About how many revolutions has the Earth made around the sun?

Question

The Earth is about 4.543 billion years old.
However, the length of a year can be changed by anything from an impact to a natural cycle of orbit changes to a slow drift toward or away from the sun to the tug of a nearby body.

So while the Earth is about 4.543 billion modern years old, it may have revolved around the sun a significantly different number of times.

What is a reasonable estimate for how many revolutions - factoring in known geologic events - that the Earth has made around the Sun?

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• This question touches on the topic of orbital changes but never really answers what I'm asking here.
• The second answer on this Quora question seems to answer my question but I don't know if that source is reputable, and the actual calculations and reasoning are not given
• This reddit discussion touches on the subject but has differing / inconclusive answers

Answer 1

Since the very early solar system, there have not been large-scale movements of the planets.

In the early solar system, while the planets were still embedded in a protoplanetary disk, there were large movements. (notably the hypothesised "Grand Tack" of Jupiter) However once Jupiter reached it's current position of 5.2AU it remained there, and the Earth formed then too.

There have not been any significant changes to the Earth's orbit since the Earth formed, otherwise there would have been significant heating or cooling. So the average year length was and is about 30 000 000 seconds, and Earth has made about 4.5 billion orbits of the sun since then.

Answer 2

The configuration of the solar system is thought to have been more or less settled after the first 10-20 million years or so. However, what governs the Earth's orbital period is it's orbital angular momentum and the mass of the Sun. Two events have certainly changed the Earth's orbital period (a) whatever collision formed the Moon and (b) the continuous process of mass loss from the Sun.

Given that (a) probably happened sometime in the first tens of millions of years and likely did not alter the Earth's angular momentum greatly - it depends on the speed, mass and direction of the impactor and the amount of mass lost from the Earth-Moon system - I will ignore it.

We know a little more about (b). It seems that the mass loss from the early Sun was much greater than it is now. A review by Guedel (2007) suggests a mass loss rate over the last 4.5 billion years that increases as $t^{-2.3}$ (with considerable uncertainty on the power law index), where $t$ is time since birth, and an initial solar mass between 1% and 7% more than it is now.

Conservation of angular momentum and Kepler's third law means that $a \propto M^{-1}$ and $P \propto M^{-2}$. Therefore the Earth's orbital period was 2-14% shorter in the past.

What this considerable uncertainty means is that your question cannot be answered to three significant figures, or perhaps even two.

If the solar wind power law time dependence is very steep, then most of the mass loss occurred early, but the total mass loss would have been greater. On the other hand, a lower total mass loss implies a shallower mass loss and the earth spending a longer time in smaller orbit.

If I have time I will run through a calculation, but it seems safe to assume that the earth has executed more orbits than its age in years, but probably not by more than a few percent.

EDIT: A further consideration could be the tidal torque exerted by the Sun on the Earth-Sun orbit, which would increase the orbital separation.

Quantifying this is difficult. The tidal torque on a planet from a the Sun is $$ T = \frac{3}{2} \frac{k_E}{Q} \frac{GM_{\odot}^{2} R_{E}^{5}}{a^6},$$ where $R_E$ is an Earth radius and $k_E/Q$ is the ratio of the tidal Love number and $Q$ a tidal dissipation factor (see Sasaki et al. (2012).

These lecture notes suggest values of $k_E/Q\sim 0.1$ for the Earth and therefore a tidal torque of $4\times 10^{16}$ Nm. Given that the orbital angular momentum of the Earth is $\sim 3\times 10^{40}$ kgm$^2$s$^{-1}$, then the timescale to change the Earth's angular momentum (and therefore $a$ and $P$) is $>10^{16}$ years and thus this effect is negligible.

Answer 3

Phil Plait recently wrote a relevant article which, while partially tongue-in-cheek, does point out the difficulty in defining a year, let alone counting how many there have been. The problem is that Sol is moving relative to the stellar background so there's no absolute reference for getting "back where we started." It gets worse when you try to map years to days, since even for a static origin, a body which rotates while orbiting executes a different number of rotations depending on whether you observe from the origin or the stellar background.

Answer 4

Back of the envelope time :

Taking a rough estimate of 35 days per half billion years as the change in orbital period over time (from that Reddit page), we get the orbital period of about 365+4.5*2*35 days 4.53 billion years ago ( about 680 days ).

The average period over that time is about 522 days, so over 4.5 billion years (with those years being our current definition of year ) that's about 3.1 billion orbits.

But it's a pretty useless number, I think.

It's worth noting that we could not accurately simulate such a large number of orbits with any accuracy.