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BASIC EXTRAGALACTIC ASTRONOMY - Part 2: Distance, Luminosity, and the Hubble Parameter
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BASIC EXTRAGALACTIC ASTRONOMY
Rudy E. Kokich, Alexandra J. Kokich, Andrea I. Hudson
26 August, 2019
Part 2: Distance, Luminosity, and the Hubble Parameter
8) PROPER DISTANCE , COMOVING DISTANCE, and the COSMIC SCALE FACTOR
The easiest model of the universe to visualize is the Hubble's model adapted for great distances and high redshifts. In this universe, which is "frozen" at any specified cosmological time, distances could be measured with a "yardstick". At the present cosmological time, the proper distance, Dp, of a galaxy is related to the observed redshift, Z, by the following simplified equations, where the results are given in millions of light years (Mly) :
Dp = ( 61094 x Z ) / ( 4.3922 + Z ) for Z between 0 and 0.2 (12)
Dp = ( 40828 x Z ) / ( 2.7985 +Z ) for Z between 0.2 and 20 (13)
Solving equation (13) for quasar APM 8279, Z = 3.911,
Dp = ( 40828 x Z ) / ( 2.7985 +Z ) = ( 40828 x 3.911) / ( 2.7985 +3.911 )
Dp = 23,799 Mly = 23.799 Bly
At the present cosmological time, APM 8279 lies at the proper distance of 23.8 billion light years, well beyond the cosmic event horizon (see section 28). Since it is presently receding at the superluminal proper velocity, Vp = 1.714 C, any photons presently emitted will never reach an observer on Earth.
In the proper distance model, as cosmological time passes to the next "frozen frame", the dimensions of the universe will increase due to expansion, while the size of the "yardstick" will remain constant.
Comoving distance model is another method for calculating distances which factors out the expansion of the universe. It assumes a universe frozen at the present cosmological time so that all galaxies within it have constant spatial coordinates regardless of the passage of time. Distances between objects can still be measured with a “yardstick”, but the length of the yardstick changes with cosmological time according to the cosmic scale factor.
Fig. 7: Comoving Distance model
In Fig. 7, the coordinate grid on the right represents two galaxies at the current cosmological time, T2, located at comoving coordinates (1,2) and (5,4), and separated by comoving distance D2=4.47 units.
The coordinate grid on the left represents the two galaxies at an earlier cosmological time, T1, when the universe was 33% smaller in diameter. The galaxies are still at the same comoving coordinates, and are still separated by comoving distance D1=4.47 units, however the unit of measurement, Y1, or the “yardstick”, is 33% shorter than Y2.
Comoving coordinates make it possible to specify the position of any object in the universe independently of expansion. This is convenient for the purposes of mapping, some calculations, and computer simulations. However, in order to convert comoving distance, Dc, to proper distance, Dpt, we need to define the cosmic scale factor, At, which compensates for the decrease in scale at higher redshifts (earlier cosmological epochs). The subscript “t” indicates that the scale factor and the proper distance change with cosmological time.
Dpt = At x Dc (14)
By definition, at the present cosmological time, T2, the scale factor is equal to 1, and proper and comoving distances are numerically identical. Therefore, in Fig. 7, distance D2 represents both the comoving and the current proper distance, while distance D1 represents only the comoving distance (not the proper distance) at time T1.
Keeping in mind that the redshift of a distant galaxy is an indicator of the cosmological time when the photons were emitted, the cosmic scale factor, At, is related to the oberved redshift as follows:
At = 1 / ( Z + 1) (15)
Applying equation (15) to quasar APM 8278 with the redshift Z = 3.911:
At = 1 / (3.911 + 1) = 0.2036
Cosmic scale factor of 0.2036 means that, when the photons were emitted, the “yardstick”, or the coordinate grid unit Y1 (Fig. 7, time T1) was about five times shorter than it is today.
At the beginning of this section we calculated the proper distance of quasar APM 8279 to be 23.8 Bly at the present time (Fig.7, time T2). By definition, the quasar’s comoving distance, Dc, is also 23.8 Bly throughout all cosmological epochs. We can then use equation (14) to calculate the quasar’s proper distance, Dpt, at the time the photons were emitted:
Dpt = At x Dc = 0.2036 x 23.8
Dpt = 4.85 Bly
When the presently observed photons were emitted, the quasar’s proper distance was 4.85 Bly, well within the cosmic event horizon.
The cosmic scale factor can also be used to estimate the proper radius of the universe when the photons were emitted. In the present epoch, the proper and the comoving radii of the visible universe, Dc, are about 45.92 Bly. Then:
Dpt = At x Dc = 0.2036 x 45.92
Dpt = 9.35 Bly
These results can also be obtained by another method described in section 12).
9) THE AGE OF THE UNIVERSE
The proper distance model also contains information on the approximate age of the universe. We know that, due to the expansion of space, the quasar's and the observer's space coordinates are presently separated by the proper distance, Dp, of 23.799 billion light years, and are moving apart at the proper velocity, Vp, of 1.714 C. Since distance travelled is equal to time, T, multiplied by velocity:
Dp = T x Vp (16)
T = Dp / Vp = 23.799 / 1.714
T = 13.89 billion years
Time in this case represents the approximate age of the universe, or the time which transpired since their space coordinates were in proximity to each other, even if the object and the observer had not yet come into existence. The estimate of 13.89 billion years agrees with the European Space Agency's (ESA) estimate of 13.82 billion years, based on the 2016 study of the cosmic microwave background radiation by the Planck spacecraft.
10) LOOKBACK TIME or LIGHT TRAVEL TIME, and LOOKBACK TIME DISTANCE
All values discussed so far were derived from a single measurable parameter, the object's redshift. It is then interesting to ask, how old are the photons presently recorded in the object's spectrum? In other words, how long did it take the light to traverse the distance between the object and the observer? Or, what distance did the light have to travel through an expanding universe in order to arrive at the present time? Or, when we observe the quasar, how far back in time do we see? This time period, Tt, is called light travel time or, more appropriately, lookback time. It represents the age of a distant object which we presently see.
The equation for lookback time is also quite complex, and is substituted here with simple equations derived by regression analysis from actual results. In these equations the result is given in billions of years (By)
Tt = ( 19292 x Z ) / ( 1.3878 + Z ) for Z between 0 and 0.2 (17)
Tt = ( 0.589 + 13.87 x Z^1.25 ) / ( 0.852 + Z^1.25 ) for Z between 0.2 and 20 (18)
Solving equation (18) for quasar APM 8279, with Z = 3.911, Tt = 12.10 By
This means that the light from the quasar we observe at the present time is about 12.1 billion years old, and that the light from the quasar took 12.1 billion years to reach us at the present time through an expanding universe. Since the quasar and the observer came into existence after the conception of the universe, lookback time has to be shorter than the total age of the universe even for the most distant objects.
Note that lookback time also describes the distance the quasar's light travelled through an expanding universe before reaching the observer. This lookback time distance, Dt, determines by how much the object's light intensity falls off due to distance (divergence of light rays through space), intergalactic medium extinction, and cosmological extinction (divergence of parallel light rays due to the expansion of space).
Dt = C x Tt where C = 1, describes Dt in light years, and (19)
Dt = ( C x Tt ) / 3.26 where C = 1, describes Dt in parsecs (19a)
11) THE AGE OF THE UNIVERSE WHEN THE PHOTONS LEFT THE QUASAR
When the light we see now left the quasar, 12.1 billion years ago, the universe was much younger and smaller. During the lookback time, the light travelled against the "stream" of an expanding universe with a gradually increasing redshift due to the expansion of space between the light waves
Knowing lookback time, Tt, allows us to estimate the age of the universe, Tthen, at which the quasar's light left in order to reach the observer at the present time, Tnow.
Tnow = Tthen + Tt (20)
Tthen = Tnow - Tt (21)
Solving equation (21) with Tt = 12.10 By, and using the ESA estimate for the age of the universe, Tnow = 13.82 By
Tthen = 13.82 - 12.10 = 1.72
When the light left the quasar, the universe was only 1.72 billion years old.
12) PROPER DISTANCE OF THE QUASAR WHEN THE PHOTONS WERE EMITTED
From the proper distance at the present time, Dpnow, and the present redshift, Z, we can calculate the proper distance of the quasar, Dpthen, when it emitted the photons we are presently detecting.
Dpthen = Dpnow / ( Z + 1 ) (22)
Solving the equation for APM 8279 with Z = 3.911, and our own value for Dpnow = 23.799 Bly:
Dpthen = 23.799 / ( 3.911 + 1 ) = 4.846
When the photons now visible left the quasar, its proper distance was 4.846 Bly.
13) THE VALUE OF THE HUBBLE PARAMETER OVER TIME
Hubble's original law was based on measurements of only nearby galaxies, within 30 megaparsecs (Hubble & Humason, 1931), and the relationship between recession velocity and the distance of galaxies appeared to be linear. The law was presented as V = H x D, or H = V/D, where H, the Hubble parameter, suggested a constant value for the expansion rate of the universe. However, on very large scales, relativity predicted departures from linearity, where the type and the degree of departure depended on the total mass of the universe. Subsequent observations of objects at great distances confirmed that the Hubble parameter was markedly higher in the early universe, and has been decreasing with the passage of time. Describing that rate of change involves a complex equation which includes still disputable values for the density of matter, radiation, and dark energy in the universe, and which may have an uncertainty up to 10%. Simplified equations presented here were derived by regression analysis from actual results, and match actual results within 1%.
H = 70.04 + 28.57z + 24.127z^2 - 3.104z^3 for Z between 0 and 1 (23)
H = 58.17 + 49.74z + 11.73z^2 - 0.5251z^3 + 0.01395z^4 for Z between 1 and 15 (24)
The Hubble parameter is clearly not constant over cosmological time, which is defined here by the observed redshift, Z. But, the parameter is still regarded as a constant in a different sense: that the universe is uniform on large scales, and expands at the same rate in all directions at the same cosmological time. Depending on the method used, recent estimates for the Hubble parameter at the present cosmological time, Ho, range between 67.4 and 74.03 km/sec/Mpc. Here we use the value of 70.4, an average of a number of different estimates, which is coincidentally consistent with the latest findings by Freedman et al. (2019) of 69.8 +/- 0.8 km/sec/Mpc.
[ https://arxiv.org/pdf/1907.05922.pdf ]
[ https://phys.org/news/2019-07-hubble-constant-mystery-universe-expansion.html ]
Solving equation (24) for APM 8279:
H = 58.17 + 49.74 x 3.911 + 11.73 x 3.911^2 - 0.5251 x 3.911^3 + 0.01395 x 3.911^4
H = 403.9754
When the light reaching us from the quasar was emitted in a young universe, 12.1 billion years ago, the value of the Hubble parameter was approximately 404 km/sec/Mpc.
Fig. 8: Change in the value of the Hubble parameter at very high redshifts
While the relationship between the Hubble parameter and redshift is a reasonable approximation, it will need further refinement upon conclusion of ongoing studies which show that the expansion rate of the universe has begun to increase in more recent cosmological epochs (see section 26).
14) APPARENT MAGNITUDE, ABSOLUTE MAGNITUDE, DISTANCE MODULUS EQUATION
Luminosity is the total amount of electromagnetic radiation emitted by an object per unit time. It is frequently referenced to the Sun which is assigned the luminosity value of 1. Brightness is defined as luminosity within a specified spectral region, such as the visible light. Apparent light intensity, L, of a light source is determined by its actual light intensity and by the distance, D, between the light source and the observer.
Fig. 9: Divergence of light rays through space, where D2 = 2 x D1
As Fig.9 shows, when light rays diverge through space (not by the expansion of space itself), an observer at distance D2 will perceive four times less light intensity than an observer at distance D1. Light intensity is inversely proportional to the square of the distance: L2/L1 = ( D1/D2 )^2
Similarly, an astronomical object's apparent brightness, or apparent magnitude, depends on its actual, intrinsic brightness and on its distance from the observer. An object's actual brightness, or absolute magnitude, is defined as the brightness it would have at the standard distance of 10 parsecs, or 32.6 light years.
From the apparent magnitude, m, and the distance in parsecs, D, it is possible to calculate the absolute magnitude, M, of an object using the following equation:
m - M = ( 5 log D ) - 5 (25)
Equation (23) can be rearranged as follows:
m - M = 5 [( log D ) - 1 ] = 5 ( log D - log 10 )
m - M = 5 log ( D / 10 ) (25a)
which is known as the distance modulus equation.
For objects which are nearby, have low redshifts, and recede at low, nonrelativistic velocities, distance can be derived from Hubble's original, linear law, where V = Ho D = CZ, and D = CZ / Ho
For remote objects with high redshifts and relativistic recession velocities, D should be based on lookback time distance, Dt. If light from APM 8279, as presently observed, travelled through space for 12.1 billion years, it was attenuated over the distance of 12.1 billion light years, or 3.71 billion parsecs ( 3.71E9 parsecs ).
It is important to understand that equation (25) describes a theoretically ideal relationship between distance and brightness. It does not take into account light amplification by lensing, light absorption and scattering by the intergalactic medium and the interstellar medium (within our galaxy), or cosmological extinction due to the expansion of space through which the light travelled.
Regarding the apparent visual magnitude of APM 8279, review of photometric data in the literature and astronomical databases reveals discrepancies beyond 2.5 magnitudes within the same photometric band. While it is likely that these variances are due to a combination of the quasar’s intrinsic variability and different data collection methods, the question arises of which apparent magnitude to use in our calculations. By comparing the image of APM 8279 to similar stars in the SDSS-II (Release 7) sky survey, we estimate the quasar’s apparent magnitude to be very nearly 16 (although it is listed in the survey as 17.3). We will accept this value since our impression agrees well with the green apparent magnitude of 15.97 listed in the SIMBAD database.
Effective wavelength midpoint for a green filter is 4640 A, which is the observed wavelength, Lo. With equation (3) we can calculate the corresponding emitted wavelength, Le, to be:
Le = Lo / ( Z + 1 )
Le = 4640 / ( 3.911 + 1 ) = 944.8 A, which lies in the extreme or ionizing ultraviolet range..
To calculate the quasar’s absolute visual magnitude, we enter the apparent magnitude, m = 16, and the distance in parsecs, Dt = 3.71E9 pc, into equation (25):
m - M = ( 5 log Dt ) - 5
M = 5 + m - ( 5 log Dt ) = 5 + 16 - ( 5 log 3.71E9 ) = 21 - 47.85
M = -26.85
Absolute magnitude of -26.85 still needs to be corrected for the effects of gravitational lensing, foreground extinction, and cosmological extinction.
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