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Treebeard's Stumper Answer
11 February 2000

A Slow Sort of World

We've been doing problems and experiments with distance, velocity, and time in science class at Dunn Middle School, and we're ready for a big question! How fast are we all traveling right now as we sit quietly reading this stumper? Well, how fast compared to what? The Earth turns on its axis once a day. We travel around the Sun once a year. We're part of the Milky Way galaxy. And beyond that? It's ok to make reasonable assumptions and round off your answers, but show your work! If we're really moving that fast, why don't we notice it? Or do we?

John Tenniel's classic illustration of Alice and the Red Queen running fast to stay put in Lewis Carroll's Through the Looking-Glass and What Alice Found There (chapter 2, page 45):
'Well, in OUR country,' said Alice, still panting a little, 'you'd generally get to somewhere else--if you ran very fast for a long time, as we've been doing.'

'A slow sort of country!' said the Queen. 'Now, HERE, you see, it takes all the running YOU can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!'


We are not at rest! The Earth turns once a day at over 1000 miles/hour on the equator. We orbit the Sun at 18.6 miles/sec and the Milky Way galaxy at 140 miles/sec. We're approaching the Virgo super-cluster at 150 miles/sec, and we're zooming towards the constellation Leo at nearly 400 miles/sec relative to the cosmic microwave background radiation. The bigger the scale, the greater our speed, but everything is moving together, so we don't notice. Our occasional motions along local earthquake faults are much more noticeable!

Notes:

We don't usually notice our movement in space because everything is moving together, including the air. Similarly, I don't feel my speed in a big airplane once its cruising unless I look out the window. Relative to the ground, we're certainly moving in a plane. But relative to my seat, everything is calm unless we encounter turbulence. (I always know exactly how fast I'm driving in my old Chevy because of the shimmies and the air leaks. The new school vans are too smooth, and I have to be careful not to drive too fast.) Maybe this is what Graybear has in mind with his answer:

I'm gonna take the easy way out and say that we are not going fast in any direction. My reasoning is as follows: On the earth's surface, we are accelerated towards the earth's center by gravity, but since the surface prevents us from moving our velocity is zero. The angular velocity of several hundred miles per hour only serves to bring us back to approximately the same point every 24 hours. Likewise, as the earth is accelerated toward the sun, its momentum keeps it from 'falling' and its angular velocity only brings us back to where we were a year ago (approximately). As our solar system orbits around the center of the Milky Way galaxy, the same principles apply. The only true velocity would be how fast our galaxy is moving from the center of the universe. But the scientists say the we really are not moving away from some point, rather the entire universe is expanding.

That said, we don't notice lots of things that are there to be noticed! We're coming to know our true place in the Universe because astronomers work hard to notice small details that we would otherwise miss. I want to face this stumper head on.

So how fast are we moving right now? This isn't my usual stumper that you can solve "with just your wits." It takes some facts. Monty Python paints the big picture in The Galaxy Song as sung by Eric Idle in Monty Python's The Meaning of Life (1983). (You can download the song in MP3 format.) Did they get it right?

Whenever life gets you down, Mrs. Brown,
And things seem hard or tough,
And people are stupid, obnoxious or daft,
And you feel that you've had quite eno-o-o-o-o-ough,

Just remember that you're standing on a planet that's evolving
And revolving at 900 miles an hour,
That's orbiting at 19 miles a second, so it's reckoned,
A sun that is the source of all our power.
The sun and you and me and all the stars that we can see,
Are moving at a million miles a day
In an outer spiral arm, at 40,000 miles an hour,          <-- Not quite!
Of the galaxy we call the Milky Way.

Our galaxy itself contains 100 billion stars
It's 100,000 light years side to side.
It bulges in the middle, 16,000 light years thick
But out by us its just 3,000 light years wide
We're 30,000 light years from galactic central point,
We go 'round every 200 million years
And our galaxy is only one of millions of billions
In this amazing and expanding Universe.

The Universe itself keeps on expanding and expanding
In all of the directions it can whizz
As fast as it can go, at the speed of light you know,
12 million miles a minute, and that's the fastest speed there is.
So remember when you're feeling very small and insecure
How amazingly unlikely is your birth
And pray that there's intelligent life somewhere up in space
'Cause there' bugger all down here on earth.

Mrs Bloke: Makes you feel so sort of insignificant, doesn't it?
First Man: Yeah yeah... Can we have your liver, then?
Mrs Bloke: Yeah. All right, you talked me into it.

Here's my list of the different ways we're moving just to "keep in the same place" on Earth. I'll consider them one by one. I've made a point of showing unit conversions with unit multipliers since that's the right way, and we're practicing showing your work at school. I won't try to add links and images for every statement, but there are some good links at the end.

  1. We are moved by tectonic forces and earthquakes.
  2. The Earth spins on it's axis once a day.
  3. The Earth's axis precesses in space.
  4. The Earth moves around the barycenter of the earth-moon system.
  5. The Earth orbits the Sun once a year.
  6. The Sun moves within the Local Cluster of stars.
  7. The Local Cluster moves around the Milky Way galaxy towards the constellation Sagittarius.
  8. The Milky Way moves within the Local Group.
  9. The Local Group is moving towards the Virgo Super-Cluster.
  10. We're moving towards the constellation Leo relative to the cosmic microwave background radiation.
  11. The Universe is expanding.

1. We are moved by tectonic forces and earthquakes.

I live in Central California on the Pacific Plate, which is sliding a few inches a year north along the San Andreas fault against the North American Plate. Unfortunately, the motion is not smooth. Pressure builds and is suddenly released by major earthquakes. San Fransisco is about 300 miles north on the other side of the fault. So:

                   5280 feet   12 inch
  d = (300 miles) (---------) (-------) ~= 19,000,000 inches
                    1  mile     1 foot

      d   19,000,000 inch
  t = - = --------------- ~= 750,000 years
      v          inch
             2.5 ----
                 year
A million years is not so much by geological time!

The last large quake on the San Andreas fault close to home was the Fort Tejon earthquake on Jan. 9, 1857 with an estimated magnitude of 7.9. In the 143 years since then, the next slip has been building up:

               inch
  d = vt = 2.5 ---- x 143 years = 358 inches
               year
That's 30 feet of horizontal displacement just waiting to happen, and it's overdue! Hopefully we'll get several small quakes instead of just one big one.

We're also being carried westward away from Europe by seafloor spreading along the Mid-Atlantic Ridge. It's interesting that these slowest global motions are the most noticeable!

2. The Earth spins on it's axis once a day.

The diameter of the almost-spherical Earth is nearly 8,000 miles, so the circumference at the equator is about:

  c = pi x diameter = 3.14 x 8,000 miles ~= 25,000 miles
The Earth turns on it's axis once a day (actually once a sidereal day, 24 hours, 56 min, and 9 sec). I know we're turning from west to east because the sun and stars seem to move from east to west, and satellite TV shows from the East Coast are 3 hours ahead of schedule. Our velocity at the equator is:
       d   25,000 miles          miles
  v =  - = ------------  ~= 1000 -----
       t     24 hours            hour 

            miles   1 hour    1 min  
    = (1000 -----) (------) (------)
            hour    60 min   60 sec

          mile
    = .28 ----
          sec
Away from the equator, we still go around once a day, but the distance is less by the Cosine of the latitude. Here in Santa Barbara at latitude 34.5 degrees north, our speed is only about:
                           mile
  v' = v Cos(34.5°) ~= 825 ----
                           hour
This difference in speed over latitude results in noticeable Coriolis effects. Trade winds blow from east to west, and winter storms move from west to east. A train running north from the equator should show more wear on the eastern track. It's easier to launch satellites into orbit to the east because they are going with the Earth's spin rather than against it. That's why most satellites are launched from Cape Canaveral in Florida, with it's clear path east over the Atlantic. Vandenberg AFB near my home has a clear path to the south (because it sits at the end of the east-west Santa Ynez Mountains), so it is used to place satellites into north-south polar orbits. A Foucault pendulum doesn't just swing back and forth. These are real physical effects of the Earth's rotation.

Slam on the brakes in a car, and everything flies forwards. If the Earth were to suddenly stop in its tracks, the same thing would happen, and we'd go flying east at about a quarter mile per second. That would put me over Ojai or Ventura in about:

      d   30 miles           
  t = - = -------- = 107 seconds
      v       mile         
          .28 ----
              sec        
What a ride!

3. The Earth's axis precesses in space.

There is a small wobble in the Earth's rotation, like a spinning top. The noticeable effect is that the north celestial pole wanders through the stars in a circular path over a 26,000 year cycle. This has been known since ancient times. One noticeable effect is that monuments like the Great Pyramid of Gizeh are no longer properly aligned with the stars. It's real motion, but its not much in miles per hour.

4. The Earth moves around the barycenter of the earth-moon system.

The Earth and the Moon go around each other as we both orbit around the Sun. If the Earth weren't also moving, the Moon's orbit would be a small ellipse. It's easy to think the Moon must sometimes be moving away from the Sun in a convex path, but the Earth is moving so fast around the Sun (relatively), and the Moon is so close (relatively), that the Moon's path around the sun is always concave. This introduces another small wobble in the Earth's path around the Sun that ephemeris calculations must consider, but we can ignore.

5. The Earth orbits the Sun once a year.

I know the Earth moves around the Sun counter-clockwise (looking down at the solar system from above the north pole) because new constellations keep appearing in the east over the year. If I measure the exact moment some star appears at the same place in the sky (a sidereal day), it's about 4 minutes less than the time it takes the sun to appear at the same place (a tropical day). This is because the Earth has moved a bit in its orbit during the day so it returns to the star before the sun. The Earth is about 93,000,000 miles from the Sun, and orbits in a near circle, so:

  d = 2 pi r = 2 x 3.14 x 93,000,000 miles

      d   2 x 3.14 x 93,000,000 miles                mile
  v = - = --------------------------- =  584,000,000 ----
      t           1 year                             year

                   mile   1  year    1 day    1 hour   1  min
    = (584,000,000 ----) (-------) (-------) (------) (------)
                   year   365 day   24 hour   60 min   60 sec

           miles 
    = 18.5 -----
            sec
Our great orbital velocity does have a noticeable physical effect besides what we see in the sky over the year, but it's not easy to notice! Dr. Sten Odenwald's Ask the Astronomer Web site explains it like this:
How do we know the speed of the Earth? Because of a phenomenon called stellar aberration. If you carefully measure the location of stars in the sky during the course of a year, you will see them move in tiny ellipses with a size of about 20 seconds of arc. This is a relativistic effect which can be related to your speed relative to these stars of (20/206265) x speed of light = 30 kilometers/second. The shape of these ellipses changes with your distance above or below the plane of the ecliptic, and the only conclusion you can draw from this is that the Earth is orbiting the Sun in the plane of the ecliptic at a speed of 30 kilometers/sec. This is the only observational evidence I know about that absolutely shows that the Sun is at the center of the Solar System, and that the Earth orbits the Sun at roughly constant speed.

6. The Sun moves within the Local Cluster of stars.

I don't know what to say about this, but the Sun is not alone. The Sun moves in a complicated dance around the Milky Way with a group of neighbor stars. Some of these close neighbors are the bright stars that dominate the winter constellations of Orion, Sirius, and Procyon. Most close stars are small and dim and unfamiliar. Our Local Cluster of stars is moving together around the Milky Way, but also around each other. Astronomers know this (and everything following) by noticing the doppler shifts of light from these stars on photographic plates. That's a subtle effect of our motion in the universe, but it's there to be noticed.

7. The Local Cluster moves around the Milky Way galaxy towards the constellation Sagittarius.

Beyond the Solar System, I'm dependent on sometimes conflicting data from the Web and my library. Monty Python got this wrong too, so at least I'm in good company!

The standard answer I find on the Web is that we're moving around the Milky Way galaxy at a speed of about 250 kilometers per second. I convert that to miles per hour with unit multipliers like this:

      km         km    1000 m    100 cm   1  inch    1 foot     1  mile
  250 --- = (250 ---) (-------) (------) (-------) (-------) (---------)
      sec        sec    1 km       1 m    2.54 cm   12 inch   5280 feet

                     1000 x 100
          = 250 x ---------------- = 250 x 0.62
                  2.54 x 12 x 5280

                miles
          = 155 -----
                 sec
I won't show the full conversion from km/sec to miles/sec again, but that's the right way to do it. It's easier to just multiply by .62! (Don't get dependent on simple conversion factors. You'll never remember them when it matters!)

That's about

       mile     60 sec     60 min             mile
  (155 ----) x (------) X (------) =  600,000 ----
       sec       1 min     1 hour             hour
That's not 40,000 miles per hour as in the Monty Python song. They got it wrong!

We're about 26,000 light years from the center of the Milky Way, and it takes about 285 million years to go around. So our orbital velocity is about:

  d = 2 pi r = 2 x 3.14 x 26,000 light years

      d   2 x 3.14 x 26,000 light years           light year
  v = - = ----------------------------- =  .00057 ----------
      t         285,000,000 years                   year
A light year is the distance light travels in a year at about 186,000 miles per second. So in a year, it goes about:

             186,000 miles            365.24 day   24 hour   60 min   60 sec  
   d = vt = (-------------) (1 year) (----------) (-------) (------) (------) 
                 1 sec                  1  year     1 day    1 hour    1 min       

          = 5,870,000,000,000 miles, about 5.8 trillion miles per second
So,
              light year   5,870,000,000,000 miles    1  year     1 day    1 hour    1 min       
  v = (.00057 ----------) (-----------------------) (----------) (-------) (------) (------) 
                 year           1 light year          365.24 day   24 hour   60 min   60 sec  

      .00057 x 5,870,000,000,000
    = -------------------------- 
        365.24 x 24 x 60 x 60

          miles
    = 106 ----- 
           sec
That's at least in the ballpark of the standard 155 miles/second or 250 kilometers/second. The sun's orbit is actually elliptical, and all stars (and planets) move slower at greater distances, which may explain the difference.

8. The Milky Way moves within the Local Group.

Structures keep repeating on larger scales. The Milky Way Galaxy moves within its own Local Group of galaxies, just like the Sun moves within it's Local Cluster of stars.

9. The Local Group is moving towards the Virgo Super-Cluster.

At this point my calculations fail, and I can only repeat what I find on the Web. We're approaching the Virgo super-cluster of galaxies at about 250 kilometers/second or 150 miles/second.

10. We're moving towards the constellation Leo relative to the cosmic microwave background radiation.

We're moving at 600 kilometers/second or nearly 400 miles/second relative to the cosmic microwave background radiation. This great motion was recorded by the COBE satellite measuring the background temperature of the sky left over from the original Big Bang. That's probably the ultimate meaningful reference that we're moving relative to. Consider this the algebraic sum of all our other motions.

11. The Universe is expanding.

Here's where it gets weird. The universe itself is expanding, but there's no center of expansion! It's easy to model this. Blow up a balloon a little bit, and use a marker to put dots on it about an inch apart. Now blow it up further. Every mark is now further from every other, and the farther the marks were originally apart, the faster they separate. That's how the universe looks from Earth. More distant objects are getting even more distant faster. The rate of expansion is known as the Hubble Constant (Ho). Estimates are constantly being revised, but the value is something between 50 and 100 km/second/megaparsec. The difference between those extremes means everything in cosmology.

The universe is big, so we need big units. A parsec is about 3.2 light years. A megaparsec is 1,000,000 parsecs. So what the Hubble Constant says is that for every 3.2 million light years you look out into space, the objects there appear to be receding from you at a rate of Ho kilometers per second. If the Hubble Constant is 100, then objects appear to recede at 100 km/second for every 3.2 million light years you look out into space. Distant clusters of galaxies are hundreds or thousands of megaparsecs away, and appear to be receding from us at speeds of many thousands of kilometers per second. That's fast! But since everything is expanding, I don't think it counts as a proper motion.

I'll keep my liver, thank you!

Whew! Like the Red Queen says, "It takes all the running you can do, to keep in the same place!" But the Earth and the Solar System and the Milky Way are running for us and carrying us along. The bigger the scale, the faster we're going. But I wouldn't sell my liver for it, and I still have to watch my speed when I drive to school on the highway. It's all relative!

There's much info about our place in the Universe on the Web. Here are a few links to get started:

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Copyright © 2000 by Marc Kummel / mkummel@rain.org