from Venus, but Venus can never reach quadrature from the Earth.

Was thinking about this -- is it true? Yes, Venus can never reach quadrature from earth, but is it at its greatest elongation when Earth is at quadrature from it? Here's a schematic from Wikipedia of Mars at quadrature -- I don't think the Earth is at greatest elongation from Mars's perspective ...


EDIT: Doh! Yes it is. Once the earth moves off that horizontal line it must move toward the sun from Mars's POV. Sorry -- dim moment.

ps200306

I had a "Keplerian moment" yesterday on Dun Laoghaire pier. I snapped this picture of sunrise:



The landmarks of "The Muglins" (just left of the Sun) and the Martello tower on Dalkey Island peeking above the promontory at the end of Scotsman's Bay, combined with my position half way along the east pier, allowed me to get a decent "fix":



Ok, I slightly cheated. I looked up the value of 128° on timeanddate.com. From there I also noted that on the winter solstice the direction of sunrise will be just 2° more, at 130°. On the summer solstice the value is 47°.

That means in mid-winter, the sun rises 40° south of east, but in mid-summer it is 43° north of east. How come the asymmetry? It came to me while I was thinking about this Kepler thread.

mickmackey1

Mainly because the Earth's orbit is an ellipse and not a circle, meaning its speed around the sun varies during the year. It's the same reason why the earliest sunset is Dec 14/15 but the shortest day is Dec 21.

Popoutman

The asymmetry is a geometric effect due to the sun having a finite size disk, the tilt of the apparent rising path of the sun, and effects of atmospheric refraction.

Because the apparent path of the sun is tilted, the sun rises in altitude as it moves to the right. As the sun is half a degree wide/tall, the highest point of the sun will appear earlier than the center, at an azimuth closer to north than when the center appears at the horizon.
Add to that the raising effect of the refraction due to the atmosphere, means that the sun is elevated in altitude compared to where it would be without the effects of the atmosphere. this also causes the sun to appear sooner, and appears closer to north.

The elliptical nature of the earth's orbit affects the timing of the sunrise but won't affect the geometry in the slightest - the path of the ecliptic is a flat line when viewed from the plane of the ecliptic.

mickmackey1

Yeah it was a bit of a stab in the dark I must admit

ps200306

mickmackey1 wrote: »

Mainly because the Earth's orbit is an ellipse and not a circle, meaning its speed around the sun varies during the year. It's the same reason why the earliest sunset is Dec 14/15 but the shortest day is Dec 21.

You sounded so confident, and I had already come to the same conclusion myself. Pity we're both wrong!

Yeah, I had convinced myself that the elliptical orbit, plus the fact that the solstices don't coincide with perihelion or aphelion, meant that somehow the two dates six months apart did not result in the earth's axis being quite "upright" at one of the solstices. That's wrong, as the definition of the solstice means that it must occur at the most southerly and northerly points of sunrise whenever they happen. In that case there is an asymmetry built into the dates which we indeed see -- approximately 184 days from the summer to the winter solstice, and only 181 days from winter to summer, due to perihelion occurring in January and the earth moving faster in that half of the year.

Popoutman wrote: »

The asymmetry is a geometric effect due to the sun having a finite size disk, the tilt of the apparent rising path of the sun, and effects of atmospheric refraction.

Because the apparent path of the sun is tilted, the sun rises in altitude as it moves to the right. As the sun is half a degree wide/tall, the highest point of the sun will appear earlier than the center, at an azimuth closer to north than when the center appears at the horizon.
Add to that the raising effect of the refraction due to the atmosphere, means that the sun is elevated in altitude compared to where it would be without the effects of the atmosphere. this also causes the sun to appear sooner, and appears closer to north.

The elliptical nature of the earth's orbit affects the timing of the sunrise but won't affect the geometry in the slightest - the path of the ecliptic is a flat line when viewed from the plane of the ecliptic.

That is pure genius! (Where do you learn these things? :pac: )

I hadn't realised that the angle the rising sun's ascent makes with the horizon is constant all year round for a given latitude. A couple of years back I used the opposite assumption to work out that twilight is longer at both solstices than at the equinoxes -- my answer was right, but I now realise the reasoning was wrong! For the present problem I should've checked the direction of sunrise at the equinoxes -- it's not due east at our latitude! Nor is sunset due west, they are both north of where they "should" be.

For visualising problems like these there is a collection of really great applets from U. Nebraska-Lincoln. The "motions of the sun" applet can be used to show sunrise at the solstices for any latitude. Here's the summer and winter solstice sunrises for Dublin:



Now that you've explained it, I can figure it out quantitatively too. Sunrise occurs when the top edge of the sun hits the horizon. The centre of the sun is 1/4 degree below the horizon then. But refraction lifts the sun by 7/12 degree. (Definitions and values can be found here). Add the two factors together and you get 5/6 degree. So sunrise is defined to happen when the geometric sun's centre is 5/6 degree below the horizon. Since the sun is sliding along that ascent line, that puts it further north. I drew a pic of it where, using small angle approximations, the 5/6 degrees is treated as a distance:



So at the latitude of Dublin we find the change in azimuth to be: ... close to the value inferred from timeanddate.com.