Electronic Music

If you haven’t experienced any of the wonderful music courses taught by Dr. Robert Greenberg, available through The Great Courses, you’re missing a lot.  In episode 1, “The Language of Music”, of Understanding the Fundamentals of Music (Course No. 7261), Greenberg describes music not only as a language, but as what I would call a superlanguage.

Music is the ultimate language, a mega-language.  A language in which our hard-wired proclivities to use successions of pitches and sounds to communicate are exaggerated, intensified, and codified into a sonic experience capable of infinitely more expressive depth and nuance than mere words alone.

Greenberg goes on to present a definition of music that is far better than any you will find in the dictionary.

Music is sound in time, or, if you prefer, time ordered by sound.  That definition isolates the two essential aspects of music, sound and time, without any qualifications.

After defining timbre, Greenberg presents the five families of instruments in the Western musical tradition.  Aside from the human voice, they are

  1. Stringed instruments
  2. Wind instruments
  3. Brass instruments
  4. Percussion instruments
  5. Keyboard instruments

And, Greenberg states,

If this course had been written back in the 1970s or ’80s, it would have included a sixth instrumental category: electronics.  There was a genuine belief back then that digitally synthesized sound was the wave of the future.  And that an entirely new vocabulary of sound, one relevant to the technocracy of the modern world, was just around the corner.  You know what?  It never happened.  As it turned out, composers prefer to write for real people playing real instruments.  And audiences would rather listen to real people playing real instruments.  Ironically, more than anything else, digital electronics are used today to imitate those “antiquated” instruments that they were purportedly going to replace.

Though I certainly agree that electronic music will never replace natural instruments played by real people, and I hope that orchestral and chamber music will be with us centuries hence, I have no doubt that new instruments will occasionally be invented and join their venerated ranks, and that electronic music will one day garner enough respect that it will take a permanent seat as a sixth instrumental category.

The world has yet to see a composer of electronic music that can be considered on equal footing with Bach, Beethoven, Brahms, Mozart, or Mahler.  But it will happen.  Perhaps, even today, there lives a young girl or boy somewhere in the world who is already on the path towards becoming the world’s first great composer of electronic music.

Isao Tomita (1932-2016), of Japan, has arguably come the closest.  Yes, his music is idiosyncratic, and his best work a reinterpretation of existing orchestral pieces, but when you listen to Tomita at his best, you get at least a sense of what is possible within the electronic idiom.  Who wouldn’t be tempted by the ability to create any tone color or instrumental timbre imaginable?  It’s not for everyone, I know.

Here is a sampling of Tomita’s best work:

Snowflakes are Dancing (1974)

Pictures at an Exhibition (1975)

Firebird (1976)

Tomita was a pioneer.  The best is yet to come.

Windows to the Earliest: Neutrinos and Gravitational Waves

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

Thesis B7…
Neutrinos and gravitational waves will in principle allow us to peer back to much earlier times (the time of neutrino decoupling and the quantum gravity era respectively), but are much harder to observe at all, let alone in useful directional detail.  Nevertheless the latter has the potential to open up to us access to eras quite unobservable in any other way.  Maybe they will give us unexpected information on processes in the very early universe which would count as new features of physical cosmology.

The cosmic microwave background (CMB, T = 2.73 K) points us to a time 380,000 years after the Big Bang when the average temperature of the universe was around 3000 K.  But there must also exist abundant low-energy neutrinos (cosmic neutrino background, CNB, CνB, relic neutrinos) that provide a window to our universe just one second after the Big Bang during the radiation dominated era.  That’s when neutrinos decoupled from normal baryonic matter.

As the universe expanded, these relic neutrinos cooled from a temperature of 1010 K down to about 1.95 K in our present era, but such low-energy neutrinos almost never interact with normal matter.  Even though the density of these relic neutrinos should be at least 340 neutrinos per cm3 (including 56 electron neutrinos per cm3 which will presumably be easier to detect), detecting them at all will be exceedingly difficult.

Neutrinos interact with matter only through the weak nuclear force (which has a very short range), and low-energy neutrinos are much more difficult to detect than higher-energy neutrinos—if they can be detected at all.  If neutrinos have mass, then they will also interact gravitationally with other particles having mass, but this interaction is no doubt unmeasurable due to the neutrino’s tiny mass and the weakness of the gravitational force between subatomic particles.

The cosmic gravitational background (CGB) points us to the time of the Big Bang itself.  Faessler, et al. (2016) state

The inflationary expansion of the Universe by about a factor 1026 between roughly 10-35 to 10-33 seconds after the BB couples according to General Relativity to gravitational waves, which decouple after this time and their fluctuations are the seed for Galaxy Clusters and even Galaxies. These decoupled gravitational waves run since then with only very minor distortions through the Universe and contain a memory to the BB.

Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.

Faessler, A., Hodák, R., Kovalenko, S., and Šimkovic, F. 2016

Saturn at Opposition: Radiant Rings

If you look at Saturn through a telescope this Thursday morning, you’ll notice something very special about the view.  Saturn’s spectacular ring system—which is presently tilted towards us near its maximum amount—will look unusually bright and white compared with the ball of the planet.  Even though you may have looked at Saturn numerous times before and noticed that the rings are brighter and whiter than the disk of the planet (as well they should be, being composed almost entirely of water ice), you will be wholly unprepared for the view that awaits you this Thursday morning.  The rings will be positively radiant!  Why haven’t you noticed this before?

Saturn, you see, reaches opposition at 5:18 a.m. CDT on Thursday, June 15.  At the moment of opposition in Iowa County, Wisconsin, Saturn will only be 2° above the SW horizon and the Sun just three minutes before sunrise in the NE.  Best to look around 3:03 a.m. at the beginning of astronomical twilight when Saturn will be a respectable 19° above the SSW horizon.

When a superior planet (like Saturn) is at opposition, the Sun, the Earth, and the planet (in that order) form very nearly a straight line.  When we look at Saturn when it is at opposition, we see sunlight reflected off of the icy ring particles pretty much along the path the sunlight took.  Put another way, when light is shining normal (perpendicular) to a reflective surface, more light is reflected back along the normal than is scattered in other directions, so the reflection seems bright.  What’s more, since our line of sight to Saturn’s rings at opposition most closely aligns with the line of incident sunlight, we “see” no shadows from the ring particles, and the rings appear even brighter because of that.  Of course, we can’t see individual ring particles, but when sunlight strikes the rings more from the side, innumerable tiny shadows are cast by ring particles on other ring particles, and the total amount of light reflected back to us is diminished.  This phenomenon is called the opposition effect.

Saturn’s equator and ring plane are tilted 26.7° relative to its orbit around the Sun, so that means we see the rings at different angles throughout Saturn’s 29.5 year orbital period.  For example, the rings were seen edge-on in 1995.  They were tipped 26.7° to the north (meaning we had the best view of the south side of the rings and the southern hemisphere of the planet) in 2003.  The rings were again seen edge-on in 2009, and this year they are tipped 26.7° to the south, meaning we presently have the best view of the north side of the rings and the northern hemisphere of the planet.

Saturn goes through seasons just like the Earth, and the tilt of the rings reflects this.  Just as the northern hemisphere of the Earth is tilted most towards the Sun at summer solstice, Saturn’s northern hemisphere and the north side of its rings are tilted most towards the Sun when Saturn is at its summer solstice.  At the autumnal equinox on Earth, the Earth’s equator lines up with the Sun, and when Saturn is at its autumnal equinox, Saturn’s equator and its rings line up with the Sun.  At winter solstice, the southern hemisphere of the Earth is tilted most towards the Sun.  Likewise, when Saturn is at winter solstice, the south side of Saturn’s rings and its southern hemisphere are tilted most towards the Sun.  At the vernal equinox on Earth, the Earth’s equator once again lines up with the Sun.  And, when Saturn reaches its vernal equinox, Saturn’s equator and its rings once again line up with the Sun.  And so on.  The only difference is our four seasons take a year, but the four seasons on Saturn take nearly 30 years.

The tilt of Saturn’s rings would thus progress peacefully and sinusoidally with two maxima and two minima every 29.5 years—if we were observing from the Sun.  But, of course, we are observing Saturn from spaceship Earth, so the tilt of the rings that we see changes due to our position relative to Saturn in addition to Saturn’s position relative to the Sun.  This causes more rapid—albeit smaller—variations in the observed tilt angle as we orbit much faster around the Sun than Saturn does.  For example, right now we see Saturn’s rings tilted 26.6°.  On July 15, it will be 26.7°.  On August 15, 26.8°.  On September 15, 26.9°.  On October 15, 27.0°.  On November 15, back to 26.9° again.

The best time, then, to see Saturn’s rings at their most radiant is when Saturn is at opposition and when Saturn’s rings are near maximum tilt. Here on Spaceship Earth, that will next occur on Thursday, June 15, 2017.  Fortunately, Saturn’s phase angle will be just 0.1° during all of Wednesday night and Thursday morning.  Saturn rises at 8:27 p.m. Wednesday evening, June 14, and reaches 10° above the SE horizon at 9:40 p.m.  Enjoy!

Habitable Zones

One common definition of the habitable zone of a star is the range of distances from the star where liquid water could exist on the surface of a planet (where the planetary surface temperature ranges between 0° and 100° C [273.15 – 373.15 K]).

Of course, atmospheric pressure affects the temperature range for liquid water.  For example, at 3% of sea level atmospheric pressure, water boils at 26.4° C, not 100° C.  But at 68 atmospheres, water stays liquid until it reaches a scalding temperature of 285° C.  At the other end of the liquid water spectrum of temperatures, the freezing point of water only increases to 0.01° C from 1 atm all the way down to 0.006 atm.  At atmospheric pressures below 0.006 atm, liquid water can’t exist: the only phases that can be present are solid and gas.  At higher pressures, all the way up to about 99 atm, the freezing point of water remains at 0° C.  Then, from 99 atm up to 2,072 atm, the freezing point of water lowers to -21.9° C.  Then it goes back up to 0° C again at 6,241 atm.  Above 70,000 atm, H2O can exist only in solid form.

So, the range of temperature where liquid water can exist is generally smaller at lower atmospheric pressure, and greater at higher atmospheric pressure.

Substances dissolved in the water, called solutes, can also change the range of temperatures where liquid water can exist.  And, who’s to say that life couldn’t exist with only water ice or water vapor in the environment?

And what about life beneath the surface of a planet, moon, asteroid, comet, etc.?  It seems reasonable to suggest that subsurface liquid water exists on more worlds than liquid water on the surface.

And does life always require H2O to exist?

Determining the “habitable zone” of a star is complicated.  That’s why we often narrow it down to just where terrestrial life could exist.

So, for now, let’s stick with that.

As you might expect, many factors enter into the equation: some relate to the star (e.g. size and surface temperature and hence bolometric luminosity), and some relate to the planet (e.g. atmospheric composition & density, and albedo).  A liberal definition might say that the habitable zone in our solar system lies between the orbits of Venus (0.7 AU) and Mars (1.5 AU).

If one accepts this, then the calculation of the habitable zone around any other star is straightforward:


R1 is the inner radius of the habitable zone, in astronomical units
R2 is the outer radius of the habitable zone, in astronomical units
r* is the radius of the star, in solar radii
t* is the effective temperature of the star’s photosphere, in Kelvin

Here’s an example that’s made big news lately: seven planets very similar in size to the Earth have been discovered orbiting the red dwarf star TRAPPIST-1, located 39 light years from our solar system in the direction of the constellation Aquarius.  The estimated size of the star is 0.117 solar radii, and the estimated effective temperature 2559 K.  Using the above equations, we get R1 = 0.016 AU and R2 = 0.034 AU. Thus, using our approach, it appears that planets TRAPPIST-1d (0.772 R) and TRAPPIST-1e (0.918 R) are most likely to be within the star’s habitable zone.

The Sachs-Wolfe Effect

The cosmic microwave background (CMB) peaks at a wavelength of 1.9 mm and frequency 160.23 GHz, if spectral radiance is defined in terms  of frequency.  If spectral radiance is defined in terms of wavelength, then the CMB peaks at wavelength 1.1 mm.  This radiation comes from all directions, and the curve of intensity as a function of wavelength very closely approximates a perfect black body having temperature 2.725 Kelvin.  Since the Big Bang 13.8 billion years ago, the universe has expanded and cooled so that today its temperature is 2.725 K.

About 380,000 years after the Big Bang, the universe had expanded and cooled enough so that for the first time it became transparent to electromagnetic radiation.  Thus when we accurately map the exact spectrum of the cosmic microwave background in different directions, we can construct a “baby picture” of the universe when it was only 380,000 years old.

Our baby picture is not smooth but has features.  At that early time, the universe had already developed into denser regions, and less dense ones. Now, it is important to note that cosmic microwave background photons that left a denser part of the universe have been gravitationally redshifted to slightly longer wavelengths (and lower frequencies) to a greater extent than elsewhere.  This is called the Sachs-Wolfe effect.


Lots of exciting cosmological information is coming out of mapping the tiny differences in the CMB spectrum as we look in different directions. I’m wondering, though, if anyone has seen temporal variations in the CMB? In other words, if you stay pointed in a particular direction and carefully measure the CMB spectrum over time, does it change or fluctuate at all (after all sources of noise have been removed)?  Even though our current understanding of cosmology might lead us to believe that the CMB would not change fast enough for us to measure, has anybody looked?

An Open Letter to an Unknown Neighbor

We haven’t met yet.  I’m a non-confrontational kind of person (a typical Midwestern trait, I’ve heard), always eager to please and not to offend.  But I want you to know how much your dusk-to-dawn floodlight bothers me.  You see, I’m an astronomer.  I even have a backyard observatory and I would love to show you the wonders of the night sky if you’re interested in seeing what’s up there.  I’m probably the only person in Dodgeville or Iowa County doing astronomical research several nights a week, weather permitting.  I accurately time when asteroids and trans-Neptunian objects pass in front of stars, blocking their light for fractions of a second up to several seconds.  There is a lot we can learn from such events.

When I moved into my house, I had to install thick curtains in my bedroom because your bright light floods into the room all night long every night.  In fact, your light floods into every window on the west side of my house.

I like it dark at night.  It helps me to sleep better and, I’ve heard, sleeping darker is sleeping healthier.  There’s even medical research that supports this.

Being an astronomer, I like to step outside and check the night sky from time to time, look at constellations—see if the northern lights are active.  All of this is a struggle for me now.  But it doesn’t need to be.

I think I know why you want to have this light.  It seems you are trying to light the stairway from your backyard to your front yard for safety reasons when using those stairs at night.  Have you considered putting those floodlights on motion sensors instead of a dusk-to-dawn timer?  You’d save money on bulbs and electricity.  Or, if you really feel you need the light to be on all night long, a better lighting system could be installed that would light your stairs without lighting up your neighbors’ houses and yards.  Can’t afford it?  I’m not wealthy either, but I’d be more than willing to pay for the lighting improvements, because I want to be a good neighbor and having a dark backyard and house at night means that much to me.  Besides, one of the benefits of living in a small town in this beautiful area of rural southwest Wisconsin is getting a decent view of the night sky.  No big city can compete with that.

I’ll even pay for us to hire a professional lighting engineer to do the job right so both you and I (and probably your other neighbors) will be thrilled with the results.  I know enough about lighting to say confidently we will have a win-win situation.  Guaranteed.

I’m looking forward to meeting you and discussing this.  Thank you.

Small Universe

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

4.3.1 Small universes
A Small Universe: a universe which closes up on itself spatially for topological reasons, and does so on such a small scale that we have seen right round the universe since the time of decoupling.  Then we can see all the matter that exists, with multiple images of many objects occurring.  This possibility is observationally testable by examining source statistics, by observation of low power in the large angle CBR anisotropies, and by detecting identical temperature variation on various circles in the CBR sky.  There are weak hints in the observed CBR anisotropies (the lack of power on large angular scales) that this could actually be the case, but this is not solidly confirmed.  Checking if the universe is a small universe or not is an important task; the nature of our observational relationship to the universe is fundamentally different if it is true.

In 1900, Karl Schwarzschild (1873-1916) was perhaps the first to suggest the idea of a small universe topology that would lead to multiple images of the same object at different points in the past.  Though most cosmologists favor the idea of a very large universe with a simple topology, the possibility of a more complex small universe topology is still not out of the question.  The universe might be measurably finite in some or all directions.

The smaller a finite topological region of space, the easier it should be to discover multiple images of the same object at different ages (except for CMB features which will all be the same age).  The distribution of distant sources might show “patterns” that are related to more nearby sources.  A comprehensive survey of sources at redshifts between about z=2 to z=6 is still needed before any conclusions can be drawn.

Another approach, of course, is to look at patterns in the CMB temperature (intensity) and polarization.  Analyses of the most recent release of Planck satellite data, however, shows no evidence of a compact topology smaller than our visual horizon.

Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.

Luminet, J.-P. 2016,  arXiv:1601.03884v2 [astro-ph.CO]

Bonner Durchmusterung und Gaia

As our civilization and technology continue to evolve, it seems we take far too much for granted.  We neglect to consider how incredibly hard people used to work years ago to achieve results we today would pass off as almost trivial.  But history has many lessons to teach us, if only we would listen.

As an example, Prussian astronomer Friedrich Wilhelm August Argelander (1799-1875) at the age of 60 began publishing the most comprehensive star catalogue and atlas ever compiled, as of that date.  From 1852 to 1859, Argelander and his assistants carefully and accurately recorded the position and brightness of over 324,000 stars using a 3-inch (!) telescope in Bonn, Germany.  Employing the Earth’s rotation, star positions were measured as each star drifted across the eyepiece reticle in the stationary meridian telescope by carefully recording when each star crossed the line, and where along the line the crossing point was.

Stars Transiting in a Meridian Telescope

One person observed through the telescope and called off the star’s brightness as each star crossed the line, noting the exact position along the reticle on a pad with a cardboard template so that the numbers could be written down without looking away from the telescope.  A second person, the recorder, noted the exact time of reticle crossing and the brightness called out by the observer.  In this way, two people were able to record the position and brightness of every star.

Each star was observed at least twice so that any errors could be detected and corrected.  In some areas of the Milky Way, as many as 30 stars would cross the reticle each minute.  What stamina and dedication it must have taken Argelander and his staff to make over 700,000 observations in just seven years!  Argelander’s catalogue is called the Bonner Durchmusterung and is still used by astronomers even today.  It was the last major star catalogue to be produced without the aid of photography.

Like Argelander’s small meridian telescope, the European Space Agency’s Gaia astrometric space observatory is currently measuring tens of thousands of stars each minute (down to mv = 20) as they transit across a large CCD array—the modern day equivalent of an eyepiece reticle.  But instead of utilizing the Earth’s rotation period relative to the background stars of 23h56m04s, Gaia’s twin telescopes separated by exactly 106.5° sweep across the stars as Gaia rotates once every six hours.  A slight precession in Gaia’s orientation ensures that the field of view is shifted so that there is only a little overlap during the next six-hour rotation.

When Gaia completes its ongoing mission, it will have measured the positions, distances, and 3D space motions of around a billion stars, not just twice but 70 times!

Though electronic computers do most of the work these days, someone still has to program them.  Some 450 scientists and software experts are immersed in the challenging task of converting raw data into scientifically useful information.

I’d like to conclude this entry with a quotation from Albert Einstein (1879-1955), who was born and died exactly 80 years after Argelander.

Many times a day I realize how much of my outer and inner life is built upon the labors of my fellowmen, both living and dead, and how earnestly I must exert myself in order to give as much as I have received.

I love that quote.  Words to live by.

Outdoor Lighting Codes and Ordinances in Wisconsin

Last Updated: 5/22/2017

Here are all the outdoor lighting codes and ordinances in Wisconsin that I am aware of.  A big thank you to Scott Lind, PE, of Hollandale, Wisconsin for initially putting together this list in 2007!

Please post a comment or contact me via email if you have additions or updates to this list.

Blue Moundsmap





See Section 23.707 Exterior Lighting Standards

Egg Harbormap

Fontana-on-Geneva Lakemap

Fox Crossingmap

Fox Pointmap


Green Lake Countymap


See Section 4.07 Artificial Light and Glare


See Section 10.085 Outdoor Lighting



Mineral Pointmap
Is this lighting ordinance still in effect?  I cannot find it on the Mineral Point website.

See Section e Lighting Standards

New Glarusmap
See Article XVIII Exterior Lighting Plans and Standards

Oconomowoc Lakemap


Shorewood Hillsmap


See sections 9.02(7) Exterior Lighting, and 9.04(7) Exterior Lighting Plan

Sturgeon Baymap
See Section 20.12.(1)(b)12

See Section 17.0608 Lighting


Whitefish Baymap
See Section 16.31 III A2

Williams Baymap
See Section 15.03 Outdoor Lighting and Advertising Signs


The Wisconsin State Law Library maintains a comprehensive list of Wisconsin Ordinances and Codes.  This will be a good resource for us to find additional outdoor lighting codes and ordinances to be added to this list, as well as to check your local government’s codes and ordinances in general.

It is interesting to note that nearly two-thirds of these ordinances are for suburban communities in very light-polluted metro areas.  Another four ordinances are no doubt in place to help protect the Yerkes Observatory (Williams Bay, Geneva, Fontana-on-Geneva Lake, and Delavan).  Where are the rural ordinances and dark sky preserves?  Since there are very few remaining locations in Wisconsin where the night sky is truly dark, shouldn’t we be aggressively protecting those areas?  Wouldn’t it be easier to save a pristine area than to restore an almost hopelessly polluted one? Another interesting point is that upscale suburban communities are much more likely to have a lighting ordinance than more affordable communities.  Some subdivisions even exclude streetlights, but these are almost never places where most of us can afford to live.

The Hidden Universe

We continue our series of excerpts (and discussion) from the outstanding survey paper by George F. R. Ellis, Issues in the Philosophy of Cosmology.

Thesis B6: Observational horizons limit our ability to observationally determine the very large scale geometry of the universe.
We can only see back to the time of decoupling of matter and radiation, and so have no direct information about earlier times; and unless we live in a “small universe”, most of the matter in the universe is hidden behind the visual horizon.  Conjectures as to its geometry on larger scales cannot be observationally tested.  The situation is completely different in the small universe case: then we can see everything there is in the universe, including our own galaxy at earlier times.

What an intriguing idea.  If the entire universe (or the self-contained section we find ourselves in) is substantially smaller than the distance light has traveled since the universe became transparent to radiation (“decoupling”, about 380,000 years after the Big Bang), we might be able to see our Milky Way galaxy (and other galaxies) at various points in the past.

The key point here is that unless we live in a small universe, the universe itself is much bigger than the observable universe.  There are many galaxies—perhaps an infinite number—at a greater distance than the horizon, that we cannot observe by any electromagnetic radiation.  Furthermore, no causal influence can reach us from matter more distant than our particle horizon—the distance light can have travelled since the creation of the universe, so this is the furthest matter with which we can have had any causal connection.  We can hope to obtain information on matter lying between the visual horizon and the particle horizon by neutrino or gravitational radiation observatories; but we can obtain no reliable information whatever about what lies beyond the particle horizon.  We can in principle feel the gravitational effect of matter beyond the horizon because of the force it exerts (for example, matter beyond the horizon may influence velocities of matter within the horizon, even though we cannot see it).  This is possible because of the constraint equations of general relativity theory, which are in effect instantaneous equations valid on spacelike surfaces.  However we cannot uniquely decode that signal to determine what matter distribution outside the horizon caused it: a particular velocity field might be caused by a relatively small mass near the horizon, or a much larger mass much further away.  Claims about what conditions are like on very large scales—that is, much bigger than the Hubble scale—are unverifiable, for we have no observational evidence as to what conditions are like far beyond the visual horizon.  The situation is like that of an ant surveying the world from the top of a sand dune in the Sahara desert.  Her world model will be a world composed only of sand dunes—despite the existence of cities, oceans, forests, tundra, mountains, and so on beyond her horizon.

Let us now define some terms that Ellis uses above.

visual horizon – the distance beyond which the universe was still opaque to photons due to high temperature and density

particle horizon – the distance beyond which light has not yet had time to reach us in all the time since the Big Bang; our particle horizon is, therefore, farther away than our visual horizon

spacelike surface – a three-dimensional surface in four-dimensional space-time where no event on the surface lies in the past or future of any other event on that surface; every point on the surface as it exists at one instant of time

Hubble scale – a cosmological distance unit equal to the reciprocal of the Hubble constant times the speed of light; see derivation below

A reasonable value for the Hubble constant H0 is 70 km/s/Mpc.  A galaxy one megaparsec distant has a cosmological recession velocity of 70 km/s, two megaparsecs distant 140 km/s, and so on.

You may notice that there are two units of distance in H0: kilometers and megaparsecs.  We can thus rewrite H0 in units of s-1 (reciprocal seconds of time) as follows:

The Hubble time is defined as the inverse of the Hubble constant:

Converting this into more convenient units of years, we get

The Hubble scale is now simply the Hubble time multiplied by the speed of light.

Converting this into more convenient distance units of light years, and then parsecs, we get

As Ellis says, we are like ants in the Sahara desert that cannot see their Earth-universe beyond the sand dunes.  Like the ant, is there a limit to our intellect as well?

Ellis, G. F. R. 2006, Issues in the Philosophy of Cosmology, Philosophy of Physics (Handbook of the Philosophy of Science), Ed. J. Butterfield and J. Earman (Elsevier, 2006), 1183-1285.