# What methods exist to calculate the ellipticity of galaxies

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What methods exist to calculate the ellipticity of galaxies and what are their drawbacks? I have asked this question about ellipticity in the SDSS but I want to know about general methods for cases where I have just an image of a galaxy.

I do so far know these methods:

• ellipticity from stoke parameters or the flux-weighted second moment as described here.
• from adaptive moments as described here (which would allow me to do shear calibration).
• fit an ellipse on a particular isophote as decribed here.

My question is whether there are further methods I am not aware of. Does there also exist a method when I fit an ellipse to all isophotes? And how would I calculate the 'average' from it (such that the spherical center does not contribute to much)?

I would like to learn about the pro and cons of each method. I am especially interessted in methods that get close to what people would estimate by eye. I would also be glad to hear about references (textbooks or papers) for the different methods.

The first two methods mentioned in the question are really similar. They both use moments of the light distribution, only the weighting is different.

Another family of methods is 2D profile fitting to the image, with more or less sophisticated luminosity profile models. A simple case would be to fit a Sérsic profile. But instead, one could make a composite model with two or more elliptical profiles (say one exponential disk and a de Vaucouleur bulge). This would allow to separate the ellipticity of the disk from the ellipticity of the bulge. A widely used software for this is GALFIT by Chien Peng.

There are three variants of flattening when it is necessary to avoid confusion, the main flattening is called the first flattening. [1] [2] [3] and online web texts [4] [5]

In the following, a is the larger dimension (e.g. semimajor axis), whereas b is the smaller (semiminor axis). All flattenings are zero for a circle ( a = b ).

(First) flattening Second flattening Third flattening f a − b a >,!> Fundamental. Geodetic reference ellipsoids are specified by giving 1 f >,!> f ′ a − b b >,!> Rarely used. n , ( f ″ ) a − b a + b >,!> Used in geodetic calculations as a small expansion parameter. [6]

The flattenings are related to other parameters of the ellipse. For example:

## What methods exist to calculate the ellipticity of galaxies - Astronomy

• Classification of Galaxies(originally by Edwin Hubble)
• Elliptical Galaxies
Similar to globular clusters, except much larger and farther away.
Mostly old, red, population II stars. Little gas, dust or star formation.
Stars orbit in random, elliptical orbits, denser toward the middle.

Subclassified according to the degree of ellipticity:

 E0 Round M87, Copyright AAO E1 10% flat M105 by NASA E2 20% flat M32, from SEDS E3 30% flat (Doctored M32) E4 40% flat M49, from SEDS E5 b=a/2 M59, by Bill Arnett E6 60% flat NGC205, by Bill Keel E7 70% flat (Doctored M59)
 Sa Large nucleus Tight arms M104, from SEDS SBa Large nucleus Tight arms Central bar M65. AAO photo by David Malin Sb Med. nucleus Med. arms Andromeda (M31), from SEDS SBb Med. nucleus Med. arms Central bar M91, from SEDS Sc Small nucleus Loose arms M74, from SEDS SBc Small nucleus Loose arms Central bar M83. AAO photo by David Malin. Scd M33. IAC photo by David Malin. SBcd M101, from SEDS. S0 (or
 Scp galaxy A spiral which has beed distorted by a large neighbor, NGC4631. NGC4656, by Michael Purcell E0-pec The body of this galaxy appears to be an elliptical galaxy which has eaten a spiral disk! NGC5128. AAO photo by Malin.

Spiral galaxies.
Rotation of stars and HII regions in spiral arms, using doppler shifts.
Gives masses ranging from 10 10 to 10 11 solar masses.
Orbit of a neighbor galaxy gives 10 10 to 10 12 solar masses.
Must be extra mass beyond the spiral arms.

## BASIC EXTRAGALACTIC ASTRONOMY - Part 6: Galaxies, Discovery and Classification

The term galaxy is derived from Greek words for a "milky circle". When capitalized, Galaxy refers to our own Milky Way galaxy. Since human beginnings, the sublime splendor of the Milky Way spanning a dark night sky served as evidence that the scope of creation is unimaginably greater than mortal affairs. The spectacle inspired mythology, religion, art, philosophy, architecture, and ultimately experimental science and technology. Throughout history, speculations about its nature ranged from metaphysical follies to surprisingly accurate scientific insight. For example, as early as the 5th century BC, Greek philosopher Democritus, the father of the Atomic Hypothesis, the Void Hypothesis (between atoms), exobiology, and remarkably factual cosmology, proposed that the Milky Way is composed of innumerable distant stars, too faint to be seen individually. It would be two thousand years before Democritus's insight could be confirmed by Galileo with the newly invented telescope.

A number of nebulous objects visible with unaided eye were known to the ancient astronomers. In his books on the motions of 1022 fixed stars, around 150 AD, Claudius Ptolemy recorded five "nebulous stars", all of which are actually open clusters. The first true nebula was documented by Persian astronomer al-Sufi around 964 AD, who described a "little cloud" at the location of the Andromeda galaxy. He also documented the two Magellanic Clouds which are not visible from Iraq, but can be seen from the Arabian Peninsula. The number of reported nebulae increased rapidly with the use of telescopes. In 1610, French astronomer Peiresc discovered the Great Nebula in Orion. By 1781, Charles Messier completed his catalogue of 103 nebulae. And by 1802, William and Caroline Herschel published three catalogs of nebulae totalling 2,510 entries. In spite of a sizable collection of identified objects, the nature of nebulae remained uncertain for over a century. The Herschels thought them to be composed of clouds of innumerable unresolved stars.

The first step toward identifying galaxies among the large set of unclassified nebulae was taken by Anglo-Irish astronomer William Parsons, 3rd Earl of Rosse. Using his 72 inch (1.83 m) telescope known as Leviathan of Parsonstown, the largest instrument in the world until 1917, Parsons observed that some nebulae had a distinct spiral structure. His drawings of spiral nebulae, especially of M51, closely resemble modern astrophotographs.

Fig. 29: Rosse's drawings of spiral nebulae observed through his Leviathan telescope.

Like the Herschels, Rosse theorized that all nebulae consist of large numbers of unresolved stars. In fact, he reported that his telescope could resolve the Orion Nebula into stars. This allegation was not entirely untrue since the region is a stellar nursery strewn with nearly 3,000 young stars belonging to the Orion Nebula Cluster, which can cause an illusion of partial resolution in a large telescope. Astronomers who hypothesized that nebulae are composed of clouds of interstellar gas from which new stars formed were met with some resistance because the idea implied a Universe which is not constant, but changing and evolving. Various interpretations of the constancy of the Universe persisted well into the middle of the 20th century.

In 1864, English astronomer William Huggins, an early pioneer of astronomical spectroscopy, photographed the spectrum of planetary nebula NGC 6543 (Cat's Eye Nebula) and found it to have a bright emission line spectrum typical of fluorescing gas. Over his career, assisted by his wife, Margaret, he discovered that approximately one third of the 70 nebulae examined showed the emission spectrum of a gas, while the rest had a continuous spectrum characteristic of stars. Incidentally, in 1868 Huggins measured a redshift in the spectrum of Sirius, and suggested it could be used to calculate the star's recession velocity relative to Earth.

A third type of nebula was identified in 1912 by American astronomer Vesto Slipher. He showed that the spectrum of the nebula surrounding the star Merope was of the absorption type, precisely matching that of the star itself. This proved that the nebula was illuminated by light reflected from the star (reflection nebula) rather than by gas fluorescence (emission nebula).

Between the 1860's and the 1920's the nature and distances of nebulae - spiral nebulae especially - remained under intense analysis and speculation. It is important to keep in mind that the scientific world of that period developed a wealth of hypotheses, but had no confirmed knowledge regarding the dimensions of the Milky Way, the size or the age of the Universe, expansion of the Universe, particle physics, the origin of elements, nuclear reactions, energy generation mechanisms within stars, the etiology and differences between novae and supernovae. Progress was facilitated by the development of photographic techniques and larger, more precise instruments, while it was hindered by dubious findings of several highly reputable astronomers. One such confused the absolute magnitude-luminosity period relation of Cepheid variables (used as standard candles) by alleging them to be occulting binary stars. Another one successfully claimed to have photographed evidence of rotation in the Pinwheel spiral nebula, which imposed a limit on the nebula's distance and size, lest the stars fly apart. In spite of some truly prescient hypotheses, the prevalent scientific opinion during these decades was that the Milky Way comprised the entire Universe, including all the nebulae, and that it was in a number of properties constant.

Progress was made gradually. In 1912, Slipher was the first to measure the shift of spectral lines of nearby spiral nebulae, allowing him to estimate their motions relative to Earth. Two years later, at the onset of World War I, he reported radial velocity of -300 km/s for M31, and demonstrated that spiral nebulae rotate by measuring that one side of the spiral arms exhibited a redshift, while the other exhibited a blueshift relative to the center.

Fig. 30: Spectroscopic detection of the rotation of spiral nebulae. For nearby galaxies, tangential velocity can be calculated by using equations (4) and (5) in Section 2).

In 1915, he published a survey of the motions of 15 spiral nebulae, and found that the average radial velocity was 400 km/s, about 25 times greater than the average velocity of stars within the Milky Way.

In 1917, American astronomer Heber Curtis discovered evidence of 12 novae in the photographic record of the Great Andromeda Nebula, and found them to be on average 10 magnitudes fainter than those detected in the Milky Way. This allowed him to estimate the distance of nearly 500,000 light years - a number five times lower than the current value, but sufficiently large to prove that the Andromeda Nebula lies far outside the confines of the Milky Way. As a result, Curtis became a major advocate of the Island Universe Hypothesis which proposed that nebulae are in fact very distant independent galaxies comparable in scale to our own. The speculation about stellar systems of equal rank outside the Milky Way actually dates back to the early 1700s. It was independently proposed by Swedish philosopher Emanuel Swedenborg and English astronomer Thomas Wright of Durham. It is frequently mentioned in the form of "sidereal systems" in William Herschel's writing. And, it was adopted by German philosopher Immanuel Kant (1755) and Swiss polymath Johann Lambert (1761), even as it was not possible at the time to observationally distinguish between aggregations of stars and gaseous nebulae.

On April 26, 1920, the Great Debate took place in the hall of the US National Academy of Sciences in Washington DC proposing to resolve the question of the size of the Milky Way, and whether it constituted the entire Universe, or was merely one of the thousands of similar spiral nebulae.

On one side was Harlow Shapley, who had been measuring the size of the Milky Way using Cepheid variables as standard candles, and had calculated its size to be at least 10 times larger than previously thought. He claimed that, if the Andromeda Nebula were of similar dimensions, it would have to lie at the distance on the order of 100 million light years - a number unacceptable to most contemporary astronomers. He pointed out that, in 1885, a nova with apparent magnitude of 5.8 had been observed in M31. It outshone the entire nebula, requiring an impossible output of energy if the nebula were extragalactic. (The "nova" was actually of a previously unknown type - a much more powerful and brighter Ia supernova, SN 1885A).

On the other side was Heber Curtis who emphasized that, according to his estimates, the Andromeda Nebula lies far outside the confines of the Milky Way, that it has prominent dust lanes which resemble those displayed by the Milky Way, and that its radial velocity and rotation speed are far faster than any star within our galaxy.

Although, already for several years, most astronomers had been favoring the idea that spiral nebulae were in fact distant galaxies, no definite winner was declared in the end because none of the arguments were regarded as conclusive.

One line of decisive evidence came from redshift measurements of spiral nebulae made with a new generation of large telescopes. In 1925, Belgian priest and astronomer Georges Lemaitre began writing a report titled A Homogeneous Universe of Constant Mass and Growing Radius Accounting for the Radial Velocity of Extragalactic Nebulae, which was published in April, 1927 in a little-known Belgian journal. Having discovered a set of solutions to Einstein's field equations which allowed for an expanding Universe, he presented mathematical proof that galactic redshifts can be explained by their recession velocity, and should be proportional to their distances. Based on available data and on his own observations, he provided the first estimate of the expansion rate of the Universe (later named the Hubble constant).

From the start, Lemaitre's idea was met with skepticism which lasted for decades. Most influential cosmologists at the time, Einstein and Hubble included, believed in a static Universe in which galactic redshifts had to be explained by a still unknown phenomenon of nature. When Lemaitre submitted his report to Einstein during their meeting in Brussels in 1927, Einstein commented, "Your calculations are correct, but your physics is atrocious." The influential man refused to accept the possibility that the Universe might be expanding, which he would later in life describe as his greatest blunder. As for Hubble, unwilling or unable to challenge the prevailing concept of a static Universe, he always prudently referred to the term recession velocity as apparent recession velocity.

In 1931, Lemaitre received wider attention when Arthur Eddington published a commentary on his 1927 article in the Monthly Notices of the Royal Astronomical Society. Lemaitre was invited to London to participate in a British Science Association meeting where he confounded the world with another radical idea. If the Universe is expanding, he reasoned, and if we project the expansion backward in time, then the Universe must have originated from a single point in space of infinite density, which he named The Primeval Atom. The hypothesis emerged to confront public scrutiny in a 1931 report published in the British journal Nature, and in an article for general readers in the December 1932 issue of Popular Science. Eddington found Lemaitre's hypothesis unpleasant. Einstein regarded it unjustifiable from the physical point of view. Even decades later, after Lemaitre's observations had been confirmed by Edwin Hubble, notable British astrophysicists Fred Hoyle, Hermann Bondi, and Thomas Gold found Lemaitre's conclusions incompatible with the scientific method. While he had accepted the idea of an expanding Universe, Hoyle declared the notion that the Universe had a beginning at which it had arisen from nothing an irrational argument in favor of a Creator. In a 1949 BBC interview, Hoyle referred to the hypothesis of the primeval atom as the Big Bang, possibly intending to convey derision. And, the name stuck.

Lemaitre's response to criticism was that the Universe must have had a beginning because the prevailing assumption of a static Universe could not be sustained into the infinite past.

Another line of convincing evidence that spiral nebulae are in fact distant galaxies arose from the work of American astronomer Henrietta Swan Leavitt who published in 1912 a close relationship between the period and the apparent magnitude of Cepheid variables in the Small Magellanic Cloud.

Since all the stars lay at approximately the same distance, their absolute magnitudes could be deduced from apparent magnitudes as soon as the scale was calibrated with parallax measured distances to some nearby Cepheids. The resulting Leavitt's Law made Cepheid variables the first "standard candles" by which distances could be measured to other galaxies too remote for parallax observations.

Fig. 31: The current relationship between the period and absolute magnitude for two major types of Cepheid variables. At the time of discovery, the existence of two major types of Cepheids was unknown, leading to errors in distance estimates of nearby galaxies.

American astronomer Edwin Hubble established himself as one of the most influential astronomers of all time by using Leavitt's Law to determine distances to nearby galaxies and conclusively prove them to lie far outside the Milky Way. Working on the recently completed 100 inch Hooker telescope at the Mt. Wilson observatory, Hubble sought evidence of Cepheid variable stars in nearby spiral nebulae, including M31 and M33. Between October 1923 and February 1924 he identified in the Andromeda nebula the first extragalactic variable star, M31-V1, whose light curve was typical of a Cepheid, with a brief peak, prolonged decline, long trough, and a rapid rise. His observation notes indicate a period of 31.415 days which, by the Leavitt's Law of those days, corresponded to an absolute magnitude of -5.0. From the median apparent magnitude of 18.5, and estimated color index of +0.9 he calculated the distance to the star and the nebula to be 220,000 parsecs, or 717,200 light years. This is much less than the distance we know today, but it was quite sufficient to prove to Hubble that the Andromeda nebula was a galaxy in its own right, far outside the bounds of the Milky Way.

Fig. 32: The second most influential star in history, M31-V1, the Cepheid variable Hubble identified in the Andromeda nebula, which allowed him to estimate its distance to be far outside the Milky Way

Actually, Hubble may have made an error in his observation notes when he initially calculated the distance of 220,000 parsecs. If we enter his values, corrected for the color index, into the distance modulus equation, the result is 331,000 parsecs, or 1,079,000 light years. He amended his result before announcing the discovery in a New York Times article on 23 November 1924, along with 35 other Cepheid variables in M31 and M33. The newspaper carelessly misspelled his name as Dr. Edwin Hubbell.

Urged by Shapley and astronomer Henry Russell, Hubble wrote a paper titled Extragalactic Nature of Spiral Nebulae which was presented in December 1924 at a joint meeting of the American Astronomical Society and American Association for the Advancement of Science. The paper shared the first prize, but did not cause a sensation because leading astronomers had been notified of his findings months earlier. Hubble's results regarding the Andromeda galaxy would not be formally published in The Astrophysical Journal until April 1929.

The current value for the distance of the Andromeda galaxy is 2.537 million light years, about 2.5 times greater than the distance estimate in Hubble's report. A major source of error was that in Hubble's time the difference between Type I (Classical) and Type II Cepheids had not been discovered, which distorted the Leavitt's Law curve. Also, Hubble worked with insensitive film emulsions, and lacked accurate photometric instruments. According to a 2011 study, M31-V1 has a period of 31.397 days, color index of 0.6, median apparent magnitude 19.0, and absolute magnitude -5.3 in the visual band.

Applying these magnitudes in the distance modulus equation results in the distance of 724,400 parsecs or 2.362 million light years, "subject to reduction if star is dimmed by intervening nebulosity," in Hubble's own words. Averaging measurements of a large number of M31 Cepheids would produce a more accurate result.

Hubble proceeded to measure distances to nearby galaxies using the Cepheid method. He then analyzed redshift measurements for 46 galaxies collected by Vesto Slipher and by his own assistant astronomer Milton Humason. When galaxy distances were plotted against their CZ recession velocities, a roughly linear relationship emerged, where the slope of the line reflected the expansion rate of the Universe (see section 5, Fig. 6). His first estimates yielded the expansion rate, later known as the Hubble Constant, of 500 km/s/Mpc, about 7 times higher than the current value, because his distance measurements had been too short by a factor of 7. But, inaccuracies aside, Hubble convincingly proved a law (now known as the Hubble-Lemaitre Law) that the recession velocity of a galaxy is proportional to its distance.

It is difficult to believe that, after years of painstakingly correlating galaxy distances with redshifts, and after estimating the expansion rate of the Universe, Hubble remained sincerely doubtful of Lemaitre's interpretation that redshift measured actual recession velocity. But, it seems that he genuinely shared with the majority of renowned cosmologists the conviction in the constancy of the Universe. In his own words, he used "the term 'apparent' recession velocity to emphasize the empirical features of the correlation," leaving the issue of the causal mechanism of redshifts to the theoreticians, "who are competent to discuss the matter with authority."

After Einstein heard of Hubble's findings, in April 1931 he published a paper for the Prussian Academy of Sciences in which he renounced the cosmological constant, which he disliked for a number of reasons, and came around to the idea of a dynamic, expanding Universe model described by Friedman and Lemaitre. He then had a meeting with Hubble, during which he failed to convince Hubble that the Universe can not be static because that would require an impossibly precise balance.

As reported in a Los Angeles Times article on 31 December 1941, Hubble announced to the American Association for the Advancement of Science that a six-year Mt. Wilson telescope survey revealed a uniform distribution of galaxies in space, which can not be consistent with the theory of an expanding Universe. "Explanations which try to get around what the great telescope sees fail to stand up. The Universe probably is not exploding but is a quiet, peaceful place and possibly just about infinite in size.”

According to American astronomer Allan Sandage, to the very end of his writing Hubble favoured "the model where no true expansion exists, and therefore that the redshift represents a hitherto unrecognized principle of nature."

If Hubble could not accept that redshift is due to actual recession velocity, he was in very illustrious company. The idea of a static Universe, infinite both temporally and spatially, first proposed by English astronomer Thomas Diggs in the 16th century, appeals to important facets of human mentality. Fritz Zwicky maintained the tired light hypothesis whereby photons lose energy by interacting with matter. In 1976, quantum field theorist Irvin Segal proposed that redshift is due to the curvature of space. A number of scientists showed that redshift can be caused by light escaping from deep gravitational wells. Throughout his career, until 2001, the father of the theory of stellar nucleosynthesis, Fred Hoyle, promoted the steady-state cosmology in which the density of an expanding Universe remains constant by continuous creation of matter. As recently as 2013, a group of Chinese astronomers headed by Ming-Hui Shao suggested that redshift is due to the interaction of photons with intergalactic electromagnetic fields. And, there are many other examples of truly brilliant minds devising interpretations outside the sphere of the simplest explanation. Skepticism, speculation, and criticism lie at the foundation of the scientific method.

31) Morphological Classification

Working with the world's largest telescope at Mt. Wilson, Hubble was in a unique position to begin morphological classification of galaxies. In 1926, he published in the Astrophysical Journal an article titled Extra-Galactic Nebulae in which he reported the properties of 400 objects, and classified them "based on the forms of the photographic images."

-Ellipticals, with featureless light distribution, labeled E1 - E7, where the integer indicates ellipticity x 10.

-Lenticulars, showing a featureless disk without spiral structures surrounding a bright nucleus, labeled E0.

-Normal Spirals, showing spiral arms surrounding unbarred nuclei, labeled Sa - Sc.

-Barred Spirals. showing spiral arms surrounding a barred nucleus, labeled SBa - SBc.

-Irregular Galaxies, which lack a dominating nucleus and rotational symmetry, labeled Irr.

He assigned subscripts a, b, and c to spiral galaxy labels based on stages he defined as early, intermediate, and late respectively. He described lenticular galaxies as a "hypothetical intermediate" between normal and barred spirals. It is probable that he thought this classification scheme, from left to right, reflects actual evolutionary pathway of galaxy formation. This assumption was generally accepted for decades, resulting in presently obsolete terms "early-type galaxies" for the ellipticals and the lenticulars, and "late-type galaxies" for the spirals.

Fig. 33: Hubble's visual galaxy classification

In the relatively small sample of spiral galaxies accessible to him, Hubble made a key observation that galaxies with larger nuclear bulges presented with more tightly bound spiral arms. Over subsequent decades this led to the density wave model of spiral arm formation whereby stars randomly move around the galaxy, but are delayed within density waves of matter in the thin galactic disk, much as waves of traffic form on a congested highway.

New models, suggested by intuition and supported by computer simulations, propose that collections of stars and interstellar matter within spiral arms are locally bound by gravity, also constituting actual comoving structures rather than only density waves.

In 1959, French astronomer Gerard de Vaucouleurs recommended that additional criteria be used in the morphological classification of galaxies. These included the size and the appearance of the nuclear bar, rings in the disks, and lens shapes of edge-on spirals.

The Hubble-Vaucouleurs classification divides lenticular galaxies into unbarred (S0A) and barred (S0B), with S0 retained for those galaxies in which it is impossible to tell. Spiral galaxies are divided into normal spirals (SA), barred spirals (SB), and intermediate spirals (SAB) with weakly barred nuclei. Galaxies without a central bulge are denoted (m). Those with rings are denoted ®, without rings (s), and transitional galaxies with partial rings (rs). Irregular galaxies are denoted (I), and dwarf galaxies as (d).

For example, an unbarred galaxy with loosely wound arms and no rings would be classified as SA(s)c. And dIm would be an irregular dwarf galaxy with no central bulge.

Fig. 34: Hubble-Vaucouleurs morphological galaxy classification diagram

In many cases galaxy classification is subjective, and in some cases impossible due to the orientation of the galaxy relative to the observer, or to the quality of its image. For this reason SIMBAD Astronomical Database specifies a data quality flag ranging from A (best) to E (worst) for every object's morphological type listing. A very prominent galaxy M 101 is classified as SABc, but only with intermediate confidence of C. Even on excellent photographs, the difference between ellipticals and lenticulars may be difficult to distinguish. The same is true for galaxies without a central bulge labeled (m).

It is important to note that the galaxy classification diagram, from left to right, does not represent morphologial evolution of galaxies. Quite the opposite is true. With numerous exceptions, galaxy evolution models follow the reverse path on the diagram - from right to left, or from the "late-type" galaxies to the "early-type". Generally speaking, primordial irregular dwarf galaxies merge to form large gas-rich Sd spirals with loosely wound arms. These deplete gas through new star formation to become tightly wound Sa spirals. Finally, mergers of large galaxies give rise to the ellipticals. Lenticular galaxies are intermediate between the spirals and the ellipticals. Two mechanisms - not mutually exclusive - have been proposed for their formation. One is the process of galaxy mergers. The other is gas depletion and loss of spiral arms in tightly wound Sa spiral galaxies.

## What methods exist to calculate the ellipticity of galaxies - Astronomy

a) From the stellar velocity dispersion.

b) From the neutral gas velocities found in the outermost region, in certain galaxies.

c) From the X-ray corona surrounding all ellipticals.

There also exist complementary methods, using observations of ionized gas in the central parts, globular clusters, gravitational lensing, theoretical considerations about the bar instability and the chemical evolution.

The general conclusion, taking into account all these studies, could be, in summary, that dark matter amounts comparable to visible matter could be present in the visible part of the galaxy, and that larger dark matter amounts, probably as large as in spirals, are present in a halo surrounding the galaxy, but that, in any case, the evidence of dark matter in ellipticals is less than in the case of spirals. Even the complete absence of dark matter cannot be easily ruled out.

Several reviews have been written on the topic (e.g. Ashman, 1992 Trimble, 1987 de Zeeuw, 1992 Kent, 1990 Bertin and Stavielli, 1993). Let us remember that the surface brightness of an elliptical galaxy can be fitted by de Vaucouleurs' law

(de Vaucouleurs, 1948), where R e is the radius enclosing half of the light and I e = I ( R = R e ) is another constant. The value of R e is often used as a parameter that normalizes all lengths as does the radial scale length in spirals. This law seems to be rather well matched, but it is just one fitting which might be less appropriate for some subtypes (Andreakis, Peletier and Balcells, 1995).

Let us comment on the three basic methods, and more briefly about other methods:

a) The observations of stellar velocity dispersion, interpreted in terms of Jeans' equation or of the Virial theorem, can provide the total mass for R < R e , or even at larger distances.

The Virial theorem for a spherical, steady-state, static isothermal elliptical galaxy reduces to the simple expression

where R is an equivalent radius. For a given R , M , because the stellar chaotic thermal velocities, quantified by the velocity dispersion, , must prevent gravitational collapse. The larger the mass, the larger the stellar velocities must be. This formula gives a first approximate mass. In practice however, much more sophisticated models than this one are used to interpret the velocity dispersions. There is a "degeneracy" between the unknown anisotropy and the unknown gravitational potential. If the anisotropy of the orbits is known the potential can be determined, but not both simultaneously. We should know if orbits are mainly circular, or mainly radial, or something in between. The anisotropy is characterized by the parameter , which is defined later, in Section 3.5.2.

Pioneering works by Binney, Davies and Illingworth (1990), van der Marel, Binney and Davies (1990) and others have concluded that no gradients in M/L were clearly appreciated and that no dark matter was needed to explain the central surface brightness and the velocity dispersions. The M/L values are of the order of 12h (Binney and Tremaine, 1987) (about 8 for h=0.65) which is comparable to the solar neighbourhood values. It is slightly higher, but this fact can be explained mainly by the absence of young stars in ellipticals. One-component models, without any halo, provide a good zeroth-order description (Bertin, Saglia and Stiavelli, 1992).

Bertin, Saglia and Stiavelli (1992) also considered two-component spherically symmetric collisionless self-consistent models, which were later used to interpret real data from 10 bright selected galaxies (Saglia, Bertin and Stiavelli, 1992) and found some evidence for dark matter to be of the order of the visible mass. The presence of rotation and of tangential anisotropy makes it difficult to draw firm conclusions.

As in the case of spirals with their rotation curve, a flat or slowly increasing velocity dispersion, ( r ), may indicate dark halos dominating the dynamics (Saglia et al. 1993) but there is a surprisingly large variety of -profiles, some of which decrease outwards relatively fast. Therefore, Saglia et al. did not find any compelling evidence of dark matter out to 1-2 R e . Carollo et al. (1995) observed flat or gently declining velocity dispersion profiles in four elliptical galaxies, concluding that massive dark halos must be present in three of the four galaxies, although no clear conclusion was obtained for the fourth. Bertin et al. (1994) found that in a sample of 6 galaxies, three of them were not suitable for reliable modelling, two of them presented no evidence for dark matter and one (NGC 7796) appeared to have a distinct dark halo. But the conclusion that some galaxies have a dark halo while others do not is problematic for understanding what an elliptical galaxy is. De Paolis, Ingrosso and Strafella (1995) found that dark matter inside R e is negligible with respect to the visible mass.

b) A small fraction of elliptical galaxies are surrounded by a ring of neutral hydrogen, for instance, NGC 1052, NGC 4278 and NGC 5128. In these cases, the determination of a dark matter halo is very similar to its determination in spiral galaxies, from the rotation curve. One of the best studied gaseous rings is that of IC 2006 (Schweizer, van Gorkom and Seitzer, 1989). The neutral gas counter-rotates at a radius of 18.9 kpc (6.5 R e ) and has a total mass of 4.8 × 10 8 M . This galaxy also has a counter-rotating central mass of ionized gas out to 5 kpc. These gaseous components of some ellipticals have either been accreted or are the remnant of a merger from which the elliptical was created.

Schweizer, van Gorkom and Seitzer (1989) found evidence for a DM halo in IC 2006 with twice the mass of the luminous matter within 6.5 R e , under the assumption that the HI ring is flat and circular. Bertola et al. (1993) analyzed five elliptical galaxies, combining the M/L ratios obtained with the inner ionized hydrogen component and the outer neutral hydrogen ring. M/L is constant out to about R e with a moderate value of 3.50.9 but becomes very large in the ring region. These authors found a similarity in the distribution of dark matter in ellipticals and in spirals, suggesting a similar picture for the origin of both.

As we will discuss later, magnetic fields may explain rotation curves without requiring dark matter in spirals. Similar arguments can be considered to interpret gaseous rings around ellipticals. In particular, a narrow ring is pushed towards the centre more easily than a disk, because the outward magnetic pressure force need not be compensated by a magnetic tension. It is to be emphasized that the IC 2006 gaseous ring is very narrow, and is not even resolved by VLA.

c) The most promising method to study dark matter in ellipticals is based on the existence of X-ray halos. A hot X-ray emitting gas typically extends out to 50 kpc (Forman, Jones and Tucker 1985). The probable origin of the gas is mass loss from stars supernovae heat it up to 10 7 K , bremsstrahlung being the main cooling process (Binney and Tremaine, 1987). Typical masses of this hot gas are 10 10 M .

Hydrostatic equilibrium is usually assumed for the gas. Then, for a spherical DM halo

where is the density of the gas. Once M ( R ) is determined in this way, we obtain the DM halo profile.

The gas is not in perfect hydrostatic equilibrium. The innermost gas in the X-ray halo is more efficiently cooled, because cooling is proportional to the electron density, which is still high. An inwards flow in the inner region is therefore to be expected (Binney and Tremaine, 1987). Cooling flows have been observed (Mushotzky et al. 1994) and models including radial flows have been developed (e.g. Ciotti et al. 1991). The equilibrium probably breaks down in galaxies with low X-ray-to-optical luminosity ratios. Nevertheless, hydrostatic equilibrium is generally assumed.

In the above formula, the temperature profile T ( R ) is not provided by the observations with enough precision. The strengths of some X-ray lines or the shape of the X-ray continuum should provide this T-profile but, in practice, this is still rather problematic. For giant cD galaxies, like M87, the temperature is exceptionally well determined and the method provides more reliable results. For M87 the data are spectacular: M ( R < 300 kpc ) 3 × 10 13 M the mass-to-light ratio reaches a value of 750 about 95% of M87 mass is dark matter (Fabricant and Gorenstein, 1983 Stewart et al. 1984 Binney and Cowie, 1981). However, cD galaxies may be exceptional as they lie at the centre of a rich cluster, the DM encountered could belong to the cluster as a whole. Below, we address this problem in Section 5.

Difficulties arise in the analysis of normal ellipticals. If T ( r ) is unknown, it is tempting to assume an isothermal distribution (e.g. Forman, Jones and Tucker, 1985), which might be justifiable. Mushotzky et al. (1994) were able to obtain 6 points of T ( R ) in NGC 4636, finding that T was approximately constant. Moreover Matsushita (1997) and Jones and Forman (1994) confirmed the constancy of T ( R ). High M/L ratios are in general obtained, in the range 10-80, especially at large distances, but Trinchieri, Fabbiano and Canizares (1986) concluded that DM halos were not absolutely required by the data. Fabbiano (1989) also found much lower masses.

Furthermore, the contribution of unresolved discrete X-ray sources, such as accreting binaries, complicates the analysis (de Paolis, Ingrosso and Strafella, 1995), which could be related to the fact that the relative amount of DM is higher for X-ray bright ellipticals.

Models often take as a boundary condition that X-ray emitting gas is confined by the cluster intergalactic gaseous pressure (Fabian et al. 1986). Other authors assume a vanishing pressure at infinity (Loewenstein and White, 1999).

The gas responsible for the X-ray emission cannot rotate very fast and hence no dynamo can generate magnetic fields capable of affecting the hydrostatic equilibrium. However, intergalactic magnetic fields could have an influence as a boundary condition. For the intracluster intergalactic space, with n 10 -5 cm -3 and T 10 7 K the thermal pressure is of the order of 10 -14 dyncm -2 . As discussed below, cluster intergalactic fields are of the order of 10 -6 G, and therefore the magnetic energy density is of the order of the thermal pressure. External magnetic fields could contribute to confining the X-ray emanating hot gas, thus reducing the large amounts of dark matter required. This external field would not act isotropically and would produce eccentric X-ray isophotes, such as for instance in NGC 720. Eilek (1999) makes suggestions about the importance of magnetic fields in the dynamics of clusters which are relevant to the dynamics of X-ray halos around giant ellipticals at the centre of clusters, where the field can provide an important part of the pressure support.

Buote and Canizares (1998) observed a different isophote geometry for X-rays and for the optical in NGC 720. The X-ray isophotes are more elongated and their major axes are misaligned by about 30 o . If the total matter were distributed as is the optical light, it could not produce the observed ellipticities of the X-ray isophotes. They interpreted this ellipticity as being produced by a dark matter halo and developed a model that did not need the T ( R ) profile, and which also favoured the existence of a large dark matter halo. Davis and White (1996) and Loewenstein and White (1999), too, developed methods not requiring the temperature profile that imply DM halos.

d) In addition to these basic methods there are others that should be mentioned. The image splitting of an individual gravitational lens system consisting of an elliptical is only slightly sensitive to the existence of a DM halo, and so, one cannot definitely discriminate between galaxies with and without halos, with some exceptions (Breimer and Sanders, 1993 Kochanek, 1995). Indeed, in three cases where the lens is clearly a single galaxy, there is no need to consider any dark matter halo. Maoz and Rix (1993), however, deduce from gravitational lensing methods that M ( R ) increases linearly with R , as is typical in isothermal halos.

Globular clusters have been considered to deduce the existence of dark matter halos in ellipticals, mainly in M87 (Huchra and Brodie, 1987 Mould et al., 1990). They support the conclusions obtained by other methods: models without dark halos do not fit the data in M87, but they cannot be excluded in NGC 4472 (Mould et al. 1990). This problem is considered in Section 2.6. Planetary nebulae have also been considered in NGC 5128 by Ford et al. (1989) and by others, who found a radial increase in ( M / L B ) reaching values of about 10, although de Zeeuw (1992) suggested a lower gradient. Ciardullo and Jacoby (1993) deduced that the non-interacting elliptical galaxy NGC 3379 has no dark matter halo, and that a constant M/L of about 7 explained the observations perfectly. Theoretical studies of bar instability (Stiavelli and Sparke 1991) and chemical evolution (Matteuci 1992) were unable to unambiguously determine the presence of a dark halo.

In conclusion, elliptical galaxies could have dark matter halos similar in mass and extent to those in spiral galaxies (Danziger, 1997), but the evidence is not so clear and it cannot even be completely rejected that they possess no dark halo at all. As exceptions, in giant cD galaxies like M87, the existence of large amounts of dark matter seems to be fully demonstrated.

## Author information

### Affiliations

Max Planck Institute for Astronomy, Heidelberg, Germany

Ling Zhu, Glenn van de Ven, Remco van den Bosch, Hans-Walter Rix & Marie Martig

European Southern Observatory, Munich, Germany

Glenn van de Ven & Mariya Lyubenova

Kapteyn Astronomical Institute, University of Groningen, Groningen, The Netherlands

Instituto de Astrofísica de Canarias (IAC), La Laguna, Tenerife, Spain

Universidad de La Laguna, Dpto. Astrofísica, La Laguna, Tenerife, Spain

Astrophysics Research Institute, Liverpool John Moores University, Liverpool, UK

Physics Department and Tsinghua Centre for Astrophysics, Tsinghua University, Beijing, China

National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China

Jodrell Bank Centre for Astrophysics, The University of Manchester, Manchester, UK

Heidelberg Institute for Theoretical Studies, Heidelberg, Germany

Dandan Xu & Robert J. J. Grand

Universitäts-Sternwarte, Ludwig-Maximilians-Universität München, Munich, Germany

New York University Abu Dhabi, Abu Dhabi, United Arab Emirates

Aura Obreja, Aaron A. Dutton & Andrea V. Macciò

Zentrum für Astronomie der Universität Heidelberg, Astronomisches Recheninstitut, Heidelberg, Germany

Instituto de Investigación Multidisciplinar en Ciencia y Tecnología, Universidad de La Serena, La Serena, Chile

Departamento de Física y Astronomía, Universidad de La Serena, La Serena, Chile

Leibniz-Institut für Astrophysik Potsdam (AIP), Potsdam, Germany

Instituto de Astrofísica de Andalucía (CSIC), Granada, Spain

Osservatorio Astrofisico di Arcetri Largo Enrico Fermi 5, Florence, Italy

Instituto de Astronomía, Universidad Nacional Autonóma de México, Mexico D.F., Mexico

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### Contributions

Text, figures and interpretation are by L.Z., G.v.d.V., H.-W.R., M.M. and S.M. Modelling is by L.Z., R.v.d.B. and G.v.d.V. Observational data are from J.F.B., M.L., G.v.d.V., J.C.W., R.G.B., S.Z. and S.F.S. Methodology is by L.Z., D.X., Y.J., A.O., R.J.J.G., A.V.M., A.A.D. and F.A.G.

## Morphology/Classification

This page provides detailed descriptions of various morphological outputs of the photometry pipelines. We also provide discussion of some methodology for details of the Photo pipeline processing please read the corresponding EDR paper section. Other photometric outputs, specifically the various magnitudes, are described on the photometry page.

The frames pipeline also provides several characterizations of the shape and morphology of an object.

Star/Galaxy Classification
The frames pipeline provides a simple star/galaxy separator in its type parameters (provided separately for each band) and its objc_type parameters (one value per object) these are set to:

ClassNameCode
Unknown UNK 0
Cosmic Ray CR 1
Defect DEFECT 2
Galaxy GALAXY 3
Ghost GHOST 4
Known object KNOWNOBJ 5
Star STAR 6
Star trail TRAIL 7
Sky SKY 8

The photometric pipeline version used for DR2 and later data (5_4) classifies objects as extended ("galaxy") or point-like ("star") based on the difference between the cmodel and PSF magnitude. An object is classified as extended if

If satisfied, type is set to GALAXY for that band otherwise, type is set to STAR . The global type objc_type is set according to the same criterion, applied to the summed fluxes from all bands in which the object is detected.

Experimentation has shown that simple variants on this scheme, such as defining galaxies as those objects classified as such in any two of the three high signal-to-noise ratio bands (namely, g, r, and i), work better in some circumstances. This scheme occasionally fails to distinguish pairs of stars with separation small enough (<2'') that the deblender does not split them it also occasionally classifies Seyfert galaxies with particularly bright nuclei as stars.

Further information to refine the star-galaxy separation further may be used, depending on scientific application. For example, Scranton et al. (2001) advocate applying a Bayesian prior to the above difference between the PSF and exponential magnitudes, depending on seeing and using prior knowledge about the counts of galaxies and stars with magnitude.

The frames pipeline extracts an azimuthally-averaged radial surface brightness profile. In the catalogs, it is given as the average surface brightness in a series of annuli. This quantity is in units of maggies'' per square arcsec, where a maggie is a linear measure of flux one maggie has an AB magnitude of 0 (thus a surface brightness of 20 mag/square arcsec corresponds to 10 -8 maggies per square arcsec). The number of annuli for which there is a measurable signal is listed as nprof, the mean surface brightness is listed as profMean, and the error is listed as profErr. This error includes both photon noise, and the small-scale bumpiness'' in the counts as a function of azimuthal angle.

When converting the profMean values to a local surface brightness, it is not the best approach to assign the mean surface brightness to some radius within the annulus and then linearly interpolate between radial bins. Do not use smoothing splines, as they will not go through the points in the cumulative profile and thus (obviously) will not conserve flux. What frames does, e.g., in determining the Petrosian ratio, is to fit a taut spline to the cumulative profile and then differentiate that spline fit, after transforming both the radii and cumulative profiles with asinh functions. We recommend doing the same here.
The annuli used are:

10.560.231
21.690.689
32.581.0321
44.411.7661
57.513.00177
611.584.63421
718.587.431085
828.5511.422561
945.5018.206505
1070.1528.2015619
11110.5044.2138381
12172.5069.0093475
13269.50107.81228207
14420.50168.20555525
15657.50263.001358149

Surface Brightness & Concentration Index
The frames pipeline also reports the radii containing 50% and 90% of the Petrosian flux for each band, petroR50 and petroR90 respectively. The usual characterization of surface-brightness in the target selection pipeline of the SDSS is the mean surface brightness within petroR50.

It turns out that the ratio of petroR50 to petroR90, the so-called inverse concentration index'', is correlated with morphology (Shimasaku et al. 2001, Strateva et al. 2001). Galaxies with a de Vaucouleurs profile have an inverse concentration index of around 0.3 exponential galaxies have an inverse concentration index of around 0.43. Thus, this parameter can be used as a simple morphological classifier.

An important caveat when using these quantities is that they are not corrected for seeing. This causes the surface brightness to be underestimated, and the inverse concentration index to be overestimated, for objects of size comparable to the PSF. The amplitudes of these effects, however, are not yet well characterized.

Model Fit Likelihoods and Parameters
In addition to the model and PSF magnitudes described on the photometry page, the likelihoods deV_L, exp_L, and star_L are also calculated by frames. These are the probabilities of achieving the measured chi-squared for the deVaucouleurs, exponential, and PSF fits, respectively. For instance, star_L is the probability that an object would have at least the measured value of chi-squared if it is really well represented by a PSF. If one wishes to make use of a trinary scheme to classify objects, calculation of the fractional likelihoods is recommended:

and similarly for f(exp_L) and f(star_L). A fractional likelihood greater than 0.5 for any of these three profiles is generally a good threshold for object classification. This works well in the range 18<r<21.5 at the bright end, the likelihoods have a tendency to underflow to zero, which makes them less useful. In particular, star_L is often zero for bright stars. For future data releases we will incorporate improvements to the model fits to give more meaningful results at the bright end.

Ellipticities
The model fits yield an estimate of the axis ratio and position angle of each object, but it is useful to have model-independent measures of ellipticity. In the data released here, frames provides two further measures of ellipticity, one based on second moments, the other based on the ellipticity of a particular isophote. The model fits do correctly account for the effect of the seeing, while the methods presented here do not.

The first method measures flux-weighted second moments, defined as:
Mxx = <x 2 /r 2 >
Myy = <y 2 /r 2 >
Mxy = <xy/r 2 >

In the case that the object's isophotes are self-similar ellipses, one can show:
Q = Mxx - Myy = [(a-b)/(a+b)]cos2&phi
U = Mxy = [(a-b)/(a+b)]sin2&phi

where a and b are the semi-major and semi-minor axes, and &phi is the position angle. Q and U are Q and U in PhotoObj and are referred to as Stokes parameters.'' They can be used to reconstruct the axis ratio and position angle, measured relative to row and column of the CCDs. This is equivalent to the normal definition of position angle (East of North), for the scans on the Equator. The performance of the Stokes parameters are not ideal at low S/N. As of DR1, frames provides measurements of adaptive moments which are better shape estimators than the Stokes parameters. Read the detailed description of adaptive moments.

Isophotal Quantities
A second measure of ellipticity is given by measuring the ellipticity of the 25 magnitudes per square arcsecond isophote (in all bands). In detail, frames measures the radius of a particular isophote as a function of angle and Fourier expands this function. It then extracts from the coefficients the centroid (isoRowC,isoColC), major and minor axis (isoA,isoB), position angle (isoPhi), and average radius of the isophote in question (Profile). Placeholders exist in the database for the errors on each of these quantities, but they are not currently calculated. It also reports the derivative of each of these quantities with respect to isophote level, necessary to recompute these quantities if the photometric calibration changes.

The adaptive moments are new quantities which were not present in the Early Data release. Read the detailed description of adaptive moments.

## A Principle Component Analysis of Galaxy Properties from a Large, Gas-Selected Sample

Disney et al. (2008) have found a striking correlation among global parameters of HI-selected galaxies and concluded that this is in conflict with the CDM model. Considering the importance of the issue, we reinvestigate the problem using the principal component analysis on a fivefold larger sample and additional near-infrared data. We use databases from the Arecibo Legacy Fast Arecibo

-band Feed Array Survey for the gas properties, the Sloan Digital Sky Survey for the optical properties, and the Two Micron All Sky Survey for the near-infrared properties. We confirm that the parameters are indeed correlated where a single physical parameter can explain 83% of the variations. When color (

) is included, the first component still dominates but it develops a second principal component. In addition, the near-infrared color (

) shows an obvious second principal component that might provide evidence of the complex old star formation. Based on our data, we suggest that it is premature to pronounce the failure of the CDM model and it motivates more theoretical work.

#### 1. Introduction

One way to understand our universe is to gain insights into the structure of galaxies. For one thing, it helps to reveal the role of dark matter in their formation and dynamics. The cosmological model that consists of a cosmological constant and the cold dark matter in addition to the ordinary baryon matter and radiation (

CDM) has been able to successfully explain the evolution of the cosmic structures especially at large scales. By measuring CMB fluctuations COBE [4, 5], WMAP [6–9], Type Ia supernovae [10], and gravitational lensing [11], this model of cosmology has by now been established as the standard model of cosmology. Among its various implications, it suggests a hierarchical, bottom-up history of structure formation that evolves from small fluctuations to galaxies, clusters, and eventually superclusters. On the other side, the success of CDM has not been as clearcut at small scales. There still exist several inconsistencies between CDM and the observations at small scales. For example, the simulations based on CDM have revealed more number of galactic satellites [12–15] and less number of disk galaxies [16] than what have been observed. Besides, the degree of emptiness in the voids is also inconsistent between theory and observation [17]. While the CDM can explain the galaxy rotational curves at large radii [18], the relatively higher density at the galactic core than that predicted by the CDM, that is, the so-called cusp-core problem, is still unresolved [19].

The hierarchical galaxy formation scenario has been actively investigated previously (e.g., [20–22]), and the tight correlation between selected galactic parameters has been studied in the past [23–30]. Moreover, the overall correlation among all major galactic variables has also been thoroughly investigated recently by Disney et al. ([31] hereafter D08) and Garcia-Appadoo et al. ([32] hereafter G09). Based on 195 galaxies, the two studies found remarkably strong correlations between six galactic variables: the 90%-light radius (

), the luminosity ( ), and the color. G09 found strong correlations among the galactic properties, including a common dynamical mass-to-light ratio within the optical radii, a correlation between surface brightness and luminosity, and a common HI surface density. D08 further showed that all the parameters can be included in one correlation and suggested that this is in conflict with the notion of the hierarchical structure formation scenario and the CDM model.

Considering the importance of this issue, we set out to reinvestigate the analysis made by D08 and G09. We note that their claim actually involves two separate subissues. First, there is the issue of whether the galactic dynamics is indeed controlled by a single parameter. Second, even if this is true, there still remains the issue of whether such a fact necessarily concludes the failure of the hierarchical structure formation scenario and the cold dark matter model. In our view the positive conclusion of the former does not necessarily lead to the affirmative conclusion of the latter. In this paper we demonstrate tentative evidence that there may exist more than one principal component among the global parameters of galaxies with regard to the first issue. As for the second issue, we argue about the importance of the nongravitational baryon physics in galactic structure formation, which renders the naive extrapolation of the hierarchical structure formation scenario from cosmic to galactic scales questionable.

In order to scrutinize the correction issue, we performed a similar analysis on global parameters of galaxies with a significantly larger database and two additional parameters, and , based on the infrared band. We include a total of 1022 galaxies from the Arecibo Legacy Fast Arecibo -band Feed Array Survey (ALFALFA [33–36]) for the atomic gas properties, and the Sloan Digital Sky Survey (SDSS [37]) for the optical properties. Recently, Toribio et al. published two papers ([38, 39] hereafter T11) with these two surveys. In T11, they assembled three samples to analyse the data. They argued that HI emission provides the most reliable way to determine the morphologies and conclude that color and HImass of gas-rich galaxies cannot be very closed. In addition, they also found the correlation between mass, radius, luminosity and ( -

) by principal component analysis (PCA [40]) for one of their subsamples. Among our 1022 galaxies, 479 of them have also been detected by the Two Micron All Sky Survey (2MASS [41]). We use these galaxies to study their near-infrared properties, which contain more stellar information.

Near-infrared studies of HI-selected galaxies had been attempted by various groups (e.g., G09 [42, 43]). Our motivation of adding the near-infrared data in our analysis is the following. The optical emission is sensitive to young stars. The near-infrared emission, on the other hand, is less affected by the young stars and is therefore a better tracer of the total stellar mass, which dominates the baryonic matter at galactic scales. We believe that the inclusion of the infrared data would provide us additional and independent information on the baryonic mass assembly history of the galaxies.

Through the PCA, we confirm that except the color, all other observables, from HI, optical to near-infrared bands, are highly correlated and dominated by a single parameter. This is true both in the optical and in the near-infrared bands and this confirms the results in D08. In addition, we also see a second component from color, especially in our near-infrared data. Because near-infrared color provides the information of integrated star formation history, it may be an evidence for complex formation history, whereas a valid structure formation theory needs to explain this observation.

The organization of this paper is as follows. We describe the data and sources in Section 2. Then several variables are adopted and applied to statistical analysis in Section 3. In Section 4, we summarize and discuss our results.

#### 2. Data

Our samples are the blind 21 cm survey from ALFALFA. This selects gas rich galaxies which also contain low luminosity and low surface brightness galaxies in higher proportion than those in an optical selection. The optical data for this study come from the SDSS DR7, which covers 12,000 deg 2 for imaging and provides spectra of 930,000 galaxies. Here we briefly describe how we select SDSS counterparts to the ALFALFA sources, and we refer to G09 for more detailed discussion about identification.

The Arecibo Telescope has a beam size of

at 21 cm. Since the majority of the HI detections have S/N

[34–36], we adopted a conservative searching radius of . We found 1233 SDSS galaxies that appear to be detected by ALFALFA. We then excluded Virgo galaxies, because their neutral hydrogen is known to undergo strong environmental impact (e.g., [44]). We also excluded galaxies whose half-light radii are too small (

) comparing to the SDSS resolution. These small half-light radii either arise from misidentifications (from stars) or would result in large uncertainties. We are left with a large sample of 1022 SDSS nearby galaxies, which have distances smaller than 254

. Among these galaxies, 889 have magnitudes that are <18 and 120 have

–20. The cumulative number counts of SDSS galaxies (e.g., [45]) are

450 at –20. Given our search radius, we therefore expect at most 1.3 misidentifications in our 889 galaxies and additional 1.3 misidentifications in our 120 galaxies. These number are sufficiently small and misidentified galaxies should not impact our analyses.

The total stellar masses of galaxies are more directly reflected by the near-infrared observations. We added the data from 2MASS to our samples. 2MASS provides , , and

-band observations of the entire sky as well as a point source catalog and an extended source catalog. We use the 2MASS All-Sky Extended Source Catalog (XSC) to find the galaxies in the HI samples. To understand the quality of the identification, we first compared the 2MASS and SDSS coordinates of the 2MASS-detected galaxies. We found that more than 90% of them have offsets between 2MASS and SDSS that are well within their half-light radii. We visually inspected all galaxies with large offsets of and found small number of cases that are likely misidentifications as well as ongoing mergers. We excluded these galaxies from our samples. Because 2MASS is shallower than SDSS, we are left with 481 reliably identified galaxies in the near-infrared. To match the same aperture of the color for SDSS and 2MASS catalog, 479 galaxies are left in our near-infrared subsamples.

From the ALFALFA-released catalog 1, 2, and 3, we obtained 1796 HI data, out of which 1265 galaxies could be found in the SDSS DR7 database. There are 32 galaxies within this set that are too faint in the optical to have reliable magnitudes and luminosities. Hence we finally used the remaining 1022 galaxies in our analyses. We have also analyzed the 195 galaxies of D08 and G09 from HIPASS [46–49] and have obtained similar results. However, since the definitions of the observational variables, such as those of the rotational velocity, are not entirely consistent between the two data sets, we only report on the results from ALFALFA. These 1022 galaxies can be regarded as a blind HI-selected sample. We deduced from the data six variables, which are (half-light radius in units of pc), (

-light radius in units of pc), (luminosity in band in solar units), (HI mass in solar units), (dynamical mass in solar units), and color ( - ).

The variables and represent the radii in the Petrosian system [45, 50, 51]. In SDSS, the parameters are petroR50 and petroR90 , respectively. Because the Petrosian system is based on circular objects, we corrected the radii with the major-to-minor axis ratios, which are the parameters deVAB_r or expAB_r in SDSS. To do this, we follow the result in [52] and [48, 49]. The authors fitted the corrections from circular to elliptical apertures as functions of major-to-minor axis ratios. We directly adopted their formulas for our corrections. By comparing the likelihoods of the de Vaucouleur and the exponential models, we chose the one with the larger likelihood between deVAB_r and expAB_r . was derived from the Petrosian system and calculated from the Petrosian magnitude, petroMag_r . In order to have the same aperture for 2MASS, we use Petrosian color, which is from petroMag_g and petroMag_i. The variable is calculated by

, in which is the rotational velocity from the HI spectra and corrected for the inclination with the major-to-minor axis ratio as what we did for and . is acquired directly from the ALFALFA database and it is derived from the HI flux. This estimation is based on the assumption that the masses from optical radius are proportional to dark matter halos.

To make sure that the masses from the HI measurement can describe the dynamical mass, we compare them with the masses based on the stellar velocity dispersion, which is

, as in [1–3]. Here, is the velocity dispersion, is the effective radius,

, and is the Sérsic index. In SDSS, the velocity dispersion can be calculated by

, is from the best fitting circularized Sérsic profile, and is the SDSS measurement within the fibers. The Sérsic data are from NYU Value-Added Galaxy catalog (VAGC [54]). After matching, we have 320 of our 1022 main galaxy sample and 164 of our 479 2MASS subsample.

Figure 1 is the comparison between the HI and the stellar dynamical masses. There is an apparent sequence, indicating that the two masses trace similar dynamics for most of the galaxies. On the other hand, the HI dynamical masses are on-average

larger than the stellar dynamical masses. This ratio is indicated by the solid line in Figure 1. This is not too surprising, since neutral gas (and ) can trace the dynamical mass to a larger distance in a dark matter halo. In our subsequent analyses, we adopted HI dynamical masses. This is consistent with the work of D08/G09 and provides a larger sample here.

Comparison between dynamical masses from HI as D08/G09 and dynamical masses from starlight [1–3]. The dots are the 320 galaxies in ALFALFA/SDSS samples and cross symbols are the 164 galaxies from ALFALFA/SDSS/2MASS subsamples. The solid line indicates that the HI dynamical masses are on-average

Because there is high degeneracy between , , and bands, we only chose band to represent the near-infrared data. Therefore, we acquired (half-light radius in band in units of pc) and (luminosity in band in solar units) of the galaxies that overlap in the ALFALFA, SDSS, and 2MASS catalogs to gain a better insight into the stars of the galaxies. More specifically, is derived from the magnitude in band, which is the parameter j_m_ext in the 2MASS database. This magnitude is based on the extrapolation of the fit to the surface brightness profile. And is the integrated half-flux radius of band, which is the parameter j_r_eff in the 2MASS database. We adopted - for the color of the near-infrared subsample. The aperture of petroMag_i in SDSS is twice of the Petrosian radius, petroRad_i. Thus we interpolate the -band magnitude in 2MASS at the same aperture as SDSS. In our 481 2MASS subsamples, 479 of them have sufficient data at different apertures to interpolate the 2MASS magnitude. Therefore our final near-infrared subsample contains 479 galaxies.

#### 3. Methods and Results

Our sample of 1022 galaxies is not only larger than that in D08 and G09 but also covers broader ranges of size, luminosity, and mass (Figure 2). Despite that our sample is still dominated by

galaxies, the minimum value of in our sample is much smaller than that of D08 and G09. As for the HI and the dynamical masses, although the median values of the two samples are similar, our sample contains a substantial amount of lower mass galaxies. Our sample also covers a broader range of the - color. Because of the larger sampling and the wider range of the galactic properties, our sample is in general more representative. However since 2MASS is shallower, our 2MASS subsample of 479 galaxies is relatively speaking less representative than that from SDSS. Even so, our 2MASS subsample is still substantially larger than that of D08 and G09.

(a)
(b)
(c)
(d)
(e)
(f)
(a)
(b)
(c)
(d)
(e)
(f) Distribution of the variables. The solid histograms are our samples and the dashed histograms are the samples of D08 and G09. We could only find 157 galaxies with rotational velocities for

It is important to investigate whether our HI-selected sample is biased against early type galaxies, since these galaxies are usually gas poor. To do so, we identify spheroidal and disk galaxies in our sample based on the morphology with a method similar to that in [55]. In the SDSS database, there are de Vaucouleur and exponential models for each galaxy. By comparing the likelihood and the fractions of the two models for our 1022 galaxies, we found that 804 galaxies are disk like and 218 galaxies are spheroidal like.

In Figure 3, we show a color-luminosity diagram for our 1022 galaxies. The spheroidal galaxies are in general more luminous and redder than the disk galaxies. This is consistent with what we expect for elliptical and spiral galaxies. Most importantly, in the color-luminosity space, the spheroidal galaxies are redder than the blue cloud although they do not yet form a complete red sequence. Our sample thus appears to include a fair number of red and elliptical galaxies. Although the bias against extremely gas-poor galaxies can be hardly avoided here, fortunately, we found no major difference between these two types in our subsequent studies. We therefore believe that the omission of extremely gas-poor galaxies should not have caused major systematic bias in our analysis.

Relations among the key parameters can be inferred from the 1022 galaxies in both ALFALFA and SDSS. For instance, it is found that the half-light radius is proportional to the 90%-light radius (

[56]) the -band luminosity is proportional to the cubic power of the half-light radius ( [57]) the HI mass is proportional to the square of the half-light radius ( [43, 58]) finally, the dynamical mass is proportional to the -band luminosity ( [25]). We found that all the correlations are evident even after including the near-infrared variables, except the color (Figures 4 and 5).

Scatter plots showing correlations between six measured variables. All the variables are in solar units and with logarithmic representation. The diagonal line is the histograms, which have vertical scales from 0 to 700.

Scatter plots showing correlations between eight measured variables, including 2MASS data and reducing to 479 galaxies. All the variables are in solar units and with logarithmic representation. The diagonal line is the histograms, which have vertical scales from 0 to 350. There are small numbers of outliers in many of the plots. They are likely misidentifications or bad photometry, and they do not impact our analyses.

As a whole, the correlations between color and other variables are much weaker than other correlations. We tested various combinations of colors and found that - gives a larger PCA correlation coefficient than other colors (e.g., - , adopted by D08). The reason could be the larger wavelength difference between and . Among the three 2MASS bands, the result based on is somewhat better in the PCA, possibly because of the better signal-to-noise ratio. Hence, we adopted - for the color when we analyzed the 479 galaxies in 2MASS. Nevertheless, our analyses show that all the correlation coefficients are smaller than 0.7, indicating that they are not so highly correlated with other parameters. Intuitionally, more luminous galaxies tend to be redder because their colors are dominated by older stars. In fact, the color is more complex than any other variable because of the bias introduced by the very luminous young stars.

We conducted PCA to find correlations among the variables. PCA typically produces a series of new variables called the principal components, namely, PC1, PC2, and so forth. The correlations between these principal components and the original variables then reveal the general correlations between the particular variable and the others. In our case we found that the first principal component, PC1, is highly correlated with the six observational variables. We notice that while the color is less correlated to PC1, possibly because of the recent star formation, it is much more correlated with the second principal component, PC2. In addition to the investigation into the diagram of correlations, the eigenvalues of the correlation matrices of the original variables give quantitative information of the degree of correlations. For the 1022 galaxies based on SDSS, the eigenvalues of PC1 through PC6 are 4.29, 0.92, 0.39, 0.20, 0.18, and 0.02 (Figure 6), respectively, where the maximum possible value is 6. Based on common PCA criteria, eigenvalues larger than 1 are considered significant. We therefore only plot PC1 and PC2. Next we conducted PCA without color, and we found the eigenvalues of PC1 through PC5 to be 4.15, 0.41, 0.22, 0.20, and 0.02 (Figure 7), respectively, where the maximum possible value is 5.

The PCA results for 1022 galaxies from ALFALFA and SDSS with colors. Here we only show the strongest ones, PC1 and PC2, because other principal components are not significant by PCA criterions. PC1 is well correlated with all the variables. In the first row, the color is still correlated with the other five variables and PC1. In the second row, the rightmost plot shows that the color is even more strongly correlated with a new principal component, PC2, which is not correlated with other variables. In this plot,

The PCA results for 1022 galaxies from ALFALFA and SDSS without colors. Here we only show the strongest ones, PC1 and PC2, because other principal components are not significant by PCA criterions. PC1 is well correlated with all the variables. In this plot,

The aforementioned observations confirm the finding of D08. All the observed parameters are tightly correlated with PC1. The eigenvalue of PC1 indicates that it can explain 83% of the variance in the data (when color is not included). Color itself forms a second principal component. This might be explained by the fact that the optical color tends to be strongly affected by recent star formation activities and thus carries extra information that is unrelated to the global formation history of the galaxies. We will come back to the issue of color in Section 4. D08 claimed that the strong dominance of PC1 implies a single physical parameter to govern the structure of galaxies. However, this may be an oversimplified view of galaxies due to limited observational parameters. To test this, we extend the PCA to the near-infrared.

We included the 2MASS -band radius and the luminosity to investigate the role of stars, which dominate the baryonic matter in galaxies. For the 479 galaxies detected by 2MASS, we found the eigenvalues of PC1 through PC8 to be 5.40, 1.01, 0.61, 0.45, 0.32, 0.16, 0.03, and 0.02 (Figure 8), respectively, where the maximum possible value is 8. Conducting PCA without color again, we found that the eigenvalues of PC1 through PC7 are 5.35, 0.62, 0.45, 0.31, 0.19, 0.05, and 0.02 (Figure 9), respectively, where the maximum possible value is 7.

The PCA results for 479 galaxies from ALFALFA, SDSS, and 2MASS with colors. Here we only show the strongest ones, PC1 and PC2, because other principal components are not significant by PCA criterions. PC1 is well correlated with all the variables. The color is correlated with other variables and PC1 in the first row as well as strongly correlated with PC2 as in Figure 6. In this plot,

The PCA results for 479 galaxies from ALFALFA, SDSS, and 2MASS without colors. Here we only show the strongest ones, PC1 and PC2, because other principal components are not significant by PCA criterions. PC1 is well correlated with all the variables, as in Figure 7. In this plot,

The overall trends in the previous near-infrared PCA are similar to those in the optical PCA. When color is not included, PC1 dominates and can explain 76% of the variance in the data. The importance of PC2 slightly increases from 8% in the optical case to 9% in the near-infrared case here. When color is included, it forms another principal component by itself. These again confirm the observations of D08 that only one physical parameter governs the dynamics of galaxies.

A subtle but surprising difference between the optical and near-infrared PCAs is the behavior of color. In the optical PCA in Figure 6, although the - color forms a second principal component, it still weakly correlates with other parameters and it is part of the first principal component. On the other hand, in the near-infrared PCA in Figure 8, the - color almost does not involve in the first principal component and itself forms a second component that is almost independent of other parameters. A potential issue here is the combination of SDSS and 2MASS . It is the reason we choose consistent apertures in SDSS and 2MASS to calculate the color. To test whether other minor differences between the two photometric system hamper the correlation, we replaced the - color with the pure 2MASS - color, and we found consistent results. In addition, the median errors in and for the 1022 SDSS galaxies are 0.015 and 0.018, respectively, and the median errors in and for the 479 2MASS galaxies are 0.011 and 0.075, respectively. These translate to typical color errors of 0.023 in - and 0.076 in - . Both values are significantly smaller than the color dynamical ranges shown in Figures 4 and 5, meaning that the distribution of the data is not dominated by measurement errors. We therefore believe that the lack of correlation between the near-infrared color and other parameters is real. We will discuss more on this result in Section 4.

#### 4. Discussion and Conclusion

For the 195 galaxies analyzed in D08 and G09, the correlations among the parameters , , , , and are obvious. By selecting significant 5 times larger, 1022 overlapping samples from SDSS and ALFALFA, we also performed the PCA and confirmed through the high eigenvalues of the correlation matrices that the correlations are similarly strong. It follows that the radius, luminosity, HI mass, and dynamical mass of those chosen galaxies are tightly correlated. D08 shows that the basic parameters of galaxies are highly correlated and that there exists only one dominant principal component. From our studies, we know that this is true in both optical and near-infrared bands. Based on the hierarchical galaxy formation assumption, the dark matter halo formation has been well studied through the merger tree process (e.g., [20, 21, 59–64]). D08 believes that this scenario may not be consistent with the simple relation between all basic galactic parameters because the processes of merger would break the original galactic structures. Although our analysis has confirmed D08’s finding, we do not find it compelling or sufficient reason to reject CDM. On the other hand, the dynamical mass-to-light ratio enclosed by the optical radius that strongly anticorrelates with surface brightness has been studied [65–67]. It is not shown in our data. It may be caused by the selection effects from galaxies and insufficient dynamic range in surface brightness with SDSS and 2MASS data. Nevertheless, the strong correlations between parameter in D09 and our gas-selected samples are obvious in both optical and near-infrared data.

However, the color appears to be much less correlated, as T11 mentioned. This indeed complicates the situation and may be explained by a more sophisticated theory that would include, for example, the influence of recent star formation and very luminous young stars. Indeed these studies have already been pursued by many authors (e.g., [55, 59, 68–71]. D08 suggests that the optical color ( - in their case) consists of two components: the systematic component that correlates with other parameters (and therefore involves in the first principal component), and the rogue component that is more or less random (and forms the second principal component). It is tempting to assume that the systematic component comes from the established stellar populations and is related to the global formation history of the galaxies, while the random component is related to the ongoing star formation activities and can be short-lived events in the formation history. Surprisingly, our near-infrared analyses suggest a different story. In our near-infrared PCA, the color is even more uncorrelated with other parameters, comparing to the optical color.

One may believe that optical colors like - can be more strongly affected by recent star formation, whereas the - color may be a better tracer of integrated star formation history because it is less affected by ongoing star formation and dust extinction. If this is correct, the result that - is more related to other dynamical properties suggests that the recent star formation history is more controlled by the current dynamical structure galaxies. On the other hand, our result that - is unrelated to other dynamical properties suggests that the formation history of the old stellar components is a more chaotic process. In other words, the stronger second component may be an indication of complex formation and evolution history of galaxies. If this is the correct interpretation of our data, then the evidence of complex merger history predicted by the CDM hierarchical formation model had been hidden in the near-infrared colors and was not revealed by D08. However, we believe that this has to be tested by more observations and simulations, and it is premature to suggest the failure of the CDM model.

To sum up, we believe that the 5 times larger sample size strengthens the claim in D08 that there is one dominant parameter in galactic structure. In addition, our near-infrared analyses also provide additional insight into the color of galaxies. The tight correlations and the uncorrelated color part between the parameters in the optical and near-infrared data provide potentially powerful observational constraints on the hierarchical structure formation theory and any other cosmology models.

#### Acknowledgments

This work is supported by Taiwan National Science Council under Project no. NSC97-2112-M-002-026-MY3 (P. Chen and Y.-Y. Chang), NSC98-2112-M-001-003-MY2 and NSC99-2112-M-001-012-MY3 (W.- H. Wang and Y.-Y. Chang), and by US Department of Energy under Contract no. DE-AC03- 76SF00515. The authors also thank the support of the National Center for Theoretical Sciences of Taiwan. They are grateful to L. Lin, H. Hirashita, M. Disney, P. Kroupa, and H.-W. Rix for useful comments and discussion. The Arecibo Observatory is part of the National Astronomy and Ionosphere Center, which is operated by Cornell University under a cooperative agreement with the National Science Foundation. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the US Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS website is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium for the participating institutions. The participating institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington. This paper makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.

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## How do we calculate distances to other galaxies?

There are a few different methods, but one of the most common is the ‘standard candle’ method. This relies on the fact that if we know how bright an object in space really is (its ‘intrinsic’ brightness), then we can estimate its distance from how bright it appears to us from Earth (its ‘apparent’ brightness).

A ‘Cepheid variable’ is one type of standard candle. Cepheid variables are a type of star that have a consistent relationship between their intrinsic brightness and how fast they pulsate – so you can watch one, and if it pulsates at x speed, you know its intrinsic brightness is y.

Measuring the intrinsic brightness of a Cepheid variable, or other kinds of standard candles such as supernovae, allows astronomers to calculate the distance to the standard candle’s home galaxy.

For the most distant galaxies, standard candles are too faint to be useful, so astronomers often use the ‘Hubble-Lemaître’ law, which shows that the further a galaxy is from Earth, the faster it is moving away from us. This is just a consequence of the fact that the Universe is expanding.

Astronomers first measure the speed of the galaxy by analysing the shift in the galaxy’s light towards the red end of its light spectrum (its ‘redshift’), and once its speed is known, they can work out its distance.

## Digging up “red nuggets” in local elliptical galaxies

Last week, this astrobite discussed the formation of spiral galaxies similar to the Milky Way. The authors of the study concluded that spiral galaxies do not initially form stars starting from the inside and working outwards, but rather that the central region (the bulge) forms in conjunction with the rest of the disk. In this Letter, Huang et al. 2013 compare local elliptical galaxies (Figure 1) to their distant progenitors and conclude that, unlike spirals, massive elliptical galaxies do build up stars from the “inside-out”: first in a compact core, then in the outer layers.

Figure 1: Local elliptical galaxies from the Carnegie-Irvine Galaxy Survey. From Huang et al. 2013a, the companion paper to this Letter.

To understand why this picture makes sense, let’s look back in time. In the high-redshift universe (read: very distant and thus much younger universe, since light takes a finite time to travel to us), astronomers have observed a population of compact, massive, and red galaxies dubbed the “red nuggets”. Based on these observed properties, the red nuggets likely are a younger version of present-day elliptical galaxies. The red nuggets must have grown quickly to be so compact and massive at such an early time, and would have required a large supply of gas to create all of their stars. The authors believe the nuggets grew either through the mergers of gas-rich galaxies/protogalaxies, or by drawing in cold gas (which could easily collapse to form stars) along filaments of the universe’s large-scale structure. Something–mostly likely radiation and/or jets from the accreting supermassive black hole at the center of the galaxy–would then push out any leftover gas and prohibit the galaxy from creating new stars, as described in these astrobites.

The color of the red nuggets suggests they are “dead”, i.e., done forming stars. If the red nuggets were in the process of making new stars, the bright, blue short-lived stars would outshine the dim, red, long-lived stars. Yet somehow they must gain more stars to match observations of similar galaxies leading up to the present day. Figure 2a shows that the spatial extent of the stars in red nugget-type galaxies, quantified by the effective radius, substantially increases over time. What’s more, the stellar mass of these distant red galaxies will double or even triple. How can elliptical galaxies obtain new stars without creating them internally? These authors and others suspect that the answer comes from outside elliptical galaxies build up their outer layers by pulling in small, gas-poor satellite galaxies, in what we call “minor mergers”.

Figure 2: (a) The increase in spatial extent of red nuggets/elliptical galaxies (y-axis) as a function of redshift, z (x-axis higher z implies a larger distance). The panel on the far left shows the spatial extents of local galaxies (z=0) broken into various components.
(b) The relationship between stellar mass (x-axis) and spatial extent (y-axis). The orange points–the combined inner and middle components of local ellipticals–fall on top of the grey points–the red nuggets, indicating their similarity.

Huang et al. find evidence for this two-phase formation scenario lurking in local elliptical galaxies from the Carnegie-Irvine Galaxy Survey. Astronomers typically use 2-D surface brightness, i.e., what we can observe, to infer information about the 3-D stellar density. Traditionally, it was thought that an entire elliptical galaxy could be well-described by a single surface brightness profile proportional to R 1/4 , where R is the projected distance from the galaxy center, perhaps indicating a single formation mechanism. Here, the authors demonstrate that the surface brightness of local elliptical galaxies is better represented by three equations: two to describe the inner components, and one to describe the outer component (Figure 3). Moreover, the combined mass and spatial extent of the two inner components is strikingly similar to entire red nugget galaxies, as shown in Figure 2b. This structural evidence strongly favors a model with the two formation phases described here: one to create the red nuggets in the state at which we observe them, and another to preferentially add material to the outskirts of the galaxy later on.

Figure 3: A model of one local elliptical galaxy. The top three panels on the left simply describe the detailed shape, ellipticity, and orientation of ellipses of equal surface brightness as a function of radius (to the 1/4 power) the bottom right panel shows the residuals. Most important is the 4th panel down on the right: the surface brightness as a function of projected radius from the center, where red, green, and blue curves show the three components in the model. On the right-hand side is the observed galaxy (top), the model galaxy (middle), and the residuals, i.e., what’s left when you subtract the model from the observations (bottom). Three colored ellipses show the effective extent of each component. From Huang et al. 2013a.

Many details remain uncertain. Why are there two, subtly distinct inner components as opposed to just one? How are the red nuggets themselves structured is mass already distributed like it is now, or did these early galaxies have other features (like disks) which were later destroyed? In the future, astronomers might answer these questions by decomposing the structures of these early galaxies. While this would be difficult since the red nuggets are so far away, any information we can gain would give direct insight into the early formation of elliptical galaxies. Still, unearthing the red nuggets within our close neighbors is an important step towards a complete understanding of elliptical galaxy evolution.

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