You are at a high spot at sunrise and point directly at the center of the sun.
Return to the same spot at noon and point again at the center of the sun.
This forms two lines in space. Where do those two lines cross?
Are they really two different lines?
Your location, tied to the dock?
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A high spot with an unobstructed view. Same spot both times.
Yes.
The earth continued its orbit around the sun between rise and noon; the two lines have the same origin (the sun) but not the same destination (the earth’s position).
Hence, these theoretical lines will cross at the sun’s center.
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What about the correction for semi-diameter?
They cross at the point where the person is standing.
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Agreed, depending on the observer. To an observer within the solar system the lines cross at the barycenter of the Sun/Earth. And, I guess, to an observer outside the solar system the probability of the lines intersecting at all is almost zero.
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Unfortunately, the barycenter of the system Sun+Earth is about 500 km off the sun’s center, which is just nothing against the sun’s radius of about 700 000 km.
Only the very distant four gas or ice giants, Jupiter to Neptune, are significant for the barycenter of the system Sun+AllPlanets.
Just now, Jupiter, Saturn and Neptune are nearly in line, and the solar system’s barycenter is 1.9 sun radii away from the sun’s center.
However, the brainteaser is about two lines of sight of the sun from a fixed point on the (moving and rotating) earth, at different times.
Not arguing that. I’m just pointing out that the intersection, to an outside observer, would not be the first person observer. It would also not be the Sun… exactly. It would be the barycenter of the two bodies (which as you correctly pointed out is within the Sun).
The original question was vague enough and teased with the word “brainteaser” that a more detailed answer was appropriate.
An even more precise answer would be that the lines would intersect between the barycenter and the Sun’s center of mass. Exactly where depends on the time of year and the first observer’s latitude on Earth. Only twice a year would that be the barycenter.
Not intuitively obvious, at least it wasn’t to me.
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The solar angle of this triangle is the equatorial parallax.
At a subsolar point, the centers of sun and earth and the viewer are in line, the angle is 0°.
At sun rise or set the angle is 8.794143 arcseconds, an IAU constant.
The nearby moon, with nearly the same angular diameter as the sun, has a much larger equatorial parallax, 0.9° to 1.03°, depending on the true distance earth<>moon.
0n the earth’s orbit around the sun, the daily angular difference is about 1°.
Taking 6 hours between sunrise and noon, this position difference would be 15 arcminutes.
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Doesn’t it depend on one’s frame of reference?
From Wikipedia Earth’s Rotation
It’s the idea that the sun travels across the sky that is misleading in this case. If we use reference #3 in the graphic to define what we mean by “point at the sun” would it be correct to not consider the earth’s orbit?
Is that a legit frame of reference? Seems like that would be the case if we use local apparent noon, which would have shifted a bit in six hours.
My guess is : That the planet has moved along its orbital path around the Sun: Then, two lines from observer to Sun meet at the centre of the Sun. A triangle might exist with the third side a curved section of the orbit.
A complication exists because the earth has turned some knowable amount on its axis. Describing this triangle might be difficult mathematically.