How high were the tides back when the Moon was much closer to Earth?

Question

In this article it is stated that the Moon was originally much closer to Earth at a distance of 22,500 km rather than the 384,400 km it is today. What was the height of the tides back then in meters? Surely they would've been orders of magnitude higher?

Answer 1

To expand on Keith's comment, the equation for the tidal force is based upon the difference between the acceleration due to the orbiting object, and the gravity of the body itself, i.e. the difference between how hard the Moon pulls on you compared to how hard the Earth pulls. This results in an overall equation of:

$$|a_{tidal}| = 2\Delta r G \frac{M}{R^3} $$

where $\Delta r$ is the radius of the Earth (6371 km), G is the gravitational constant (6.674 x 10$^{-11} m^3kg^{-1}s^{-2}$, M is the mass of the Moon (7.342 x 10$^{22}$kg), and R is the distance between the two (here either 22,500km or 384,400km). To check our equation we can calculate the current tidal acceleration: using 384,400km, we arrive at a value of 1.099 μm/s$^2$, which is close to the accepted value of 1.10 μm/s$^2$ (2). Calculating the tidal force using the distance of 22,500km, we arrive at a value of 0.0055 m/s$^2$, or about 5,000 times the current tidal force. That could make for some massive tides, however, were there oceans?

Current research suggests that the Moon formed as the result of a major impact, around 4.5 billion years ago (3), while the first oceans formed around 4.4 billion years ago (4). This leaves a gap of around 100 million years between that 22,500km distance and the formation of the oceans, but it is close enough that the tidal force is likely not that different. However, translating the height of the tides is difficult. As the average tide is about 0.6m (5), a tide 5,000 times higher would be almost 3km high, which seems impossible. Additionally, the height of the tide appears to be more strongly affected by geography than by the tidal force, as the current range of tides is from 0-15m in height. The earliest evidence we have of tides is about 2.5 billion years ago (6), but unfortunately, the geologic record only records the pattern of the tides, not the height. We can tell that the Earth was spinning faster then, with a day 4.5 billion years ago being about 4-5 hours long, but it may have been too cold to melt the oceans.

So, to sum up: The tidal force with the Moon at that distance would be 5,000 times greater than it is today, but there may not have been oceans, and the tide height is affected more by the shape of the ocean than the tidal force itself. However, something like the water planet in Interstellar, with massive, 3,000km-high tides sweeping over the Earth every couple of hours is possible, although a boring ice planet is possible too.