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For instance, consider the image below. The location of the planets seem to be so random relative to Earth. If the solar system was indeed flat, then I would expect the planets to lie on the same 'plane', and relative to us, it would look something like the second image. I mean, maybe Jupiter and Mars and Saturn could be on the same plane, but I fail to see how Venus would be.

What is wrong with my understanding here?

enter image description here

enter image description here

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    $\begingroup$ There is a difference between perfectly and nearly. Zoom out to see the whole sky and show the orbits (press 'o' in stellarium) $\endgroup$ – planetmaker Aug 10 at 4:27
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    $\begingroup$ The flat plane looks likea curve when projected onto the sky. $\endgroup$ – Steve Linton Aug 10 at 7:01
  • $\begingroup$ The diagram at the top of en.wikipedia.org/wiki/Celestial_coordinate_system shows the ecliptic plane (the plane of the Earth's orbit) on the celestial sphere, but I admit that it's not a great diagram. A planetarium program like Stellarium is probably more useful. $\endgroup$ – PM 2Ring Aug 10 at 9:12
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The first image is from a July article by TV meteorologist Brent Watts. Approximating that view in Stellarium:

Jul 19 AM sky, center alt 20 deg

The yellow arc is the ecliptic, the plane of the Earth's orbit around the Sun. It looks curved due to the map projection. If the view is centered on it, the ecliptic looks straight:

Jul 19 AM sky, centered on ecliptic

Unlike the Sun, which appears on the ecliptic at all times, the planets can appear a few degrees north or south of it because their orbital inclinations are all slightly different. The more another planet's orbit is inclined relative to ours, and the closer it approaches us, the more its ecliptic latitude varies.

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A plane looks like a curve when viewed on large angular scales on the sky (how could it be otherwise, think about a plane that cuts the north and south poles).

That plus the planets aren't in exactly the same plane.

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