# How quickly was the Earth rotating 250 million years ago?

The Earth is slowing at a rate of $4.7\times10^{-4}$ miles per second every 100 years due to tidal forces of the moon.

See:

250 million years ago, it would have been spinning at 4.2 million miles per hour! The dinosaurs would have flown off the earth.

If the deceleration rate is inaccurate by 95% (assume it is slowing more slowly), then 250 million years ago, it would be spinning at 213,000 miles per hour.

Also, given the fact that the moon is receding from the Earth, the effect the moon has on the Earth would have been greater in the past.

I don't understand; can someone explain?

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Just out of curiosity, this sounds exactly like the kind of flawed numbers your local preacher would tell you (it sounds suspiciously like one YEC documentary I've seen some time ago). Did someone else tell you those numbers, or have you just made a simple mistake when doing the conversion? Also, be careful about the numbers in your links. Some talk about time, some talk about orbital velocity, there's angular velocity and more, those aren't all necessarily compatible. –  Luaan May 12 '14 at 14:42

The Wikipedia article you linked states:

Atomic clocks show that a modern day is longer by about 1.7 milliseconds than a century ago

If we take this change of 1.7 ms/century and multiply by 2.5 million centuries (250 million years) then we get a change of 4,250 seconds or 1.18 hours. So 250 million years ago the day length would have been 22.82 hours.

The circumference of the Earth around the equator is 40,075 km, so the speed of rotation at the equator would have been about 1,750 km/hr or about 1,092 mph. The current speed is 1,670 km/hr or about 1,040 mph.

Later:

If you're interested, the paper "Geological constraints on the Precambrian history of Earth's rotation and the Moon's orbit", Reviews of Geophysics 38 (1): 37–60, 2000, by George E. Williams discusses the day length changes since the Precambrian. There is a PDF available here. From his figure 2 the estimate of 22.8 hours 250 million years ago looks pretty close.

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...If we take this change of 1.7 ms/century and multiply by 2.5 million centuries... This assumes the rate of change itself was constant throughout this length of time, correct? –  BigHomie May 12 '14 at 13:20
@BigHomie: yes. The paper I linked suggests the rate of change was roughly constant. –  John Rennie May 12 '14 at 15:18
The rate of change depends on the tides, which depend on the configuration of the continents, so it almost certainly isn't exactly constant. But I don't think the rate of change would vary by that large a factor. –  Peter Shor May 13 '14 at 0:26
@PeterShor What about if Pangea were in existence? –  BigHomie May 13 '14 at 15:11
@BigHomie: that's the question, isn't it? Looking at maps of the current tides, it looks to me like the height really depends on resonance phenomena, so I would guess that it is heavily dependent on the shape of the oceans, and less dependent on their size. But I'd guess that unless the continents gave exactly the perfect shape, the rate of change wouldn't vary by more than 50% or so (probably less). –  Peter Shor May 13 '14 at 15:24

First of all it is a bit strange to express the change of Earths rotation in miles per second every 100 years, since the speed due to Earths rotation depends on your position on Earth. It would be better to express it as an angular deceleration, so for example in radians per second squared.

But lets assume you mean the velocity at Earths equator, which has a radius of $6378.1\ km$ ($3963.2\ miles$). However I am unable to find or derive your $4.7\times 10^{−4}$ miles per second every 100 years. It only states that:

Atomic clocks show that a modern day is longer by about 1.7 milliseconds than a century ago

Current angular velocity of Earth is $7.292115 \times 10^{−5} rad/s$, which means that its rotation period is equal to $8.616410 \times 10^{4} s$. So the average angular acceleration of Earth, $\alpha$, over the last 100 years is equal to: $$\alpha=7.292115 \times 10^{−5} - \frac{2\pi}{8.616410 \times 10^{4} - 1.7 \times 10^{-3}}={-2.0\times 10^{-12}\ rad/s}\ \text{per 100 years}$$ Which is therms of change of equatorial velocity only yields $7.9\times 10^{−9}$ miles per second every 100 years

The angular velocity, $\omega$ which would be required to make dinosaurs fly off the Earth, assuming that the Earth would not change shape much (it probably would though, but is hard calculate), is equal to: $$\omega > \sqrt{\frac{GM}{r^3}} = 1.24 \times 10^{-3}\ rad/s$$

This would make the duration of a day take less than 1 hour and 24 minutes. Using the average angular deceleration you would have to go back 58 billion years to reach those angular velocities. However Earth is only roughly 4.54 billion years old.

There is also geological and paleontological evidence that the Earth was rotating faster, namely by looking at sedimentary layers of sand and silt laid down offshore by tides:

This geological record is consistent with these conditions 620 million years ago: the day was 21.9±0.4 hours, and there were 13.1±0.1 synodic months/year and 400±7 solar days/year.

This means that the average angular acceleration of Earth the last 620 million years is equal to: $$\alpha=\left(7.292115 \times 10^{−5} - \frac{2\pi}{21.9 \times 60^2}\right)\frac{100}{620\times 10^6}={(-1.1\pm 0.2)\times 10^{-12}\ rad/s}\ \text{per 100 years}$$ This means that Earths angular acceleration has decreased (the deceleration has increased). I did not expect this, since since the Moon was closer to Earth and the rotation was bigger, which both would lead to bigger tides and thus a larger torque slowing down Earth rotation. But perhaps the lower angular velocity lead to the Earth becoming more spherical and therefore have a lower moment of inertia, so initially making it harder to slow down its angular velocity.

PS: This does makes me wonder how close the Moon was to the Earth and how short a day would have been when the earliest life forms roamed the Earth about 3.6 billion years ago.

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Neither article that is quoted shows "$4.7 \cdot 10^{-4}$ miles per second".

The Wikipedia article claims that a day grows longer by about $1.7$ milliseconds per century, that is say $86,400.0017$ instead of $86,400.0000$ seconds. Around the equator, the distance covered in a day is exactly $40,000$ Km (that's how the kilometre was initially defined). That's a change from $462.962963$ to $462.962954$ meter per second, or $9 \cdot 10^{-6}$ meter per second. A mile is $1609$ meters, so $4.7 \cdot 10^{-4}$miles per second are $0.76$ meters per second. That's about $84,000$ times what Wikipedia says.

Looks suspiciously as if someone messed up "$4.7 \cdot 10^{-4}$ miles per day" and turned it into "$4.7 \cdot 10^{-4}$ miles per second".

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I thought the meter was defined based on the length of a meridian, rather than the equator, not that it matters at all here. –  Chris White May 13 '14 at 4:26