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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
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 high redshift SN Ia 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.
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).
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)
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.
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