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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, T*t*, allows us to estimate the
age of the universe, T*then*, at which the quasar's light left in order to
reach the observer at the present time, T*now*.

**T now = Tthen + Tt
(20)**

** **

**T then = Tnow - Tt
(21)**

** **

Solving equation (21) with T*t* = 12.10 By, and using
the ESA estimate for the age of the universe, T*now* = 13.82 By

T*then* = 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, Dp*now*,
and the present redshift, Z, we can calculate the proper distance of the
quasar, Dp*then*, when it emitted the photons we are presently detecting.

**Dp then = Dpnow / ( Z + 1
)
(22)**

** **

Solving the equation for APM 8279 with Z = 3.911, and our
own value for Dp*now* = 23.799 Bly:

Dp*then* = 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.

__ __

- eros312, barbarosa, dUbeni and 4 others like this

## 5 Comments

great pair of articles

thanks!

Just to see if I could, I tried (and fairly easily succeeded) in imaging a single star in the Andromeda galaxy. M31_V0619 or as you may know, commonly M31_V1.

As I looked at it, and the nearby stars that looked just exactly like it, or dimmer, it struck me that it was entirely it's apparent brightness, compared to what it should be, that distinguished it as Millions of light years away, or foreground. Simply looking at the apparent "same" nearby stars, I couldn't tell one from the other. E Hubble did, which made him great. We've understood the cosmos different ever since.

I would be interested to see your photo. M31 V0619 is a Cepheid variable, so Hubble knew its absolute magnitude and distance based on its period.

About 3 years ago, I inadvertently imaged extragalactic stars in NGC 206, a large OB Association in M 31, with a small 100 mm astrograph.

https://www.cloudyni...-m31-andromeda/

Thanks for this. I really appreciate this kind of article, making cosmology more accessible.

CORRECTION:While writing a cosmological calculator program, I noticed an error.

Equation 17) solves for MILLIONS of years, and equation 18) for BILLIONS of years. The corrected section should read as follows:

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 millions of years (My)

Tt = ( 19292 x Z ) / ( 1.3878 + Z ) for Z between 0 and 0.2 (17)

Tt = 1000( 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 = 12,102 My = 12.102 By

Apologies.