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Final Answers
© 2000-2014   Gérard P. Michon, Ph.D.

Sand and Stars

Nesque on porroit les estoiles nombrer
[We could never count the stars.]
Song by  Guillaume de Machaut (c.1300-1377) 

Related articles on this site:

  • The Sand Reckoner  (De Arenae Numero)  by Archimedes of Syracuse.
  • Cosmology 101:  Studying the physical Universe as a whole.
  • The Solar System: Our home star (the Sun) and what revolves around it.
  • Stars:  From brown dwarves to blue supergiants and supernovae.
  • Galaxies  and the large-scale structure of the Universe.

Related Links (Outside this Site)

Chandra Resources : Cosmology Q & A (at Harvard).
Pretty Pictures of Galaxies (USAFA : United States Air Force Academy)
On starry grains by Chris Flynn.   |   Perspectives by Evan Couche.
Religion and the Size of the Universe   |   Double Hubble by Gary Suttle
More stars than sand particles?  Google Answer by Richard-ga
To see the Universe in a Grain of Taranaki Sand  by  Glen Mackie  (May 1999)
 
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Galaxies, Stars, Sand, Particles, Photons...

There are some, King Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. [...]
Archimedes of Syracuse
(The Sand Reckoner, c. 213 BC)
The beach reminds us of space. Fine sand grains, all more or less uniform in size, have been produced from larger rocks through ages of jostling and rubbing, abrasion and erosion, again driven through waves and weather by the distant Moon and Sun [...]
      A handful of sand contains about 10,000 grains [sic; 100,000 rather], more than the number of stars we can see with the naked eye on a clear night. But the number of stars we can see is only the tiniest fraction of the number of stars that are. What we see at night is the merest smattering of the nearest stars. Meanwhile the Cosmos is rich beyond measure: the total number of stars in the universe is greater than all the grains of sand on all the beaches of the planet Earth. [sic; almost so, but beach sand may still win...]
      We know the number of stars in the Milky Way Galaxy fairly well, by careful count of stars in small but representative regions of the sky.  It is a few hundred billion; some recent estimates place it at 4´10 11 [...]  The great majority [of these stars] have lifetimes of billions or more years in which they are shining stably [...] 
Carl Sagan (1934-1996)
(Cosmos, 1980. Chapters VIII and XII)
 
Too Close to Call ?   Our current estimates of Sagan's number  (the total number of stars in the observable universe)  happens to be close to the total number of grains of sand there are on Earth; on beaches, deserts and elsewhere.

At 32 grains per mm3, 1022 grains of  medium  sand would just form a fairly thin layer over the entire surface of the Earth  (0.6 mm in thickness).


 The Andromeda Galaxy
 (M31 or NGC 224)
The Andromeda galaxy harbors
about 500 000 000 000 stars.
(2002-04-14)   Sand Reckoning
Are there more stars in the Universe than grains of sand on the beach?

Yes, but a beach is only a small amount of sand. A heap of sand with as many grains as there are stars in he Universe would be just about the size [and shape] of the Fujiyama.

 Sand from the
 Sahara desert 
 (Saqqara, Egypt)
 Scale
Sahara Sand
collected in Egypt, near the Step Pyramid of Saqqara ( the world's oldest stone structure, built around 2650 BC )
Courtesy of
Dr. Dave Douglass, Pasadena City College

Huntington Beach
 Scale
 Sand from Huntington beach, CA 
(collected under the pier)
 

First, let's consider sandSand is the disaggregated [or clastic] type of sediment whose grains [clasts] are intermediate in size between gravel (2 mm or more) and silt (2 or 4 mm to 62.5 mm).  Only the coarsest silt particles are visible to the naked eye. Clay is anything finer than silt.  Pure clay does not even feel gritty on your teeth. Mud is a wet mixture of silt and clay.

Sand Grades : vcs, cs, c/mds, s, mds, md/fs, etc.
GradeMin.Max. per mm3
Very Coarse Sand1 mm2 mm0.5
Coarse Sand0.5 mm1 mm4
Medium Sand0.25 mm0.5 mm32
Fine Sand125 mm 250 mm256
Very Fine Sand62.5 mm 125 mm2048

The above is the Udden-Wentworth grade scale based on a 1 mm standard grain size and a geometric ratio of 2 between grades. Geologists commonly call it the Wentworth scale, and it extends to coarser gravel as well as finer silt or clay.  It was proposed in 1898 by Johan A. Udden and made popular around 1922 by C.K. Wentworth. It has been the basis for the modern logarithmic f (phi) scale devised by W.C. Krumbein in 1934:

-1f  is  2 mm  1f  is  0.5 mm  3f  is  125 mm
  0f  is  1 mm 2f  is  0.25 mm 4f  is  62.5 mm   etc.

In the last column of the previous table, the number of grains per cubic millimeter was obtained by considering the densest packing [density p/Ö18, see below] of perfect spheres with diameters equal to the geometric mean of the two extremes shown. 
 Packing Spheres  
Luckily, this number turns out to be a whole number  (a power of two).  Such a "mathematical" estimate gives grain densities fairly typical of experimental data for the various sand grades.  We shall consider, therefore, that there are  32 grains  in a cubic millimeter of [medium] sand.

The tighest packing of spheres is the familiar cubic-centered lattice, whose density is equal to p/Ö18 = 0.740480489693... This much was first conjectured in 1611 by Kepler, but proven only in 1998 by Thomas C. Hales (then at the University of Michigan).

In the 1983 movie Local Hero, oil executive  Mac MacIntyre  (Peter Riegert)  is out to buy the entire Scottish fishing village of Ferness.  Only one person refuses to sell:  Old  Ben Knox  (Fulton Mackay),  a recluse who owns the local beach.  In a delightful scene,  Ben teases Mac about being "good with numbers" and offers to sell his beach for a price proportional to the number of grains of sand in a handful.  In the end,  Mac backs down from what would have been a very small price to pay for the beach...

If there are 32 grains of sand in a cubic millimeter, we have 32 000 per cubic centimeter (cc), 32 000 000 per liter, 32 000 000 000 per cubic meter.
   The Pinwheel Galaxy in Triangulum 
 (M33 or NGC 598) may contain 
 about 40 000 000 000 stars.
A cubic meter of such sand has therefore about as many grains in it as there are stars in a typical galaxy:  Our own Milky Way galaxy is larger than average; it's estimated to harbor roughly 400 000 000 000 stars, which is less than the great Andromeda galaxy (M31) but about 10 times more than the Triangulum galaxy (M33), the third largest in our "local group" of about 3 dozen galaxies.  With 30 or 40 billion stars, the Triangulum galaxy [at right] may thus be a fairly typical galaxy.

The latest estimates indicate that the total number of galaxies is at least 100 000 000 000.  There are this many cubic meters in a cube 4642 m on a side (about 3 miles). Picture such a cube of sand; it contains roughly as many grains of sand as there are stars in the Universe.  That's an impressive amount of sand.  This is an impressive Universe.

 Fujiyama Actually, a heap of dry sand cannot have a slope exceeding 34°...  With that slope, the volume of a circular cone of height h is about 2.3 h3.  Our heap could therefore resemble a large volcano culminating at 3515 m over a surrounding plain and extending 5212 m from the center in all horizontal directions.  In fact, an actual volcanic  cinder cone  (formed by dry debris called cinders, deposited near the center, rather than fluid lava)  would also have a slope hovering around  34°,  because the physics involved does not depend on grain size.  In other words, a heap of sand with as many grains as there are stars in the Universe would be just about the size and shape of the  Fujiyama  (3776 m).

Yet, the Sahara desert (the World's largest) has an area of about 9000 000 square kilometers and even this much sand would represent only a rather thin coat (about 11 mm thick) over its entire surface.

Our estimate (3.2 1021 = 3200 000 000 000 000 000 000) of the number of stars in the Universe could easily be off by a factor of 2 (in either direction), and the height of the corresponding heap of sand may then vary by 26% or more... However, we could then decide to "use" a different grade of sand, so the whole thing would still fit exactly the volume of the Fujiyama: It's such a nice mountain!


To see a world in a grain of sand
and a heaven in a wildflower,
hold infinity in the palm of your hand
and eternity in an hour.
(William Blake)


It's only very recently that we have been able to estimate with any confidence the total number of stars in the Universe.  For centuries, humanity could only observe the 6000 stars or so that are visible to the naked eye...

On the other hand, there is a distinguished history to the exercise of counting grains of sand, starting with a famous essay by Archimedes of Syracuse (c.287 BC - 212 BC), which is known by the title of The Sand Reckoner.  For Archimedes, a major hurdle was to express large numbers at a time when a proper system to do so was not yet in common use.  In fact, the main point of the essay was to introduce such a system and to convey the idea that very large numbers could be grasped and "named" with relative ease.

On the Web and in the Media :


(2002-05-08)
How many galaxies in the Universe?  How many stars?

That's a popular question, which too many people give up on.

Around 1980, one of the people who did not give up was the late Carl Sagan (1934-1996):  Sagan estimated that there are about 100 000 000 000 galaxies and that each typically harbors around 100 000 000 000 stars.  The total number of stars in the Universe would thus hover around 1022 (Sagan's Number).

The number 1022 also happens to be roughly equal to the number of molecules in a human breath and, coincidentally, also to the number of such breaths in the entire atmosphere of the Earth (there are about 1.068 1044 molecules in the atmosphere).  In the folklore of physics, this observation is often expressed by stating that each time you inhale you take in about one of the molecules from "Caesar's Last Breath"...

More than 20 years after Sagan, we are in a position to confirm his rough estimate and give a somewhat more precise number:

Let's start with our own neighborhood.  There are 33 stars whose distance from the Sun is less than 12.5 light-years.

A light-year is precisely equal to a whole number of meters, namely 9460730472580800 m or approximately 9.46073 1015 m. That's the distance traveled by light in a vacuum, at a speed of 299792458 m/s, during a "scientific year" of 31557600 s. All these numbers are exact...  In particular, "Einstein's Constant" is exactly c = 299792458 m/s, because of the latest definition of the meter, officially adopted in 1983.

From what is observed at this scale, or a slightly larger one, it's estimated that 80% of the stars are red dwarfs.  Typically, a red dwarf is ten times less massive than the Sun, and a hundred times less luminous.  Less massive (and more numerous) than red dwarves are the so-called brown dwarves, which are not stars at all, since they are not massive enough to ignite nuclear fusion in their cores (about 8% of the mass of the Sun is required for that).  Brown dwarves are typically 15 to 80 times as massive as Jupiter.  They shine by gravitational contraction [like Jupiter does, at a faint level] rather than nuclear fusion.  In spite of their large number, the total mass of all brown dwarves in the Milky Way is thought to contribute less than 0.1 % of its halo mass.

Our local group of galaxies is dominated by two large spiral galaxies:  the Milky Way, which harbors our Solar System, and the Andromeda Galaxy (M31 or NGC 224). Which of these two is larger depends on which measure you use.  The diameter of Andromeda (200 000 light-years) is about twice that of the Milky Way (100 000 light-years), but the Milky Way is much denser and turns out to have a larger mass:  The total halo mass of the Milky Way is estimated to be 3.8 1042 kg, whereas the Andromeda Galaxy is only 2.5 1042 kg (respectively 1.9 and 1.23 trillion solar masses).

The rest of the local group is not as well known as one might expect.  This is due, in part, to the fact that our own galaxy blocks our view of more than 20 % of the celestial sphere.  The blocking is less thorough with infrared light than it is for the visible part of the spectrum. This has allowed the fairly recent discovery of Galaxies behind the Milky Way, including one whose center is only 78000 light-years away, which makes it our closest neigbor yet:  It was discovered in 1994 and goes by the name of "Sagittarius Dwarf Elliptical Galaxy", or "SagDEG" (not to be confused with the Sagittarius Dwarf Irregular Galaxy, abbreviated SagDIG). The previous record holder was the prominent Large Magellanic Cloud, which is conspicuous to the naked eye from the southern hemisphere, and is located at a distance of about 179000 light-years.

The Local Group (galaxies ranked by mass)
Designation Mass
(/109 Suns)
Diameter
(/103 ly)
Stars
(/109)
Milky Way 1900100400
Andromeda (M31) 1230200500
Triangulum (M33) 2006040
  Large Magellanic Cloud   63520

The masses listed in the above table are the most recent estimates we could find for the total masses of the listed galaxies.  A large galaxy [like the Milky Way or the Andromeda Galaxy] often has a massive dark halo, which contributes to most of its mass.  The presence of such a halo is revealed by studying how the orbital speeds of stars vary with their distances from the galactic center.  Other galaxies, like the Large Magellanic Cloud (LMC), appear to have a less massive halo (a "mass to light" ratio of about 4)...

Up until April 2002, our deepest picture of the Universe was provided by two dramatic pictures from the Hubble Space Telescope (HST).  The first one was a deep view of a tiny patch of the Northern Sky obtained from 342 exposures taken with the Wide Field and Planetary Camera 2 (WFPC2) for 10 consecutive days between December 18 and 28, 1995.  It became known as the Hubble Deep Field (HDF).  A similar "picture" was taken in October 1998 for the benefit of southern observers (Hubble Deep Field South, HDF-S).  The WFPC2 used in both cases was installed  Field of view of the 
 WFPC2 instrument on the 
 Hubble Space Telescope. on the HST to correct the spherical aberration of the primary mirror; it replaces an earlier version which did not expect the aberration (hence the "2" in the denomination).

The instrument consists of 4 separate CCD cameras, each with a resolution of 800 by 800 pixels.  A splitter in the shape of a square pyramid is used, so that each of the 4 cameras may handle a quarter of the field of view.  The so-called planetary camera (PC) has a higher resolution than the other three "wide field" cameras, and thus covers a smaller patch of the Sky.  This gives the total field of view the strange "chevron" shape pictured above.  It is customary to express the resolution of a telescopic CCD camera in milli-arcsecond (mas) per pixel.  This is 45.5 mas/pixel for the planetary camera (PC) and 96.6 mas/pixel for the wide field cameras (WF2, WF3 and WF4).  800 times the angle per pixel give the angular width of the square field of view of each instrument (respectively 36.4 and 77.28 arcsec).  Expressed in steradians (sr), the entire field of view of the WFPC2 is therefore:

[(36.4)2 + 3 (77.28)2 ](p/648000)2   »   4p / 27.8 106   (»   5.345 arcmin2 )

This would be subtended by a disk roughly 0.66 mm in diameter at a distance of 0.75 m; which the media described as "a [large] grain of sand at arm's length".

In other words, the entire celestial sphere (4p sr) is about 27.8 million times larger than the field of view of WFPC2.  1686 galaxies have been found in the HDF picture (slightly less than in the subsequent HDF-S), but it's estimated that about 4500 would be detected with better sensitivity.  This guess translates into a grand total of about 125 billion (125 000 000 000) galaxies.  At cosmological distances, only 2 galaxies  (the Milky Way and Andromeda)  would be detectable by WFPC2 among the three dozens of our Local Group, so we may guess that the total number of galaxies in the observable Universe may well be 20 times as large, if smaller galaxies are to be tallied.  (Also, young galaxies may collide to form larger ones, so that galaxies are expected to be more numerous at very large distances where we observe a younger Universe.)

 Come back later, we're
 still working on this one...

In March 2002, the so-called Advanced Camera for Surveys (ACS) was installed aboard the NASA/ESA Hubble Space Telescope, in the the space vacated by the Faint Object Camera (FOC).  The ACS is an instrument with finer resolution (49 mas/pixel) than WFPC2 and a field of view (202" ´ 202") about 2.12 times as large.  The CCD detectors consist of two butted arrays of 2048 ´ 4096 pixels, each 15 mm on a side (1/10 the width of a human hair). The instrument is also about 5 times more sensitive than WFPC2, allowing deep sky observations to be completed much faster.   The Tadpole Galaxy and 
 6000 galaxies in the background.  On April 1 and 9, the newly installed ACS obtained a dramatic picture of the  Tadpole Galaxy  (UGC 10214, at a distance of 420 million light-years, in the constellation Draco) via 3 separate exposures through near-infrared, orange and blue filters.  The resulting color picture was released on April 30, 2002 and shows a background of about 6000 individual galaxies.  For a field of view about twice as large, this translates into the same density as the estimated 3000 galaxies seen in either of the "Hubble Deep Field" pictures (HDF and HDF-S) obtained with the WFPC2 in 1995 and 1998.  (Note that the ACS total exposure for the Tadpole picture was 12 times shorter than the total exposure for either WFPC2 picture.)

 Come back later, we're
 still working on this one...

Photometric redshifts may be used to obtain an overall distribution of the number N(z) of galaxies observed at a certain redshift.  From such a distribution, the number of undetected galaxies may be better estimated.

 Come back later, we're
 still working on this one...


(2002-05-29)
How many grains of sand are there on Earth?

A poet [Rolf Jacobsen] once said that "the grains of sand grow constantly in number, and the deserts are getting bigger".  At first sight, the poet seems to be telling the truth:  Everytime a grain of sand breaks, the number of grains increases by at least one (let's ignore, for now, the fact that very fine sand may become technically silt, mud or clay in the process).

On a geological timescale, however, this nice poetic observation falls short of correct accounting, for there are processes which decrease the number of grains of sands as well.  Over long periods of time, sand may become sandstone, siltstone, mudstone or shale...  Over longer periods still, the material of some of those sedimentary rocks may be slowly recycled and eventually reappear as solid rock from inside the Earth.  This is what plate tectonics eventually implies:  With the possible exception of a few zircon crystals in limited regions of some continental plates [most notably in Australia and Greenland], every microscopic grain of every rock ever observed is very much younger than the Earth itself.  The oldest sea floor, in particular, is not much older than 200 million years (less than 5 % of the age of the Earth).

Let's ignore, therefore, the poet's qualms and consider only the sand that's currently on the face of our mature Earth.  The number of grains has been just about constant for quite some time...

 Come back later, we're
 still working on this one...


(2002-05-11)
How much matter in the Universe? How many elementary particles?

The total mass of a galaxy may be estimated very precisely from the speeds of stars that orbit it at a certain distance from its nucleus.  Furthermore, the way such speeds vary with distance indicates how the mass is distributed within the galaxy.  The problem is that this mass is found to be about 10 times larger than the total mass of everything we see or guess (stars and interstellar gas or dust).  90% of the mass in or around galaxies is thus unaccounted for and has become known as dark matter.  Since obvious possible solutions to the problem (like numerous barely detectable brown dwarves) are being ruled out, some are suggesting that ordinary matter (so-called baryonic matter) is not all there is.  On the contrary, most of the stuff in the Universe might be something else which we have not yet been able to detect because of its apparent lack of interaction with everything else we see, except for the [huge] gravitational effects...  The nature of dark matter may still be unclear, but recent advances do confirm the basic fact that about 90% of the total mass in the Universe is dark matter. 

 Come back later, we're
 still working on this one...

Density of the Universe

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 (c) Copyright 2000-2014, Gerard P. Michon, Ph.D.