What if Earth’s orbit were twice as wide?

Earth goes once around the Sun every year. That is 365 days.

What if Earth's path were twice as big? How long would one trip around the Sun take?

A year is the time Earth takes to go once around the Sun. It depends on how big Earth's path is and how fast Earth moves along it.

So what if Earth's orbit were twice as wide, with Earth twice as far from the Sun? Would a year simply be twice as long? Make a guess, then test it in the model. The answer has a surprising twist.

Orbital period depends on the size of the orbit and the mass of the central body. Planets farther from the Sun have longer years, but the relationship is not a simple proportion.

This lab doubles Earth's orbital radius while keeping it on a circular orbit, and asks why the period grows by more than a factor of two. It connects the model's result to Kepler's third law and to the real planets.

Earth is very far from the Sun: about 150 million kilometers.

It zooms around the Sun about 30 kilometers every second. One trip takes a year.

Earth orbits about 150 million kilometers from the Sun, moving at about 30 kilometers every second. Even at that speed, one lap takes about 365 days, because the path is enormous: going once around, Earth travels roughly 940 million kilometers.

In the model, Earth starts at today's distance with exactly the right speed for a circle. Its distance stays at 149.6 million kilometers, and one orbit takes 365.26 days.

NASA lists Earth's semimajor axis as 149.598 million km, its sidereal period as 365.256 days and its mean orbital speed as 29.78 km/s. The circumference of a circle of that radius is about 940 million km, and 940 million km ÷ 29.78 km/s ≈ 365 days.

The model's start reproduces this: r = 1 AU, v = √(G(M+m)/r) = 29.785 km/s, T = 365.26 days.

In our pretend lab, Olly moves Earth twice as far from the Sun.

Now Earth has a much longer path to travel. It also moves more slowly. One trip takes almost three of our years!

Choose 'twice as far'. The model puts Earth on a circle about 300 million kilometers from the Sun, with the right speed to stay on it. That speed is only about 21 kilometers per second, because the Sun's pull is weaker out there.

Now Earth has twice as far to go, at about 71 percent of the speed. One orbit takes about 1,033 days, about 2.8 times as long as today. Try three times as far: the model's year becomes about 1,898 days, more than five of our years.

At 2 AU the model's circular speed is 21.061 km/s (29.785 ÷ √2) and the period is 1,033.1 days, 2.828 times today's. At 3 AU: 17.20 km/s and 1,897.9 days (×5.20 = 3^1.5). At half the distance: 42.12 km/s and 129.1 days.

The ratios follow T ∝ r^1.5 exactly, since the path length scales as r and the speed as r^−0.5. The model keeps the starting speed at the circular value for the new distance; it does not say how Earth could be moved there.

Far from the Sun, the Sun's pull is weaker. So Earth only needs to go slowly to stay on its path.

A longer path and a slower speed make a much longer year.

Two things add up. First, a circle twice as wide is twice as long to go around. Second, the Sun's pull gets weaker with distance: at twice the distance it is only a quarter as strong. A weaker pull can only bend the path of a slower-moving planet into a circle, so Earth must move slower to stay in orbit.

Twice as far, at about 71 percent of the speed, makes the year about 2.8 times as long. Johannes Kepler spotted this pattern in the planets more than 400 years ago, before anyone knew about gravity's rules.

Setting gravity equal to the centripetal force, GMm/r² = mv²/r, gives v = √(GM/r), and T = 2πr/v = 2π√(r³/GM): Kepler's third law, T² ∝ a³. Doubling r quarters the force, reduces v by √2 and doubles the circumference, so it multiplies T by 2√2.

NASA notes that Kepler formulated the law empirically and that it helped Newton derive universal gravitation. The model's limits: two bodies, Newtonian gravity, and a circle set by hand; real planets' orbits are slightly elliptical.

Planets far from the Sun have very long years. Saturn takes about 29 of our years to go around once!

Mercury, the closest planet, takes only 88 days.

The real planets follow the same rule. Mercury, closest to the Sun, takes 88 days. Mars, about one and a half times as far as Earth, takes 687 days. Saturn, about nine and a half times as far, takes about 29 of our years.

Earth's orbit at twice the distance would sit between the orbits of Mars and Jupiter, the two planets that most known asteroids orbit between. Sunlight there would be only a quarter as strong, because light spreads out with distance. The lab 'What if Earth were farther from the Sun?' explores what that might do to Earth's temperature.

NASA's planetary fact sheet: Mercury 57.9 million km and 88.0 days; Mars 228.0 million km and 687.0 days; Jupiter 778.5 million km and 4,331 days; Saturn 1,432.0 million km and 10,747 days. Each obeys T² ∝ a³.

An orbit at 2 AU (299 million km) lies between the orbits of Mars and Jupiter, between which most known asteroids orbit (NASA). By the inverse-square law (OpenStax), sunlight there would be a quarter as intense. The climate consequences are beyond this orbit model and are explored with the greenhouse model in a related lab.