The Speed of the Barycenter

Here is a puzzle to amuse you.  It’s a smallish question on which you might like to test your math.  My mathematical skill is low, or slow; I work things out cautiously and am in danger of missing a step.  Perhaps this is a case where I did.

John Lash is a philosopher of comparative mythology (http://www.metahistory.org/siteauthor.php).  He says he has made long use of my Astronomical Companion, is particularly interested in the motion of the Moon, and is fascinated by the thing called the barycenter.  We think of the Moon as revolving around the Earth, but, as described by me in this paragraph,

“A truer picture… is that both Moon and Earth revolve around the barycenter (center of mass) of the Earth-Moon system; and it is this barycenter that revolves in a smooth orbit around the Sun.  The Earth has 81.3 times as much mass as the Moon.  So their barycenter is 1/81.3 of the distance from the center of the Earth to the center of the Moon.  This proves to be inside the Earth, 4728 km from the center and 1650 km (1025 miles) below the surface.  There is no one particle that is the barycenter: the Earth rotates, so the barycenter keeps traveling (at an average speed of XXX km/hr) through the Earth’s mantle, staying always below the Moon.  If you happen to be in a tropical country, then at some time in the month the Moon will pass over your head, and at that moment you can tell yourself that the barycenter of the Earth-Moon system is a thousand miles under your feet, gliding through the rock at the speed of a fast jet plane.”

Detail from the Astronomical Companion illustration.  The Moon, about 60 Earth-radii away. is about three picture-widths off to the left.

“XXX” stands for the speed I gave.  John says it can’t be correct, in fact is vastly too great.  Since I didn’t keep a note of how I arrived at my figure, whereas he has devoted hours or years to pondering such matters, I’m predisposed to expect that he’s right.

These are the figures I think you need:

–The Moon goes 360 degrees around the sky, that is, returns to the same place against the starry background, in 27.32 days on average (its sidereal period).

–The Earth rotates once (that is, turns 360 degrees) in 24 hours.  Actually that’s the solar day; the sidereal day, in which a point on Earth comes around to facing the same direction among the stars, is 23.93 hours; but I think that if you use 24 your answer will at least be of the right order of magnitude.

–The Earth’s equatorial radius is 6378 kilometers.

–The barycenter is 4728 kilometers from the center of the Earth.  (Another average, varying slightly with the Moon’s distance, but that doesn’t matter.)

–The circumference of a circle is the radius multiplied by 2 multiplied by pi (3.1416).

If you want to convert kilometers to miles (as John did), divide by 1.609344.

25 thoughts on “The Speed of the Barycenter”

1. For me the tricky part of this little exercise is to conceptualize and picture how earth and moon pivot on the barycenter (BC) which defines the orbital path of the earth around the sun. Hence the velocity of the BC is practically identical to the velocity of the earth along its orbital track.

However, my interest all these years has been on the velocity of the BC mass-point within the material body of the earth, considered independently of orbital velocity. The “tunnelling” effect came to my attention through the Astronomical Companion in 1981 and I have been fascinated by it ever since.

Thanks to reading through your various responses, and with some help from the cat sitting on my head, I think I may have been able finally to picture the tunnelling and resolve this question. With the barycenter there is a double motion involved, and one tends to get in the way picturing of the other. The first type of motion I call the proper BC shift by which it is displaced through the mantle of the earth (depth approximately 1000 miles) by about 13 degrees per day, due to the moon’s revolution around the earth in 29 days. As the BC is always below the moon, the proper motion is slow, taking a full month for one full revolution.

However, due to the fact that the earth rotates on its axis beneath the moon every 24 hours, there is another shift of the BC, relative to terrestrial axial rotation. I call it the terraxial BC shift. As the BC is always below the moon, and the earth’s axial rotation brings the moon full circle around the planet each day, the BC mass-point within the earth is displaced in 24 hours. That would be the motion involving the velocity I was seeking. The velocity then would be compatible with Guy’s original figure.

I have no skill in astronomical math at all, but if that concept is correct then it is relatively easy to compute the velocity of displacement due to the rotation of the earth that moves the moon around it circumferentially each day — and so the barycenter, placed always under the moon, moves as well. The tricky part is picturing how the earth rotates on its axis causing the moon to circle it entirely in 24 hours, and simultaneously, the barycenter fixed beneath the moon tunnels through the rotating mass of the planet.

That’s as good as it gets for me. I never wanted Guy to be wrong and was not every fully convinced that he was. A student threw me off with a remark based on displacement of the BC over a 29 day period, giving a very slow velocity of shift. But that movement is independent of its displacement in 24 hours due to terraxial BC shift.

I am happy the original figure is correct as I have staked a lot on it. The fine points of calculation you have offered are excellent. I am still curious if and how the three-body problem may play into this phenomenon.

2. Hello to All

JLL checking in here. It was a real treat to connect with Guy Ottewell after using his Astronomical Companion (1981 edition) for so many years. Then he kindly invited me to this blog. Generally I don’t do blogs, simply don’t have the time, but the topic of the barycenter is a paramount interest of mind. So much so that I based a three-year-long teaching project on it, the Gaian Navigation Experiment (GNE) on gaiaspora.org. I had assumed since 1981 that the velocity of the barycenter mass-point within the earth was 1196 km/hr as Guy stated. Abut 718 mph, just below the speed of sound.

Fine, well, but around 2014 it was suggested to me that I had erred in adopting that figure. The argument was perhaps too naive: if the barycenter is always beneath the moon, and the moon goes around the earth in 29.5 days (synodic period), then the barycenter within the mantle at an average depth of 1000 miles depth also goes around the full circle in the same period of time. But as the circumference of that supposed circle — with radius of 2824 miles (radius 3824 miles less 1000) — would be about 17,775, then the velocity needed to cover this distance in 29.5 days would obviously be very low: 740 miles a day or about 30 mph.

I now believe that this calculation is spurious and wrongly-conceived. In fact I am hugely pleased that Guy’s original figure is correct and holds up, as I based the entire GNE on it! However, I still have difficulty picturing how the motion of the earth adds velocity to the barycenter as it rotates around the earth within the mantle. Of course, that concept is also spurious: the barycenter does not rotate around the center of the earth. Vice versa, right?

So let’s see if I have this straight: moon and earth both rotate around the baycenter, the point of shared mass, and the barycenter rotates around the sun. Hence the barycenter pivot carries the earth-moon coupling as a unity. Nevertheless, the barycenter pivot of shared mass somehow tunnels through the mantle of the earth, always remaining directly beneath the moon. I still fail to perceive how the barycenter acting in this manner does not inscribe an intraterrestrial/underground orbit of 18000 miles, and if so, how can its movement in that orbit be attributed with such a high velocity? I can’t picture it, although I do accept that your consensus of figures for the “velocity of the barycenter” is correct.

JLL

1. I think blog etiquette is that I keep quiet, if I can, and let others respond. If no one does, I’ll re-explain to John.
I have one other confession to make about this paragraph in the Astronomical Companion, but I won’t make it until another edition.

2. Shannon Templeton says:

The high velocity is mostly from the earth’s daily rotation, we see the moon rise and set each day. For the circumference of 18,000 miles, the barycenter travels this distance every 24 hours beneath our feet… if I am understanding your question correctly.

3. Eric David says:

I’m glad that Mr. Lash responded because I really wanted to understand the problem if it turned out that Guy’s crowd-sourced solution was wrong. Shannon nailed it in her response below. The high speed is mainly because the question was about the barycenter’s speed “through the Earth’s mantle.” Maybe that’s why there is so much geothermal energy that we could be tapping, an object is ripping through the mantle at high speed ~ it must generate a lot of friction!

3. Anthony Barreiro says:

Hmm. The Earth and the Moon are going around their common center of mass. They’re also orbiting the Sun. The Sun and the rest of the solar system are orbiting the center of the galaxy. The Milky Way is falling toward Andromeda. The Local Group is falling toward the Virgo Cluster. The Virgo Supercluster is falling toward the Great Attractor. The entire universe is expanding. So, let’s see … 42!

But seriously, I also got 1196 km/h, just as others did, and just as in the Astronomical Companion.

What is John Lash including that the rest of us are overlooking?

4. Okay, I cheated. I modeled the Sun-Earth-Moon system on Maya and had the computer calculate the speed of the rotating “camera” that I placed at the barycenter of the Earth-Moon system.

First, I modeled all orbits as circles. Second, I modeled all bodies as single points. Third, I modeled the camera as a point in space located a fixed distance away from the Earth’s point.

For all intents and purposes (or “for all intensive porpoises” as my wife would say), I got a speed that was effectively 1200km/h, which jibes with what everyone else seems to be getting. Indeed, the speed of a fast jet plane (http://hypertextbook.com/facts/2002/JobyJosekutty.shtml).

5. Eric David says:

I agree with Bruce, Richard, William, and Shannon. My answer was 1,196.1 km/hr for the speed of the barycenter through the solid Earth. I did not use the radius of the Earth in any of my calculations, however. Unfortunately, I too am prone to leaving out important steps, so my answer could be suspect.

6. Michael Barr says:

I get 1196 km/hr. I was not very precise, but all the answers around 1200 are clearly in the same ballpark. It has to be about 3/4 the tangential speed at the earth’s surface. Where on the surface? The point directly below the moon which will be near the equator but, but most of the time, not on it. Then it has to be corrected for the moon’s orbital rotation which reduces it by about 3%. The speed is not constant so too much precision is unwarranted.

7. Shannon Templeton says:

I came up with approx 1,193km/he.

Given a radius of 4728 km, the circumference will be 4728 x 2 x 3.14159 = 29,707 km.

Using a 24hr day instead of a sidereal day, 29,707 / 24 = 1,238 which is barycenter speed for a stationary moon.

Next, I calculated the barycenter speed for a stationary earth.

27.32 days is 655.7 hours.

29,707 / 655.7 = 45km/hr barycenter center speed for stationary earth.

Since they are going the same direction, the 45km/hr is subtracted from 1,238km/hr for a total barycenter speed of 1,193km/hr. Not taking into account the more accurate sidereal day, of course.

It looks like most of us are coming up with values close to 1,200km/hr.

What value was in the calendar? This has been a fun problem, thanks!

8. Howard Shaw says:

I’m with Guy on this. Relative to a person standing on the earth (“tropical country”, with the moon overhead), the main factor is the earth’s rotation. This is about 1000 miles per hour (25000 miles in 24 hours), and since the barycenter is about 3/4 of the way from the earth’s center to the surface (4728/6378), it moves at about 750 miles per hour (yes, “the speed of a fast jet plane”). The moon’s rotation around the earth / barycenter is much less significant (1/29 as much as the previous factor).

Aside to Guy: I’ve used the Astro Calendar since around 1980. Thanks for a spectacular job; you’ve enriched my life!

9. Mike Foster says:

I’m getting (first run) about 759 km/hr to the west. I assumed for the ease of mental calculation;

A month is thirty days

The radial distance of the epicenter is about 4800 km, so the circumference is about 30,000 km

So the barycenter moves 1000 km per month or about 33km per day, or about 1 1/2 km/hr, to the east.

The earth’s surface rotates at about 1040 km/h, so the barycenter would move about 760 km/hr to the west to stay under a stationary moon.

To stay under the moving moon, the barycenter then would move just under 759 km/hr to the west.

But maybe I’ve dropped a decimal point or something somewhere.

1. Mike Foster says:

Oh, I mistakenly converted to mph…

10. John Lukens says:

No, forty-two ; )

Forty-two what?
Forty-two sir!!

11. Bruce says:

I think I know where xxx (= 1196) in the astronomical companion comes from…

The circumference of the barycenter circle is 2 * pi * 4728 = 29707 km
If the moon was stationary and the earth turned in 24 hours, the barycenter point would have a speed of 29707 / 24 = 1238 km/h

But in 24 hours the moon moves and the earth has to do a little catching up. The moon’s synodic period is 29.5 days, so in one day, it moves 360 / 29.5 = 12.2 degrees, or 12.2 / 15 = 0.81 hours

The barycenter speed is then 29707 / 24.81 = 1197 km / hour. If you use 3.14 for pi, you get 1196 km/hour

12. William Murray says:

The value I get is about 1200 km/hour.

There are two motions occurring here. The motion of the barycenter as it revolves around the center of the Earth and the rotation of the Earth on its axis.

If the barycenter is 4728 km from Earth’s center it describes a circle of circumference 2 x pi x 4728 km = 29,707 km. If it moves through this distance in a period of 27.32 days ( = 656 hours ) then its velocity is:

Distance / time = 29,707 km / 656 hours = 45.2 km/hour

This would be the speed of the barycenter through the mantle if the Earth were not rotating.

The velocity of a point 4728 km from Earth’s center due to the rotation of the Earth is:

Distance / time = 29,707 km / 23.93 hours = 1241.4 km/hour

This would be the speed of the barycenter through the mantle if the moon was fixed in space above a rotating Earth.

The speed relative to a point on the Earth’s surface would be the difference of the two speeds (since the rotation of the Earth
and the revolution of the Moon about the Earth are in the same direction – toward the East)

1241.4 km/hour – 45.2 km/hour = 1196.2 km/hour

(neglecting effects such as the tilt of the Earth’s axis)

13. Richard Bentley says:

Ignoring things like seasonal factors, tilt of moon’s orbit relative to Earth’s, etc., the distance that the barycenter must travel to once again be directly under the same point on the surface of the Earth is (4728km)(2)(3.1416) = 29,707km. Since the Moon rises an average of about 50 minutes later each day, it should take the barycenter 24.83 hours to travel this distance, so the speed of the barycenter should be, on average, 29,707km / 24.83hrs = 1,196.4 kph, or 743.41 mph. Can’t wait to see other answers to find out what I am overlooking or misunderstanding. Fun puzzle! Thanks!

1. This is the simple and elegant solution I was always seeking. What struck me those many years ago was the proximity of the mean velocity of the barycenter within the earth to the speed of sound, 767 mph. Perhaps that looks like a trivial coincidence, and could well be, but I was at the time deeply immersed in Hindu Tantric cosmology which provides many fascinating insights on physics and psycho-dynamics (the co-dynamic of mind and matter) in the universe. In particular, it asserts that of the five senses — sight, touch, taste, and smell, and hearing — the latter occupies a unique category. How so?

The four secondary qualities “which affect the corresponding senses are odour, taste, colour, and the touch sense. Sound is not regarded (exclusively) as a property of the discrete sensible things there being no such thing from which sound can be entirely eliminated…” (Sir John Woodroofe, The World As Power: Power as Consciousness, p. 19.) Additional to this remarkable yet common sense observation, Hindu Tantra gives considerable attention to the factor of mass (Adrista in Sanskrit: “that which settles, remains”. The barycenter is not an object but a point of mass, a dwell-point I believe it can be called. Mass must carry sound, and so perhaps its possibility of exceeding the speed of sound might be worth noting.

In fact, I waited for over 30 years anticipating some kind of event that might happen if the barycenter persistently broke the sound barrier. I was alerted to the possibility that such an event might be happening by reports that were widely discussed ongoing from 2010 or so, concerning bizarre sounds coming from the earth. The sounds that caught my attention resembled the groaning and wracking of the massive bulkheads of a cargo ship in high seas. Having lived through a typhoon in the South China Seas I knew exactly how that sounds. Well, it occurred to me that the earth might be making such sounds due to excessive acceleration by fits and starts of the barycenter — which acceleration I attributed to interaction with systemic events especially jupiter as it crossed its perihelion, with saturn opposite in its aphelion. Hence it was march, around the moment of the Fukushima disaster and a massive supermoon that caught the attention of the entire world — that I dared to confirm the event. Subsequently I based an entire three-year learning experiment on that supposed event.

Such is my long tale of interest in the barycentered triggered by Guy’s image of tunneling. Thanks to all for helping me settle this question once and for all.

1. Anthony Barreiro says:

But John, the speed of sound depends on the medium the sound is travelling through. Increased density, rigidity, and temperature of a substance will all increase the speed at which sound moves through the substance. 767 mph is the speed of sound in air at sea level at 68 degrees F. If you had been washed overboard in the typhoon (and I am grateful you weren’t!) you would have heard sound travelling through the water at 3340 mph. Sound (e.g. from a distant earthquake or nuclear explosion) travels several times faster through the much denser, stiffer, and hotter medium of the Earth’s mantle. A quick online search gave several different values, from 13,000 mph to 29,000 mph.

I once read an article that referenced the speed of sound in the interstellar medium. It’s really really slow.

14. Shannon Templeton says:

This is a fun problem I will tackle after the long drive to the beach today. Another thing to factor in is the direction the earth rotates and the direction of the moon’s orbit – both counter clockwise as viewed from above the north pole.

1. Shannon Templeton says:

That’s *always* the correct answer! I hope you brought your towel. :)

2. Richard Bentley says:

:-)

3. John Doe says:

Head the size of the planet and here we are trying to figure out what’s going Wheeee! inside the gourd. Call that job satisfaction because I don’t…