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BASIC EXTRAGALACTIC ASTRONOMY - Part 1: Redshift and Recession Velocity

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Rudy E. Kokich, Alexandra J. Kokich, Andrea I. Hudson

26 August, 2019


Part 1: Redshift and Recession Velocity



The only primary evidence available to an astronomer about a very remote object consists of photometric measurements, a spectrogram, and an image which is in many cases no more than a pinpoint of light. In this article we present basic cosmological concepts and simplified mathematical methods which allow an amateur to derive from this meager data a surprising number of physical properties of distant extragalactic objects with a precision of several percent within professional results. To illustrate the process we use the remote, gravitationally lensed quasar APM 8279+5255, also designated as QSO J0831+5245 and IRAS F08279+5255, which  is possibly the most remote object accessible to modest optical instruments.


Fig. 1: Quasar APM 8279 imaged by the authors in 2017 with a Meade 8in ACF telescope at f7 and a Canon T3i camera; 16 x 360 sec exposures at ISO 1600.


It must be mentioned that even professional results are only very approximate due to inevitable measurement errors, modeling errors, and uncertainties regarding physical constants of the universe, such as its precise age, dimension, density, total mass, and rate of expansion. For example, literature review of APM 8279 shows a distance estimate range of 12%, and the luminosity amplification factor due to lensing ranging between 4x and 100x. Its intrinsic luminosity estimates range by several magnitudes. Such widespread estimates are common in astronomy, and do not invalidate the scientific method. They simply represent the best possible interpretation of insufficient information. Over time, the estimates become more accurate as astrophysical models are refined, and additional observations become available from larger optical telescopes, space telescopes, radio telescopes, and infrared, X-ray, and gamma ray observatories.


Original scientific data about extragalactic objects, such as photomery and redshift, can be accessed at the SIMBAD Astronomical Database,

[ http://simbad.u-strasbg.fr/simbad/ ]

and at the NASA/IPAC Extragalactic Database (NED),

[ http://ned.ipac.caltech.edu/ ]


If comparing our estimates to results from online cosmological calculators, cosmological parameters used in our equations are as follows:

Omega Matter = 0.272 (normal + dark matter fraction of the total density of the universe, Ot)

Omega Radiation = 8.12E-5 (radiation fraction of the total density of the universe, Ot)

Omega Lambda = 0.728 (dark energy as a fraction of the total density of the universe, Ot)

Hubble Constant now = 70.4 (see section 13)


Total (mass-energy) density of the universe, Ot, is equal to the sum of Omega matter, Omega radiation, and Omega lambda. Critical density, Oc, is the total density required for a “flat universe” which stops expanding only after infinite time, and does not recollapse. It is equal to approximately 10^-26 kg/m^3, or 10 hydrogen atoms per cubic meter. The Ot/Oc ratio, or the density parameter, determines if the rate of expansion of the universe is slowing and leading to recollapse after finite time (Ot/Oc > 1), accelerating (Ot/Oc <1), or constant (Ot/Oc = 1). Most theoreticians anticipated a stable or slowing expansion rate, but ongoing studies of distant type Ia supernovas suggest that over the last 5 billion years the expansion rate of the universe has been actually accelerating.





If an object emitting pure green light is stationary relative to the observer, the observer will percieve it as having green color. If the object is approaching at very high velocity, the waves of its light will be compressed, the wavelength will decrease, frequency will increase, and the energy of its photons will increase. The phenomenon is called blueshift because the observer perceives the object's spectrum shifted in the blue direction. If the green object recedes at very high velocity relative to the observer, the waves of its light will be stretched, the wavelength will increase, the frequency decrease, and the energy of its photons will decrease. The observer will perceive the redshift phenomenon, where the spectrum is shifted toward the red color.


Fig. 2: Blueshift and Redshift


Fig. 3 shows how redshift appears in a spectrogram of a rapidly receding object. Compared to a stationary reference spectrum, its specral lines G and F are shifted toward longer wavelengths by approximately 4nm.


Fig. 3: Redshifted spectrum of a rapidly receding object


There are three main mechanisms which produce redshifts (and blueshifts):

1) Kinematic motion of an object through space (Doppler Effect),

2) Expansion of the universe, or the space itself between the object and the observer, and

3) Passage of light through distorted space-time caused by strong gravity fields.


In 1929 Swiss astronomer Fritz Zwicky proposed yet another possibility: The Tired Light Hypothesis, whereby photons lose energy and become redder through collisions with material particles in the intergalactic medium. Although observational evidence repeatedly disagreed with this hypothesis, it kept resurfacing over the decades as recently as 2013 when Shao et al. used it in an attempt to discredit the Big Bang theory. The hypothesis was finally put to rest by the undeniable existence of the cosmic microwave background radiation as evidence for the Big Bang. Further, the most remote galaxies detectable exhibit no prominent image blurring due to light scatter, which indicates that the vast majority of photons passing through the intergalactic medium do not experience collisions with particles of matter


Regardless of the causal mechanism, redshift, Z, is calculated from the observed wavelength, Lo, and the wavelength emitted by the object, Le, by the following relation:


Z = ( Lo - Le ) / Le                                                                                                                        (1)


Le is obtained from the known wavelengths of spectral lines on the reference spectrum, and Lo is measured on the object's spectrogram compared to the reference.


From an object's redshift, Z, we can calculate the resulting change in wavelength between Le and Lo by rearranging equation (1):


Lo = Le ( Z + 1 )                                                                                                                           (2)


Le = Lo / ( Z + 1 )                                                                                                                         (3)


Redshift of the quasar APM 8279 is 3.911. To illustrate the magnitude of this value, we will apply equation (2) to the ultraviolet H-epsilon line, Le = 3970 A:


Lo = Le ( Z + 1 )

Lo = 3970 ( 3.911 + 1 ) = 19,497 A


The line would be shifted from the lower limit of the visible spectrum far into the infrared region at Lo = 19,497A. In fact, all the visible and near-infrared (NIR) light observed from this quasar is shifted light originally emitted in the deep ultraviolet (UV) region.


Fig. 4: Optical spectrum of APM 8279 with emitted wavelengths of prominent lines


The optical spectrogram of APM 8279 is dominated by prominent lines of neutral Hydrogen (Lyman Series), ionized Nitrogen, and ionized Magnesium, all redshifted from the deep ultraviolet.


Let us stress that the redshift does not reveal if the light source is moving through space away from the observer, or if the space itself is expanding between the light source and the observer. For nearby objects, relative movement through space is the major component of wavelength shifts in the red or the blue direction. For very distant objects, the predominant cause of redshift is the expansion of the universe, or of the space itself.





Redshift is caused by the movement of a light source away from the observer. For nearby galaxies, receding at non-relativistic velocities, the approximate relation between the recession velocity of the light source, V, and the redshift, Z, is linear:


V = C x Z                                                                                                                                      (4)


C represents the speed of light, and recession velocity is expressed in fractions of the speed of light. To solve for velocity in terms of km/sec (Kilometers per second):


V = 299,792 x Z                                                                                                                            (5)


Applying equation (5) to APM 8279:


V = C x Z = 299,792 x 3.911

V = 1,172,487 km/sec


This CZ velocity suggests that at the present time APM 8279 is receding from us at nearly four times the speed of light. This is an invalid, greatly exaggerated value. Although astronomical databases list the CZ velocity for all objects, near and very distant, the value becomes increasingly meaningless for relativistic redshifts above 0.1.





We are familiar with the conclusion of special relativity which states that no physical object, field, or message can travel faster than the speed of light. However, superluminal recession velocity is possible, and is allowed by relativity when it is due to the expansion of space between an object and the observer, not to the movement of an object through space.


Fig. 5: Superluminal space expansion beyond the cosmic event horizon


Fig. 5 shows four galaxies which are at cosmological time T1 receding from the observer at velocities lower than the speed of light. At time T2, the radius of the universe doubled, and the distances to the galaxies also doubled. But galaxy D has moved at four times the average speed of galaxy A, demonstrating that more distant galaxies accelerate faster as the universe expands.


At time T2, galaxies C and D had already accelerated beyond the speed of light, and crossed the cosmic event horizon. Any photons emitted by C and D at time T2 will never reach the observer because space between them and the observer expands faster than the speed of light. However, at time T2, galaxies C and D will remain visible to the observer for billions of years because the photons they had emitted before crossing the event horizon will take billions of years to reach the observer, albeit with an ever increasing redshift and decreasing apparent magnitude.





Another model for describing recession velocity is the apparent recession velocity, Vr, defined by the following relativistic relation:


Z = [ (1 + Vr/C) / (1 - Vr/C) ]^1/2 - 1                                                                                             (6)


Solving for Vr, the relation becomes:


Vr =  C ( z^2 + 2z ) / ( z^2 + 2z +2 )                                                                                             (7)


And, solving for quasar APM 8279, Z = 3.911,


Vr =  0.9204 C = 275,929 km/sec


This is NOT the recession velocity used in Hubble’s plot.


In astronomical databases, Vr is listed simply as radial velocity, and represents the observer's perception of the object's spectrum, whether the object is receding through space, or by the expansion of space. Note that Vr can approach, but can never exceed the speed of light because it describes a galaxy which is presently visible.


This model describes the situation of galaxies C and D in Fig. 5, at time T2. Although the galaxies themselves have already crossed the cosmic event horizon, they remain visible (with an ever increasing redshift) for billions of years as the photons emitted within the event horizon continue reaching the observer.





In 1912, American astronomer Vesto Slipher discovered that light from some nebulae was redshifted, suggesting that they are receding from the Earth at high velocities. Theoretical framework for the expansion of the universe was derived in 1922 by Alexander Friedmann from Einstein’s general relativity equations. Then, in 1927, Georges Lemaitre presented the first observational evidence for a linear relationship between the recession velocity and the distance to the galaxies, as well as an estimated value for the expansion rate. In 1929, Edwin Hubble observationally confirmed Lemaitre’s findings that more distant galaxies have higher redshifts, and therefore higher recession velocities.


Fig. 6: Hubble's plot of distance vs CZ recession velocity for nearby galaxies. Recession velocities in the Virgo Supercluster deviate from the mean because they also orbit around the supercluster's center of gravity.


Since he was able to study only galaxies which are relatively close, within 30 Mpc (megaparsecs), receding at a low fraction of the speed of light, the Hubble's Law, describing the relationship between the recession velocity, V, and the distance, D, appeared to be linear:


V = Ho x D                                                                                                                                    (8)


where Ho represents the Hubble Constant, or the expansion rate, at the present cosmological time. Since Ho = V / D, where V is given in the units of km/sec, and D in Mpc (megaparsecs, or 3.26 million light years), Ho is given in the units of km/sec/Mpc. Recent values for Ho range between 67.4 and 74.03 km/sec/Mpc. In our calculations we use the value of 70.4.


The slope of the regression line drawn through Hubble's plot represents the Hubble Constant. Galaxies lying precisely on that line are said to recede by Hubble Flow, or exclusively by the expansion of the universe. Galaxies which lie outside the line are said to have peculiar velocities which deviate from Hubble Flow as the result of their motion through space. The most distinctive are the galaxies of the Virgo Supercluster which move through space at high velocities around the supercluster's center of gravity.


For non-relativistic galaxies we can define a relationship between the Hubble Constant and redshift by combining equations (4) and (8):


V = CZ = Ho x D                                                                                                                           (9)

Z = ( Ho x D ) / C                                                                                                                        (9a)

Ho = CZ / D                                                                                                                                (9b)


Within the Hubble model, V is sometimes referred to as the proper recession velocity, D as the proper distance, and the rate of expansion, Ho, is constant.


However, subsequent studies of more distant galaxies revealed that the expansion rate was markedly higher in the early universe than it is today (see section 13). And, contemporary studies are convincingly showing that the expansion rate of the universe is again increasing in the recent cosmological epochs (see section 26). The Hubble "constant" is not constant at all, and a more appropriate term might have been the Hubble parameter. Describing the relationship for objects with high redshifts required formulation of more complicated mathematical models to account for the properties of space, such as curvature and density of matter, radiation, and dark energy.





Astronomical databases and cosmological calculators also list proper recession velocity, Vp, based on more recent, mathematically complex models of the Hubble's Law. These models regard the universe as hypothetically "frozen" at selected cosmological times, so that distances within it could be measured with a “yardstick”. The universe expands between selected time frames, and the galaxies recede from each other at a rate described by the Hubble constant, H, defined in equation (8). At the present cosmological time, H is labeled as Ho, and we estimate it to be 70.4 km/sec/Mpc (megaparsec). When the universe was only one billion years old, that rate was around 660 km/sec/Mpc.


The following simplified but fairly accurate approximations, derived by regression analysis of actual results, relate the proper recession velocity, Vp, to the observed redshift, Z, in the units of the speed of light, C.


Vp = C x ( 4.871 x Z ) / ( 4.8834 + Z )               for Z between 0 and 0.2                                 (10)


Vp = C x ( 2.9365 x Z ) / ( 2.788 +Z )                for Z between 0.2 and 20                               (11)


Solving equation (11) for quasar APM 8279, Z = 3.911,


Vp = ( 2.9365 x Z ) / ( 2.788 +Z ) = ( 2.9365 x 3.911 ) / ( 2.788 + 3.911 )

Vp = 1.714 C = 1.714 x 300,000 = 514,200 km/sec


Notice that proper recession velocity is superluminal at 1.714 times the speed of light. This model describes a universe frozen at the present cosmological time, similar to galaxy D in Fig. 5, at time T2. Photons emitted by the galaxy at the present cosmological time will never reach the observer. For objects beyond the cosmic event horizon, space expands faster than the speed of light. However, the galaxy remains visible at time T2 as the photons it emitted before crossing the event horizon continue to reach the observer for billions of years.





So, which value is "correct": CZ, Vr, or Vp? The answer is that each one is correct within the context of its own model of the universe. Properties of distant objects can not be compared between models. CZ works well for nearby objects with low, nonrelativistic redshifts. Vr describes the appearance of distant objects receding at relativistic velocities, whether by kinematic motion through space or by the expansion of space. And Vp is correct within the variation of the Hubble's Law adjusted for great distances.



  • deepspace56, eros312, KiranKumar and 4 others like this


I received several questions regarding the difference between the Relativistic Recession Velocity and the Proper Recession Velocity.


Relativistic Recession Velocity, Vr, is an objective value obtained from the target's redshift. It describes the recession velocity a distant light source had in the remote past, when the photons we are presently observing were emitted.


Proper Recession Velocity, Vp, represents an estimated recession velocity of an object in the present cosmological epoch, calculated from its redshift and the currently accepted expansion rate of the universe. If Vp is greater than the speed of light, the object has already crossed the event horizon.

    • gustavo_sanchez and Eclipsed like this

Extragalactic Cosmological Calculator, CosmiCalc.exe, is finally finished. It derives a number of physical properties of remote galaxies and quasars based on objective data, such as redshift, apparent magnitude, and apparent diameter. Such data is readily available for all named objects at:
the SIMBAD Astronomical Database, [ http://simbad.u-strasbg.fr/simbad/ ]
and at the NASA/IPAC Extragalactic Database (NED), [ http://ned.ipac.caltech.edu/ ]
The program is based on equations in the CloudyNights.com series of articles titled Basic Extragalactic Astronomy. These equations are accurate within several percent of professional results for extragalactic objects which recede primarily by Hubble flow, or by the expansion of the universe. They will not be accurate for very nearby galaxies, like M31, whose motion relative to us is mostly due to the "peculiar velocity" through space.

The program is free and compatible with all Windows versions. It is self-contained, with no need for INI files or registry entries. It is "portable", which means it can be run from any location in the file system, including the Desktop, or from an external USB drive.


The program can be downloaded from Google Drive at



The source code BAS file can be opened with Notepad, and is available at



The first screenshot is for quasar 3c 273.

The second screenshot is for the distant quasar APM 8279 which is used as an example in the articles.

Suggestions on improving the program will be appreciated.

    • donstar and greywulf4570 like this

Extragalactic Cosmological Calculator v 2.0 information and download links:


Thank you, I learned some new things and reinforced some that I already knew. This series represents a lot of work, tip-o-the-cap to all of you. Steve

    • rekokich likes this

This is a nice article, but unfortunately it promotes some widely-held misconceptions, most notably (1) that the distance at which recession velocity is c (the Hubble sphere) is an event horizon beyond which we can't detect light, and (2) that special relativity (SR), as in formulae 6 and 7, can be used to interpret cosmological redshifts.


We, as comoving observers, do see galaxies that are and always have been receding faster than c, because the distance to the Hubble sphere also increases due to the metric expansion of the universe: a photon emitted towards us in a superluminal region recedes from us at Vr-c, where Vr>c is the superluminal recession velocity, but if the rate at which the radius of the Hubble sphere increases exceeds Vr-c the photon will eventually enter the subluminally receding region, and will reach us.


In lambda-CDM cosmology galaxies with redshifts z>~1.5 are receding from us superluminally now, and for galaxies with z>~2, dz/dt<0 (i.e., z decreases with time). Therefore, a galaxy with z>~2 such as APM 8279 (z=3.911) was receding superluminally at the time of emission of the radiation we see now (its redshift was >3.911 then because dz/dt<0), and in fact it always has been receding superluminally.


The interpretation of cosmological redshifts requires general relativity (GR), not SR. The latter applies to kinematic motion (motion through space in an inertial frame, but not to the metric expansion of space (it is a good approximation to the GR velocity-redshift relation only for small redshifts, z<~0.3 or so). According to GR with the lambda-CDM cosmology, the present-day recession velocity of APM 8279 is about ~1.7c.


It's not correct to say that "relativistic recession velocity" (as in SR formulae 6 and 7) describes the recession velocity a distant light source had in the remote past, when the photons we are presently observing were emitted. The calculation of recession velocity at the present (proper) time, Vobs, and at the (proper) time the detected photons were emitted, Vem, must be done (1) within GR, and (2) for an assumed cosmological model. As an example, for the Einstein-de Sitter cosmology (it's analytically simple compared with lambda-CDM, and is a reasonable representation of the universe in the redshift range ~2-300) we have in terms of observed redshift, z


Vobs = 2c[1-1/sqrt(1+z)]  (~1.1c for APM 8279)
Vem  = 2c[sqrt(1+z)-1]    (~2.4c for APM 8279)


Finally, note that the redshift of the cosmic microwave background we detect now is ~1100, and in lambda-CDM cosmology this corresponds to a present-day recession velocity of Vobs~3.2c, and Vem~50c at the time of emission! Clearly a recession velocity of c isn't an event horizon.

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