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# BASIC EXTRAGALACTIC ASTRONOMY - Part 4: Luminosity Distance, Cosmic Dimensions, Cosmic Magnification

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**BASIC
EXTRAGALACTIC ASTRONOMY**

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

26 August, 2019

### Part 4: Luminosity Distance, Cosmic Dimensions, Cosmic Magnification

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**22)
LUMINOSITY DISTANCE**

**Luminosity
distance**, DL, is yet another distance measure in cosmology which is
derived from the **distance modulus**, *the difference between an object's
apparent and absolute magnitudes*, using equation (25) in section 14).

Distance Modulus = m - M = ( 5 log DL ) - 5 (25)

Solving for DL, we get:

5 Log DL = m - M + 5

Log DL = ( m - M + 5 ) / 5 = 0.2 m - 0.2 M + 1

DL =
10^(0.2 m - 0,2 M + 1), where *distance is measured in
parsecs*, or

**DL =
3.26 x 10^(0.2 m - 0,2 M + 1) where distance is measured in light
years (31)**

As
mentioned in section 14), *the distance modulus describes a theoretically
ideal relationship between distance and brightness, in which light travels
through perfectly transparent and perfectly uniform Euclidean space.* It
does not correct for light absorption and scatter by the intergalactic or the
interstellar medium (**foreground extinction**). And, it does not correct
for the expansion of the universe (**cosmological extinction**). As a
result, *Luminosity Distance in equation (31) gives a fair approximation of
distance only for nearby objects. It deviates exponentially from Light Travel
Time Distance, DT, of remote objects, which is derived from their redshift, and
therefore does contain information on the expansion of the universe.*

*Fig.
14: Luminosity Distance deviates exponentially from Light Travel Time Distance
at increasing redshifts*

While the lack of correction for large distances may at first seem to be a serious drawback, Luminosity Distance is in fact essential for estimating the expansion rate of the universe at different epochs.

After correcting for foreground extinction and for the distance modulus, the remaining difference between absolute and apparent magnitudes is caused by cosmological extinction resulting from the expansion of the universe. Luminosity distance, DL, does not contain information on cosmological extinction (expansion of the universe). Redshift, Z, and light travel time distance, DT, do contain that information. Plotting DL against Z (or DT), demonstrates a curve which uniquely describes the expansion rate of the universe.

*Fig.
15: Luminosity Distance vs Redshift plot calculated for a flat universe with
specified cosmological parameters*

The DL
- Z plot in Fig. 15 is *theoretical*, based on cosmological parameters
used throughout this article, and on the assumption of a flat universe in which
total mass-energy density of the universe (the sum of the three Omega values)
is equal to 1. Plotting points of experimental data onto the theoretical DL - Z
graph allows us to compare this cosmological model to reality, and to adjust
cosmological parameters accordingly.

Experimental
data needed for the plot include the *redshift* and *apparent magnitude*,
which can be precisely measured, and an accurate estimate of *absolute
magnitude* of a class of distant objects, which is necessary for the
calculation of luminosity distance. Galaxies and quasars do not qualify for
this purpose because their intrinsic luminosities have very high variance.
However, all * nearby type Ia supernovae* have peak absolute
magnitude of 19.5 (+/- 1.5) which can be precisely calibrated based on the
light curve decline rate over 15 days following the maximum. SN Ia therefore
serve as excellent

**standard candles**for the determination of luminosity distance and expansion rate of the universe at different cosmological epochs. Most theoreticians anticipated a stable or slowing expansion rate. However, a number of studies since 1998 found that

*are about 25% fainter than expected in a stable universe, leading to the announcement that the expansion rate is actually accelerating in the recent cosmological epochs. Lingering questions remain whether the nature and luminosity maximum of high-redshift Ia supernovae born in the early universe are somehow different from those of the nearby type.*

__high redshift SN Ia__https://www.pnas.org/content/96/8/4224

Another
application of the Lumnosity Distance involves *comparative luminosity
studies* of galaxies and quasars lying at similar redshifts (light travel
time distances, DT). Since they are measuring one object relative to another,
such studies do not require knowledge of precise absolute magnitudes.

**23)
COSMIC DIMENSIONS**

Due to
daily experiences, we have an intuitive understanding that more distant objects
appear smaller. * For small angles*, at twice the distance, objects
are half the size; at four times the distance, they are one quarter the size.
At a very large distance, objects turn into a single point - the so called

**vanishing point**in artistic

**linear perspective**drawing. Over comparatively short, nonrelativistic distances up to the redshift of approximately 0.1, or 1.297 billion light years, this linear perspective also holds true in cosmology. As we recede from a star, like the Sun, it becomes ever smaller, until it eventually contracts into a single point of light. The

**apparent size**of an astronomical object can be used to calculate its distance, or its

**actual size**, with the same equations used in terrestrial surveying.

*Fig.
16: Calculating actual dimensions of a distant galaxy*

In Fig. 16 there are three variables:

A = **apparent
angular radius** of the galaxy,

*in degrees*.

**Apparent angular diameter**is then 2 x A.

L =
distance to the galaxy in light years, using the *lookback time (light travel
time) distance*.

D = **actual
diameter** of the galaxy in light years (or *the same distance units used
for distance L*).

The actual diameter of a galaxy is then given by:

**D = 2
x L x Tan ( A ) for angle A measured in
degrees (32)**

Since the apparent size of galaxies is most commonly given in ArcMinutes, the equation can be rewritten as::

**D = 2
x L x Tan ( A / 60 ) for angle A measured in
ArcMin (32a)**

For apparent angular sizes given in ArcSeconds, the equation would be:

**D = 2
x L x Tan ( A / 3600 ) for angle A measured in
ArcSec (32b)**

Given
any two of the variables, we can calculate the third. Fortunately, apparent
angle, A, is easy to accurately measure, while lookback time distance, L, can
be reliably calculated from the redshift, Z, using equation (17) in section
10), where *lookback time (light travel time) value, Tt, also describes the
distance, L, in light years.*

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

As an example, let us consider a galaxy with a redshift of Z = 0.01, and apparent diameter, 2xA, of 6 ArcMin.

- Its apparent radius, A = 6' / 2 = 3 arcmin

- Substituting Z in equation (17), its lookback time Tt = 138.02 My

- And, its distance in light years L = 138,020,000 ly

Substituting these values in equation (31a):

D = 2 x 138,020,000 x Tan( 3 / 60 ) = 240,891 ly

Our hypothetical galaxy is about 241,000 ly in diameter, or twice as large as the Milky Way.

Knowledge of a galaxy's actual size and redshift derived distance, with the addition of its apparent magnitude, allows us to calculate its absolute magnitude, and to estimate its mass, number of member stars, mass consumption, internal dynamics, and gravitational effects on nearby objects.

**24)
TELESCOPE IMAGE SCALE and APPARENT SIZE OF AN OBJECT**

A very
similar set of equations can be used to calculate the **image scale** of a
telescope on the camera sensor, **field of view** of the telescope-camera
combination, and the **apparent angular size** of an object on a photograph.

*Fig.
17: Calculating the image scale on a camera sensor and the apparent size of a
target object on a photograph*

In this case, the variables are as follows:

D =
long side of the camera sensor *in mm. D = 2xR*

R = D
/ 2 *in mm.= one half the sensor size*

L =
telescope's focal length *in mm.*

A =
angle between the center and the edge of the long side of the camera sensor *in
degrees.*

S =
size of an object's image on the camera sensor *in mm.*

To calculate the field of view, we need to rearrange above equations, and solve for A:

**A =
ArcTan ( R / L ) for angle A measured in
Degrees (33)**

**A = 60
x ArcTan ( R / L ) for angle A measured in
ArcMin (33a)**

**A =
3600 x ArcTan ( R / L ) for angle A measured in
ArcSec (33b)**

As an example, let us find the field of view in ArcMin of a 1,000mm focal length telescope (L = 1000), on an APS-C camera sensor (D = 22.3mm, R = 11.15mm). Using equation (33a):

A = 60 x ArcTan ( R / L )

A = 60 x ArcTan ( 11.15 / 1000 )

A = 38.33 Arcmin = field of view per half the sensor

The **field
of view, FOV, in ArcMin** over the entire sensor is equal to 2 x A = 2 x
38.33 = 76.66 ArcMin.

The **image
scale per millimeter = FOV / D **= 76.66 / 22.3 = 3.44 ArcMin / mm.

The Canon APS-C sensor has 5,184 horizontal pixels.

The **image
scale per pixel in ArcMin** = FOV / 5184 = 76.66 / 5184 = 0.01479 ArcMin /
pix

The **image
scale per pixel in ArcSec** = 60 x FOV / 5184 = 60 x 2 x 38.33 / 5184 =
0.8873 ArcSec / pix.

There
are several ways to determine the **apparent size of an object**, which is **angle
B** on Fig. 15. The simplest method is to divide the size of the object's
image, S, by the length of the longer side of the sensor, D, then multiply the
result by the field of view, FOV.

**Apparent
Size of Object = angle B = ( S / D ) x
FOV (34)**

In Fig. 17, D is four times longer than S. Therefore S = 1, D = 4, and FOV = 76.66 ArcMin in the telescope example above.

Apparent Size of Object = angle B = ( S / D ) x FOV = ( 1 / 4 ) x 76.66 = 19.167 ArcMin

When the photograph is displayed on a computer monitor, S is the size of the object in millimeters, D is the full width of the photograph in millimeters, and FOV is calculated by substituting telescope focal length and camera sensor size in millimeters into equation (33a).

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**25)
COSMOLOGICAL MAGNIFICATION**

The **linear
perspective** mentioned in Section 23) states that * for small angles*
an apparent angular size of an object becomes linearly smaller as the distance
between the object and the observer increases. At seven times the distance, the
object appears seven times smaller. The linear perspective holds true in our
daily experience and, in cosmology, for relatively nearby objects receding at
nonrelativistic velocities. In a stable, non-expanding universe, it would hold
true for any distance.

*Fig.
18: Apparent Diameter - Distance graph for a non-expanding universe
shows the decrease in apparent size of an object with increase in distance.
Note that the X-axis is drawn on the same scale as in Fig. 19 for the sake of
comparison. It is linear for redshift, which is NOT linear for distance.*

Fig.
18 shows the decrease in size with distance in a *non-expanding universe*
of a galaxy whose initial apparent size is 10 ArcMin, and initial distance is
1,297 Mly (corresponding to the redshift of 0.1 in our universe). Of course, *in
a universe which is not expanding there would be no redshift except for the
Doppler effect, and there would be no cosmic event horizon, no cosmological
extinction, and the light travel time distance would be equal to luminosity
distance.* For the purpose of comparison, the X-axis in the graph is drawn
to the same scale as in Fig. 19, and is not linear for distance. As a result,
the curve seems to approach the vanishing point very slowly.

In an expanding universe, the apparent Diameter - Distance graph looks dramatically different.

*Fig.
19: Apparent Diameter - Distance graph for an expanding universe shows
an increase in apparent size for high redshift objects caused by the
expansion of space.*

At relativistic distances, above the redshift of approximately 0.1, something remarkable happens. As the object is moved from Z = 0.1 to greater distances, the expansion of the universe begins to play an ever increasing role, and the linear perspective trigonometry discussed in section 23) begins to fail. Initially, the object decreases in apparent size, as expected, but at a slower rate than would happen in a non-expanding universe. Around Z = 0.8, the decrease effectively stops. Near Z = 1.0, the object's apparent size actually begins to increase. And, around Z = 9.2, its apparent angular size starts becoming larger than it was at the initial position of Z = 0.1, albeit much fainter and much redder.

This
phenomenon of **cosmological magnification** is due to the expansion of
space during the time period the emitted light travelled to reach the observer.

*Fig.
20: The effect of cosmological magnification due to the expansion of space on
the apparent size, wavelength of emitted light, and the apparent brightness of
a distant galaxy with a redshift of 3.*

The
concept of *expanding unit volume* was discussed in some detail in section
18) on cosmological extinction. As a unit volume containing electromagnetic
flux travels through expanding space, the length of each side of the cube
increases by one factor of ( Z + 1 ). Consequently, the wavelength of light
inside the unit volume increases by one factor of ( Z + 1 ), leading to a
proportional *decrease* in *photon wave energy density* (redshift).
The * apparent angular diameter of the emitting object increases by one
factor of ( Z + 1 ), resulting in cosmological magnification.*. The
apparent surface area of the emitting object increases by the square of the
distance, or by ( Z + 1 )^2. The volume of the unit volume cube increases by
the cube of the distance, or by ( Z + 1 )^3, leading to a proportional

*decrease*in the

*photon number density*. Recall from section 18) that cosmological extinction is proportional to the product of the decrease in photon number density and the decrease in photon wave energy, or ( Z + 1 )^4.

These concepts yield the following equations for cosmological magnification, CM:

**CM =
Dcm / Dad = Z +
1
(35)**

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**Dad =
Dcm / CM = Dcm / ( Z + 1
) (35a)**

where *Dcm**
is the cosmologically magnified apparent angular diameter of an object, and Dad
is the apparent diameter of the object corrected for magnification (as it
would appear by linear perspective in a non-expanding universe).*

When calculating the actual diameter of a galaxy with equation (32), the apparent diameter should be corrected for cosmological magnification with equation (35a). As Fig. 20 illustrates, the effect of cosmological magnification on apparent size is quite dramatic for high-redshift objects.

In practice, modest optical telescopes are essentially unable to photograph high-redshift objects. Quasars are an exception, however they show as pinpoints with no apparent diameter. A galaxy the size of the Milky Way, with a redshift Z = 0.075, would lie at the light travel time distance of 990 Mly, and show as a small 0.4 ArcMin image with apparent magnitude of 17. Not correcting for magnification would result in the calculation error of 7%, well within measurement errors on such a small target, which would appear even smaller because the full extent of the spiral arms would be too faint to record.

On the other hand, correcting for cosmological magnification is imperative in the case of high redshift objects photographed with large telescopes. For example, the earliest protogalaxies have redshifts around 11, which means that their recorded images are 12 times magnified by the expansion of the universe.

- fiston and PartlyCloudy like this

## 1 Comments

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

https://drive.google...iew?usp=sharing

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

https://drive.google...iew?usp=sharing

The first screenshot is for quasar 3c 273.

https://www.cloudyni...2694_118274.jpg

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

https://www.cloudyni...2694_216952.jpg

Suggestions on improving the program will be appreciated.