How we know there's no counter-Earth hiding behind the Sun.

For centuries there has been speculation about another planet on the same orbit as the Earth, but hiding behind the Sun. Sometimes this is used as a plot device in science fiction or by some conspiracy theorist to explain something. The reality is we know there isn't a planet hiding behind the Sun and here's why:

First off we'd see it. I don't mean in telescopes, but it would be an unaided eye object. The reason why is quite simple and has to do with orbital mechanics. Now for an object to remain hidden behind the Sun it would have to be moving the same speed the Earth does in it's orbit. Which would work if the Earth orbited the Sun in a circle. It doesn't. Planets orbit according to Kepler's Laws of Planetary Motion the first of which is that the planets orbit the Sun in ellipses. Now the Earth's orbit is nearly circular but is elliptical enough that the difference between our closest approach to the Sun and our furthest (perihelion and aphelion) is about 5 million kilometres.

Kepler's second law basically states that the closest a planet is to the Sun, the faster it moves and the further it is from the Sun the slower it moves. So the Earth is constantly speeding up and slowing down in its trip around the Sun. A counter Earth would have to do the same. Now since the counter Earth would be opposite the Sun from the Earth, it would be at perihelion when the Earth is at aphelion and vice versa. Some quick calculations here at Castle Evil show that the difference in orbital velocity for the Earth between perihelion and aphelion is about 1 km/s (0.995 km/s for the pedants out there). This means that at part of the orbit our counter-Earth twin would be racing to catch up with us at 1km/s and then running away from us at the same speed 6 months later. Now to be visible the counter-Earth would have to be out from behind the solar disk. At 1 km/s it would take about a month for such a body to be one solar diameter away from the solar disk at perihelion/aphelion. This means that for all practical purposes a counter-Earth would only be actually hidden from us for a few months a year as it appeared to swing from side to side behind the Sun.

Since such a body would only be hidden for few months or so and would clearly be visible to the naked eye (Mars is smaller in diameter and is visible when it is farther away than a counter-Earth) most of the time. So from this alone it is safe to say there is no planet in the same orbit as the Earth hiding behind the Sun.

Even if a counter-Earth were able to stay behind the Sun for some reason, we'd still see it's effect on the orbits of other planets. As the planet Venus, for example, lapped a twin Earth, the twin's gravity would tug at Venus causing measurable perturbations in Venus' orbit. Since no such perturbations have ever been observed, it is further evidence that there just isn't a planet hiding behind the Sun.

So the lack of a visible counter-Earth and no perturbations in Venus' orbit are evidence that it's not there, but further against the idea is that it's dynamically impossible. The counter Sun point is one of five Lagrange points that exist as part of the Sun-Earth system. Of the five Lagrange points, L4 and L5 are stable. They are analogous to a valley where if you put something there it takes a fair bit of energy to dislodge it from the point so no station keeping is needed. L1, L2 and L3 on the other hand are saddle points. This is analogous to being on a ridge. Though something can stay on the ridge it doesn't take much to push it off the ridge and down the slope below. As such any object at the L1, L2 or L3 points is unstable and requires some form of station keeping to maintain it's position. The counter-Sun point is L3. This means that any planet at L3 will have problems staying there, as the gravitational perturbations of the other planets (Jupiter for instance) will tend to push an L3 planet off the point and into a different orbit. Thus not only would we see a counter-Earth, it wouldn't stay at that point for very long before it was pushed off the L3 point into a completely different orbit. Which would obviously make such a planet visible.

Of course our Earth would be in the counter-Earth's L3 point as well, making the Earth's orbit unstable for the same reasons. Since the Earth is in a stable orbit, nothing the size of the Earth is hanging out at our L3 point.

So the notion of another planet sharing our orbit about the Sun but hidden behind the Sun is an interesting idea and fodder for questionable science fiction (and even more questionable conspiracy theories) however there is nothing there. At the end of the day it's a dynamic impossibility and even if it were possible we'd see it and it's effects on the other planets in our solar system.