Cosmologically Distant Objects Appear Magnified

George F. R. Ellis writes in section 2.3.3 of his outstanding survey paper, Issues in the Philosophy of Cosmology:

…there is a minimum apparent size for objects of fixed physical size at some redshift zc = z depending on the density parameter and the cosmological constant.  The past light cone of the observer attains a maximum area at z; the entire universe acts as a gravitational lens for further off objects, magnifying their apparent size so that very distant objects can appear to have the same angular size as nearby ones.  For the Einstein-de Sitter universe, the minimum angular diameter is at z = 1.25; in low density universes, it occurs at higher redshifts.

Electromagnetic radiation such as visible light that we observe from a source that is in motion relative to us (the observer) experiences a change in wavelength that is given by

This is called redshift and is positive for a source that is moving away from us and negative for a source that is moving towards us.  The higher the relative speed toward or away from us, the greater the magnitude of the redshift.  Superimposed upon the kinematic velocities of individual galaxies relative to our Milky Way galaxy, since 1929 we have known that there is a cosmological redshift (called the Hubble flow) that is always positive and increasing in magnitude with increasing distance between any two galaxies.  In the nearby universe, the redshift (or blueshift) from kinematic velocities (often referred to as “peculiar velocities”) swamp the contribution from the Hubble flow, so some galaxies are actually approaching each other.  A good example of this is M31 and the Milky Way galaxy.  For more distant galaxies, however, the cosmological redshift swamps any contribution from the kinematic velocities.  Thus, redshift becomes a useful proxy for distance at cosmological distances.

From our everyday experience, we know that the further away an object is, the smaller is its angular size.  However, there comes a point where the angular size of an object reaches a minimum, and at even greater distances, its angular size increases!  As George Ellis states above, the entire universe acts as a gravitational lens to magnify distant objects.

Michael Richmond presents an equation for angular size as a function of redshift (based on some classical assumptions about the structure of the universe).  In his equation, the angular size of an object also depends upon the value we choose for H0, the Hubble constant, the matter density parameter, ΩM, and, of course, the physical size of the object of interest.

Let’s work through an example using this equation.  The visible part of the Andromeda Galaxy is estimated to be about 220,000 light years across.  In megaparsecs, that is 0.0675.  This is the value we will use for S.

For the Hubble constant, H0, let use a recent result: 71.9 km/s/Mpc.

And, for the matter density parameter, ΩM, let’s use 1.0.  This indicates that we live in a universe that has just enough matter for the universe to eventually recollapse, were it not for dark energy.  Though Richmond’s equation above only applies to a matter-dominated universe where the dark energy density parameter ΩΛ is zero, as George Ellis indicates above, a minimum angular diameter is still reached in a universe with dark-energy (i.e. low density universe), only this occurs at a higher redshift than that presented here.

I have not been able to find or derive a more general equation for angular size as a function of redshift that will work for a dark-energy-dominated universe (perhaps a knowledgeable reader will post a comment here providing some insight into this issue), but it will be a useful exercise to continue with the calculation assuming the matter-dominated Einstein-de Sitter universe.

Casting Michael Richmond’s equation into the following SAS program, I was able to find that the Andromeda galaxy would reach a minimum angular size of 11.3 arcseconds at z = 1.25, as shown below.

In principle, measuring the angular size of a “standard” object at various redshifts could allow us to determine what kind of universe we live in.  But there’s a problem.  As we look further out into space we are also looking further back in time, so there is no guarantee that a “standard” object in today’s universe (say, a spiral galaxy such as M31) would have looked the same or even existed billions of years ago.

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.

Richmond, Michael, Two classic cosmological tests

M31, M33, and the Milky Way

We live within a small cluster of at least 54 galaxies (most of them small), given the unassuming name “The Local Group” by Edwin Hubble (1889-1953) in 1936. The largest of these, M31 (the Andromeda Galaxy) is a barred spiral galaxy 2.5 million light years from Earth.  The second largest galaxy in the Local Group is our Milky Way, a barred spiral galaxy whose center lies about 26,000 light years away in Sagittarius.  The third largest galaxy in the Local Group is M33 (the Triangulum Galaxy), a spiral galaxy (possibly barred) located 3.0 million light years from Earth.  There are no other spiral galaxies in the Local Group.

M31 and our Milky Way are moving towards each other, and will pass through one another (or at least graze: shall we call it star grazing?) in about 4 billion years.

M33, however, is only about 860,000 light years from M31.  Isn’t M33 in even greater danger of colliding with M31?  The answer is no, we don’t think so, because M33 appears to be orbiting M31.  M33’s eventual role in the Battle of the Titans remains to be determined.

Gentle Giant?

Pollux (Beta Geminorum) is the nearest giant star to Earth, between 33.7 and 33.9 light years away.  Its spectral type, K0III, indicates its photosphere is cooler than the Sun’s.  Relative to our Sun, Pollux is 8.8 times wider, 2.0 times more massive, and 43 times more luminous.  Many giant stars are larger than Pollux.

Beginning its life as an A-type main sequence star, but now evolved to a K-type giant, Pollux is only about 724 Myr old.

Pollux has the current distinction of being the brightest star in the night sky known to harbor at least one planet: a super-Jupiter 2.9 times as massive as Jupiter, and orbiting Pollux every 590 days at a distance of 1.7 AU.  The planet’s name is Thestias.

If you have trouble remembering which star is Castor and which star is Pollux in Gemini, here’s an easy way to remember: Castor sides with Capella, and Pollux sides with Procyon.  Another way to tell: Pollux is half a magnitude brighter than Castor.

Aldebaran Crossing

On the evening of March 4, 2017 (5 Mar 2017 UT), the Moon passed in front of the 0.9-magnitude star Aldebaran (α Tau).  Currently, this is the brightest star the Moon ever crosses (excepting the Sun, of course).  The favorable first quarter moon (with the nearside 45.9% illuminated) was at a respectable altitude of 31° in the western sky at the time of the dark-limb disappearance.  It is not an instantaneous event.  The effects of diffraction plus the non-zero angular size of the star ensure that the star disappears over several frames of a high-speed camera, as shown in my video recording of the event below.

At my location, the middle of Aldebaran’s disappearance occurred at 3:56:00.570 ± 0.001s UT.  Thin clouds and wind interfered with a pristine recording of the event, but even so you can see in the trace below that the star took about nine frames (~0.017s each) to completely disappear.

The bright limb reappearance was more difficult to time.  In my video recording below, you can see that Aldebaran first reappears at about 4:29:39.828 UT.  By that time, the Moon had descended to an altitude of 25° and was accompanied by a fair amount of atmospheric turbulence.

All in all, I felt very lucky to have observed both of these events.  Soon afterwards, it completely clouded up!

The Nearest Stars

Within 5 light years (ly) of the Earth, there are 4 stars known (just the Sun and the Alpha Centauri system).  Within 10 ly, there are 15.  Within 15 ly, there are 58 stars.  The number goes up—rapidly!  Undoubtedly, more stars will be discovered within 15 light years of the Sun.

And, cool is the rule when it comes to nearby stars.  Of the 58 known stars within 15 ly of Earth, an amazing 37 (64%) are class M stars.  The remaining 36% include one A star, one F star, three G stars, six K stars, one L infrared dwarf, five very cool T infrared dwarfs, one extremely cool Y infrared dwarf, and three white dwarfs.

The hottest (and bluest) star within 15 light years of the Sun is none other than Sirius (α Canis Majoris)—the brightest star in the night sky—just 8.65 light years distant.  Sirius A is an A1V (main-sequence) star, twice as massive as our Sun, 71% wider, 25 times more luminous, and only 237 to 247 million years old—just a single orbit around the galactic center.  Sirius rotates much faster than the Sun, too, spinning around once on its axis every 5.4 days.  Think about all these things the next time you look up and see Sirius chasing Orion across the meridian these late-winter eves.  And that Sirius has a white dwarf companion that orbits it once every 50 years, too.

All but two of the nearest 48 stars that are not white dwarfs or infrared dwarfs have a luminosity class of V, meaning they are dwarf or main-sequence stars.  The first exception is Procyon (α CMi A).  Its luminosity class of IV-V indicates it is bright for its temperature and spectral type (F5) and beginning to evolve into a subgiant star on its way towards becoming a giant star.  The other exception is Kapteyn’s Star, a red subdwarf star of spectral type and luminosity class M2VI.  A subdwarf star is underluminous for its temperature and spectral type.  This is caused by low metallicity.  The scarcity of elements other than hydrogen and helium in the star results in a more transparent stellar photosphere and thus a star that is a little smaller than it normally would be.  Incidentally, the fact that we have three white dwarf stars within just 15 light years of us suggests that white dwarfs are copious throughout our galaxy.

You might be wondering how many planets have been discovered orbiting these 58 nearest stars.  Beyond the eight planets orbiting our Sun we find another eighteen confirmed planets, plus at least three more unconfirmed planets.  This is a rapidly advancing field and no doubt many more planets will be added to the list in the coming decade.

The masses of the confirmed planets include one 55% more massive than Jupiter, one a little more massive than Neptune, one a little less massive than Uranus, thirteen super-Earths (1.14 M up to 7.7 M), and two less massive than Earth (0.75 M and 0.98 M).  Their orbital periods range from 2 up to 636 terrestrial days, and then one planet (the super-Jupiter) orbiting once every 6.9 years.  Orbital eccentricities range from circular (0.00) to 0.55, with the super-Jupiter in a very elliptical orbit having an eccentricity of 0.702.  The super-Jupiter is orbiting Epsilon Eridani (K2V, 10.48 ly), with all the rest of the confirmed exoplanets orbiting M-dwarf stars except for the four close-in planets orbiting Tau Ceti (G8.5V, 11.89 ly).

NASA Exoplanet Archive
“The Nearest Stars” by Todd J. Henry, Observer’s Handbook 2019, RASC, pp. 286-290.

Stars Like Our Sun – II

Last time we looked at the brightest G2V stars in the nighttime sky.

Now, we’ll focus on a more sophisticated approach to identify stars that are most like our Sun.  A solar twin is currently defined as a star with the following characteristics (Adibekyan et al. 2017):

Teff = 5777 ± 100 K

log g = 4.44 ± 0.10 dex

[Fe/H] = 0.00 ± 0.10 dex

Teff is the effective temperature of the star.  The effective temperature is the uniform temperature of a black body (which stars closely approximate) that would have the same radiant energy at all wavelengths as the star.

log g is the surface gravity, the base-10 logarithm of the gravitational acceleration, at the photosphere of the star.  The surface gravity is presented logarithmically because the gravitational acceleration at the surface of a star ranges over many orders of magnitude depending on the type of star (for example, a red dwarf vs. a white dwarf or neutron star).

[Fe/H] is the metallicity of the star, giving the ratio of iron to hydrogen atoms in logarithmic units relative to the Sun.  So measured, metallicity as the iron content of a star’s photosphere is often a reasonable proxy for the total metal content of the star (all elements except for hydrogen and helium).

Looking at a recent list of 21 solar twins in the solar neighborhood (Nissen  2016), we find that HD 20782 has the closest Teff match to the Sun, HR 2318 has the closest log g match to the Sun, and HD 222582 has the closest [Fe/H] match to the Sun.  The star with the closest match to all three solar twin characteristics is 18 Scorpii.

HD 20782
Teff = 5776K, log g = 4.345, [Fe/H] = -0.058
Age = 8.1 ± 0.4 Gyr, Mass = 0.97 M
mv = 7.38, mb = 8.03, B-V = 0.65, G1.5V
α2000 = 03h 20m 04s, δ2000 = -28° 51′ 15″
116 – 118 ly
Single star with one known planet, 1.4 – 2.4 MJ, 592d orbital period, in a highly eccentric orbit (e = 0.97).

HR 2318
Canis Major
Teff = 5871 K, log g = 4.445, [Fe/H] = 0.047
Age = 2.7 ± 0.5 Gyr, Mass = 1.05 M
mv = 6.39, mb = 7.01, B-V = 0.62, G1.5V
α2000 = 06h 24m 44s, δ2000 = -28° 46′ 48″
71 – 72 ly
Single star with one known planet, 87% the mass of Uranus, 5.89d orbital period, in a mildly eccentric orbit (e = 0.3).

HD 222582
Aquarius (below the Circlet of Pisces)
Teff = 5784 K, log g = 4.361, [Fe/H] = -0.004
Age = 7.0 ± 0.4 Gyr, Mass = 1.00 M
mv = 7.69, mb = 8.34, B-V = 0.65, G5V
α2000 = 23h 41m 52s, δ2000 = -05° 59′ 09″
136 – 140 ly
Single star with one known planet, 7.1 – 8.4 MJ, 572d orbital period, in a very eccentric orbit (e = 0.725).

18 Scorpii (18 Sco)
Scorpius (just below the “coffee pot” asterism of Ophiuchus)
Teff = 5809 K, log g = 4.434, [Fe/H] = 0.046
Age = 4.0 ± 0.5 Gyr, Mass = 1.03 M
mv = 5.50, mb = 6.15, B-V = 0.65, G5V
α2000 = 16h 15m 37s, δ2000 = -08° 22′ 10″
45.1 – 45.6 ly
Single star, very similar to our Sun.

An additional solar twin in the solar neighborhood has been added recently (Yana Galarza 2016): HD 195034.  It has an even closer match to the Sun’s [Fe/H] than HD 222582 does.

HD 195034
Teff = 5818 K, log g = 4.49, [Fe/H] = -0.003
Age = 2.0 ± 0.4 Gyr, Mass = 1.03 M
mv = 7.09, mb = 7.74, B-V = 0.65, G5
α2000 = 20h 28m 12s, δ2000 = +22° 07′ 44″
91 – 92 ly
Single star.

Adibekyan, V., Delgado-Mena, E., Feltzing, S., et al. 2017, arXiv:1701.05737
Nissen, P.E. 2016, A&A, 593, A65
Yana Galarza, J., Meléndez, J., Ramírez, I., et al. 2016, A&A, 589, A17

Stars Like Our Sun

The spectral type of our Sun is G2V, that is to say, a G2 main-sequence star.

Zodiacal Constellations
mv = -26.75, mb = -26.10, B-V = 0.65
0.0000158 ly
Single star

Here are the brightest stars visible in the nighttime sky that have the same spectral type and therefore are, arguably, most like our Sun.  All have an apparent visual magnitude brighter than +6.00.

Rigil Kentaurus A, Alpha Centauri A (α Cen A)
mv = 0.01, mb = 0.72, B-V = 0.71
α2000 = 14h 39m 36s, δ2000 = -60° 50′ 02″
4.30 – 4.34 ly
Triple star system

Alula Australis B, Xi Ursae Majoris B (ξ UMa B)
Ursa Major
mv = 4.73, mb = 5.38, B-V = 0.65
α2000 = 11h 18m 11s, δ2000 = +31° 31′ 46″
28 – 30 ly
Quintuple star system

HR 4523 A
mv = 4.88, mb = 5.55, B-V = 0.67
α2000 = 11h 46m 31s, δ2000 = -40° 30′ 01″
30.0 – 30.1 ly
Binary star system; exoplanet

Eta Coronae Borealis A & B (η CrB A & B)
Corona Borealis
A: mv = 5.577, mb = 6.123, B-V = 0.546
B: mv = 5.95, mb = 6.48, B-V = 0.53
α2000 = 15h 23m 12s, δ2000 = +30° 17′ 18″
57 – 59 ly
Triple star system

HR 8323
mv = 5.58, mb = 6.18, B-V = 0.60
α2000 = 21h 48m 16s, δ2000 = -47° 18′ 13″
51.9 – 52.5 ly
Single star

Mu Velorum B (μ Vel B)
mv = 5.59, mb = 6.10, B-V = 0.51
α2000 = 10h 46m 46s, δ2000 = -49° 25′ 12″
116 – 119 ly
Binary star system

HR 7845 A
mv = 5.65, mb = 6.34, B-V = 0.69
α2000 = 20h 32m 24s, δ2000 = -09° 51′ 12″
79 – 80 ly
Binary star system

HR 3578
mv = 5.86, mb = 6.39, B-V = 0.53
α2000 = 8h 58m 44s, δ2000 = -16° 07′ 58″
68 – 69 ly
Single star

HR 2007
mv = 5.97, mb = 6.61, B-V = 0.64
α2000 = 5h 48m 35s, δ2000 = -4° 05′ 41″
49.2 – 49.8 ly
Single star with exoplanet

The Eta Coronae Borealis system is noteworthy in that its two primary components are both G2V stars orbiting each other every 41.6 years.  The third component of this system is a distant infrared dwarf, spectral type L8V.

Two of these G2V stars host at least one exoplanet: HR 4523A in Centaurus and HR 2007 in Orion.

HR 4523A has a planet midway in mass between Uranus and Neptune orbiting every 122 days between 0.30 and 0.62 AU from the star (similar to orbital distance of the planet Mercury in our own solar system).  The other stellar component of this system. HR 4523B, is a distant M4V star currently orbiting about 211 AU from HR 4523A.

HR 2007, a single star like the Sun, has a planet about 78% more massive than Neptune, orbiting every 407 days, more or less.  If this planet were in our own solar system, it would range between the orbits of Venus and Mars, roughly.

Two Places, Same Meteor?

A good friend of mine, Paul Martsching, records meteor activity many nights a year for the American Meteor Society near Ames, Iowa, and has been doing so for many years.  On some of those nights, I am also recording meteor activity near Dodgeville, Wisconsin.  Is it possible for both of us to see the same meteor?

Paul’s observing location near Ames and my observing location near Dodgeville are separated by 180 miles.  Meteors burn up in the atmosphere at an altitude of about 50 miles.  Using a little simple trigonometry, we can find that the parallax angle between where Paul and I see the meteor is about 122°.  So, a meteor at either of our zeniths would be below the horizon at the other location.  If, on the other hand, Paul saw a bright meteor 29° above his NE horizon, I might be able to see the same meteor 29° above my SW horizon.

In general, if two observers are separated by a distance d in miles, then they will see the location of the meteor in the sky shifted by approximately s°, as given in the following equation:

This equation assumes that the curvature of the Earth is negligible, a reasonable assumption only when the two observers are relatively close to one another.

A more generalizable equation, taking into account the curvature of the Earth, though still assuming a spherical Earth is:

Plugging in the numbers, we get

We essentially get the same answer—a parallax angle of 122°.  In fact, using the small angle approximation tan x ≅ x for x << 1 (where tan x is in radians), the equation above simplifies to

If this looks a little familiar, it is.  Assuming the meteor burns up at an altitude of 50 miles, the equation immediately above becomes

which is our original equation!  So, for distances on the order of 200 miles or so (or less) we can completely ignore the curvature of the Earth.

Bringing Home Pieces of the Moon

The astronauts on Apollo 11, 12, 14, 15, 16, and 17 between 1969 and 1972 brought back a total of 840 lbs of moon rocks and soil.  Each successive Apollo mission brought back a larger amount of lunar material.

The Soviets brought back a total of 0.7 lbs of lunar soil through their robotic sample return missions Luna 16 (1970), Luna 20 (1972), and Luna 24 (1976).

So, excluding lunar meteorites that have befallen the Earth, a total of 840.7 lbs of lunar material has been delivered to research laboratories here on Earth.

It has been over 40 years since we have brought anything back from the lunar surface.  There are many interesting areas yet to be explored.  Why not send a series of robotic geologists to the Moon in advance of human missions? The success of the Spirit, Opportunity, and Curiosity rovers on Mars show us the exciting work that can be done at a fraction of the cost of human missions.  One enhancement would be the ability of the lunar robotic rovers to collect moon rocks and soil and return them to the mother ship for delivery to Earth.

But our 40+ year wait for additional lunar material may soon be over!

China plans to launch the Chang’e 5 lunar lander in November of this year.  It is expected to land in the Oceanus Procellarum (“Ocean of Storms”) region of the Moon, scoop up at least 4.4 lbs of lunar soil and rock—including some at least six feet below the surface!  The lunar haul will be launched into lunar orbit, where it will rendezvous with the sample return module that will bring it back to Earth.  After a high-speed entry into Earth’s atmosphere, the sample return module will rapidly decelerate, then gently parachute down to the Earth’s surface, presumably somewhere in China.

Chang’e 5 promises to be one of the most exciting and important space missions this year.  Stay tuned!


No, it’s not the name of a rock band. Astronomers (unlike everybody else) consider all elements besides hydrogen and helium to be metals. For example, our Sun has a metallicity of at least 2% by mass (Vagnozzi 2016). That means as much as 98% of the mass of the Sun is hydrogen (~73%) and helium (~25%), with 2% being everything else.

Traditionally, elemental abundances in the Sun have been measured using spectroscopy of the Sun’s photosphere.  In principle, stronger spectral lines (usually absorption) of an element indicate a greater abundance of that element, but deriving the correct proportions from the cacophony of spectral lines is challenging.

A more direct approach to measuring the Sun’s elemental abundances is analyzing the composition of the solar wind, though the material blown away from the surface of the Sun that we measure near Earth’s orbit may be somewhat different from the actual photospheric composition.  The solar wind appears to best reflect the composition of the Sun’s photosphere in the solar polar regions near solar minimum.  The Ulysses spacecraft made solar wind measurements above both the Sun’s north and south polar regions during the 1994-1995 solar minimum.  Analysis of these Ulysses data indicate the most abundant elements are (after hydrogen and helium, in order of abundance): oxygen, carbon, nitrogen, magnesium, silicon, neon, iron, and sulfur—though one analysis of the data shows that neon is the third most abundant element (after carbon).

The elephant in the room is, of course, are the photospheric abundances we measure using spectroscopy or the collection of solar wind particles indicative of the Sun’s composition as a whole?  As it turns out, we do have ways to probe the interior of the Sun.  Both helioseismology and the flux of neutrinos emanating from the Sun are sensitive to metal abundances within the Sun.  Helioseismology is the study of the propagation of acoustic pressure waves (p-waves) within the Sun.  Neutrino flux is devilishly hard to measure since neutrinos so seldom interact with the matter in our instruments.  Our studies of the interior of the Sun (except for sophisticated computer models) are still in their infancy.

You might imagine that if measuring the metallicity of the Sun in our own front yard is this difficult, then measuring it for other stars presents an even more formidable challenge.

In practice, metallicity is usually expressed as the abundance of iron relative to hydrogen.  Even though iron is only the seventh most abundant metal (in the Sun, at least), it has 26 electrons, leading to the formation of many spectral lines corresponding to the various ionization states within a wide range of temperature and pressure regimes.  Of the metals having a higher abundance than iron, silicon has the largest number of electrons, only 14, and it does not form nearly as many spectral lines in the visible part of the spectrum as does iron.  Thus defined, the metallicity of the Sun [Fe/H] = 0.00 by definition.  It is a logarithmic scale: [Fe/H] = -1.0 indicates an abundance of iron relative to hydrogen just 1/10 that of the Sun.  [Fe/H] = +1.0 indicates an abundance of iron relative to hydrogen 10 times that of the Sun.

The relationship between stellar metallicity and the existence and nature of exoplanets is an active topic of research.  It is complicated by the fact that we can never say for certain that a star does not have planets, since our observational techniques are strongly biased towards detecting planets with an orbital plane near our line of sight to the star.

Vagnozzi, S. 2016, 51st Recontres de Moriond, Cosmology, At La Thuile