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BASIC EXTRAGALACTIC ASTRONOMY - Part 3: Luminosity Corrections, Cosmological Extinction, and Mass to Luminosity Conversion
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BASIC EXTRAGALACTIC ASTRONOMY
Rudy E. Kokich, Alexandra J. Kokich, Andrea I. Hudson
26 August, 2019
Luminosity Corrections, Cosmological Extinction,
15) CORRECTING FOR LUMINOSITY AMPLIFICATION BY GRAVITATIONAL LENSING
The difference between magnitudes, M2 and M1, of two objects is related to the difference in brightness, B2 and B1, by the following relation:
M2 - M1 = -2.5 log (B2 / B1) (26)
Quasar APM 8279 appears to have the absolute magnitude M2 = -26.85. However, due to lensing, its observed brightness is amplified by a factor of 4. Therefore B2 = 4, and B1 = 1. Applying these values in equation (26):
M2 - M1 = -2.5 log (B2 / B1)
M1 = 2.5 log (B2 / B1) + M2 = 2.5 log (4 / 1) - 26.85 = 1.51 - 26.85
M1 = -25.34
The absolute optical magnitude of APM 8279 corrected for lensing amplification is -25.34. This value has to be further corrected for foreground and cosmological extinction.
16) THE STRUCTURE OF THE INTERGALACTIC MEDIUM
Interstellar space within galaxies, or Interstellar medium, is filled wirh ionized and neutral Hydrogen and Helium, as well as more complex elements, molecules, and dust ejected from supernovae and envelopes of some stars. In the galactic plane, this interstellar matter is visible to the naked eye in the form of gas and dust lanes which completely obscure the center of the Milky Way in the optical band. Even in the Solar neighborhood, at the periphery of the galaxy, the effect of starlight extinction is quite substantial, reducing apparent magnitude by approximately 1.8 magnitudes (81%) per kiloparsec (3260 light years).
Intergalactic medium (IGM), also called warm-hot intergalactic medium (WHIM), is far less dense, with 1-10 baryon (ordinary matter) particles per cubic meter, and chemically more simple. It is composed mostly of primordial Hydrogen, Helium, and small amounts of Lithium, mixed with hot plasma at temperatures between 100,000 and 10,000,000 K. High temperatures result from collisionless shock heating due to strong gravity waves emanating from extreme events such as accretion around supermassive black holes, and black hole and galaxy collisions. Computer simulations suggest that WHIM is arranged into a giant web of tenuous filaments, where galaxy clusters form in higher density regions at intersections between the filaments.
Fig. 10: Structure of the warm-hot intergalactic medium. Light from a distant object passes through numerous filaments of matter on the way to the observer. More distant filaments have higher reshifts.
One line of evidence for the filamentous structure of the intergalactic medium is a feature in the spectrum of quasars called the Lyman-alpha Forest, a crowded series of lines resulting from the absorption of the quasar's ultraviolet Lyman-alpha light by the intervening neutral Hydrogen clouds, each with a different redshift. The Lyman Forest lies to the left (toward shorter wavelengths) of the observed Ly-alpha line because intervening clouds are successively closer to the observer than the source of light, and therefore recede at lower velocities, with lower redshifts.
Fig. 11: The Lyman-alpha Forest of quasar APM 8279, Z = 3.911
The Lyman-alpha Forest is an important feature in the study of the distribution and temperature of neutral Hydrogen clouds in the intergalactic medium. The number of lines in the Lyman Forest generally increases for more distant quasars, up to the most ancient quasars with a redshift around 6. Above that point, the Forest changes into a broad absorption band called the Gunn-Peterson absorption trough, indicating that the earliest galaxies and quasars formed in a young universe filled with misty neutral Hydrogen. Later on, during the reionization phase, ultraviolet light from the first stars ionized nearby Hydrogen clouds, and clarified the universe.
Fig. 12: Density of the Lyman Forest increases with increasing redshift
17) CORRECTING FOR INTERGALACTIC MEDIUM EXTINCTION
Although the density of baryonic particles in intergalactic space is very low, distances between galaxies are immense, which results in substantial starlight extinction. However, estimating intergalactic extinction is a highly complicated process because total extinction is due to a number of variables.
1) Extinction due to the opacity of intergalactic medium (IGM) is distance (redshift) dependent, and non-linear. At greater distances, looking further back in time, space becomes significantly more opaque to short wavelengths due to higher concentrations of neutral Hydrogen at redshifts above 1. Menard et al. (2010) estimated extinction rates to be three times higher at Z=1 than at Z=0.5.
2) Light from a remote quasar passes through numerous foreground galaxies, galaxy halos, and clusters of galaxies which are frequently invisible. These contain chemically complex dust and gas, and have signifficantly higher light attenuation properties. Additionally, foreground objects may actually amplify light by gravitational lensing.
3) Extinction is due to absorption and scattering of electromagnetic radiation, and is greatly dependent on the wavelength of emitted light. The absorption component selectively attenuates specific wavelengths resulting in dark spectral absorption lines. The scattering component disproportionately disperses and exponentially attenuates shorter wavelengths.
4) On its way to the observer, distant quasar light inevitably passes through various regions of the Milky Way, where it is additionally subjected to much higher extinction rates than in the intergalactic medium. Extinction estimates are highly dependent on the object’s galactic coordinates, or its position relative to the Milky Way galaxy. Objects observed near the galactic plane suffer disproportionately higher total extinction rates due to the effect of local interstellar extinction.
In the end, accurate determination of IGM extinction is a major problem in cosmology, and beyond the ability of these authors. We will use a mean intergalactic extinction value, which is accoring to Wszolek et al. (1988) 0.0001 visual magnitudes/Mpc, with the understanding that it is only a broad approximation, and that it is probably markedly higher for the most remote objects formed before the reionization phase.
For APM 8279, where lookback time distance, Dt = 3.71E9 = 3.71x10^9 pc, IGM extinction correction in visual magnitudes, Ec, is then given by the proportion:
Ec / Dt = -0.0001 / 10^6
Ec = -( 3.71x10^9 x 0.0001 ) / 10^6
Ec = -0.371 visual magnitudes
Note that Ec is negative because this correction increases the absolute magnitude of the distant object.
This value is almost certainly a gross underestimate for an object residing in an early, less ionized universe, but we will accept it for the sake of illustration.
Extinction correction is defined as the difference between the corrected, M2, and uncorrected, M1, magnitudes.
Ec = M2 – M1, or (27)
M2 = M1 + Ec (27a)
Using equation (27a), where M1 = -25.34 is the absolute magnitude of APM 8279 corrected for lensing amplification, and M2 is the absolute magnitude "corrected" for IGM extinction:
M2 = M1 + Ec = -25.34 - 0.371
M2 = -25.71
Absolute optical magnitude of APM 8279 “corrected” for IGM extinction is –25.71. This value needs to be further corrected for cosmological extinction.
18) CORRECTING FOR COSMOLOGICAL EXTINCTION
The dual nature of light has been well documented experimentally. Under one set of conditions light behaves as wave-form energy, and under another set as a stream of discrete particles, or photons. Considering a unit volume of space which contains electromagnetic radiation and travels with it at the speed of light, photon number density is the number of photons per unit volume, and photon energy density, is the number of wavelengths contained in the unit volume. In the case of visible light, the human eye interprets photon number density as brightness, and photon wave-energy density as color. Photon energy is directly proportional to radiation wave frequency, and inversely proportional to wavelength. Higher energy photons have shorter wavelengths and bluer color, while lower energy photons have longer wavelengths and redder color. By definition (see equation (2)), redshifted photons have lower energy and longer wavelengths when observed than when they were emitted.
When Hubble documented in 1929 that more distant galaxies have higher redshifts, it was not clear to the astronomers whether redshift is caused by the kinematic recession of galaxies through space (Doppler effect), or by the expansion of space itself.
Within a year of Hubble’s discovery, American physicist Richard Tolman (1930, 1934) developed a theoretical method to test for the expansion of the universe.
If the universe were expanding, as shown in Fig. 13, the unit volume of space would increase by the cube of the distance, and the photon number density would decrease by the cube of the distance. Surface brightness of a “standard galaxy” would decrease by (z+1)^4, where three (z+1) factors come from the decrease in photon number density due to expanding unit volume, and one (z+1) factor comes from the decrease in photon energy density, or photon reddening, manifested as the redshift.
If the universe were not expanding, and the redshift of distant galaxies were caused by kinematic recession alone, unit volume of space in Fig. 13 would remain constant between distances D1 and D2, as would the photon number density. Surface brightness of a “standard galaxy” would then decrease by only one factor of (z+1) due to a decrease in photon wave-energy density related to the Doppler redshift.
[ https://pdfs.semanticscholar.org/28e3/7018fdfe9da023c2f96c2d46bb93827ecb73.pdf ]
Fig.13: Divergence of parallel light rays resulting from the expansion of space. In an expanding universe unit volume increases by the cube of the distance. At twice the distance, photon number density, F, decreases by a factor of 8, and photon energy density decreases by a factor of 2
While Tolman’s surface brightness test was well founded in theory, it has been difficult to implement due to a number of practical reasons, such as inadequate telescope resolution for high redshift objects, and a precise definition of a “standard galaxy” type. However, after expansion of the universe was proven by other means, Tolman’s method became useful in quantifying cosmological extinction, or the extinction of light due to the expansion of space itself.
In the NASA/IPAC Extragalactic Database (NED) cosmological extinction for an object is listed as Surface Brigtness Dimming in the Quantities Derived from Redshift section. In decimal fractions, cosmological extinction, Ec, is related to redshift (distance) as follows:
Ec = 1 – [ 1 / (z+1)^4 ] (28)
For APM 8279, z = 3.911:
Ec = 1 – [ 1 / (3.911+1)^4 ] = 1 – 0.00172 = 0.99828
This means that the expansion of the universe allows only 0.172% of the quasar’s light to reach us, while 99.828% of the light is attenuated, or literally diluted by the increase in volume.
We can use equation (26) to calculate cosmological extinction in stellar magnitudes, where B1 represents the attenuated (observed) brightness, and B2 the corrected (emitted) brightness:
M2 - M1 = -2.5 log (B2 / B1)
M2 - M1 = -2.5 log (1/0.00172) = -2.5 (2.764) = -6.911 = Ec in stellar magnitudes
In stellar magnitude units, Ec becomes the cosmological extinction correction to be applied to the apparent or absolute magnitude of a remote object in order to compensate for the expansion of the universe.
For APM 8279, uncorrected absolute magnitude, M1 = -25.71, and Ec = -6.911. Ec is a negative number because it increases the absolute magnitude of the quasar. Using equation (27a):
M2 = M1 + Ec
M2 = -25.71 -6.911 = -32.62
The corrected absolute optical magnitude of -32.62 qualifies APM 8279 as a hyperluminous quasar, and one of the most luminous objects in the universe. Coincidentally, our estimate agrees fairly well with the absolute magnitude of -32.2 calculated for the quasar upon its discovery in 1998, at which time it was the most luminous persistent object known in the universe.
19) BRIGHTNESS COMPARISON WITH FAMILIAR OBJECTS
The Sun’s visual absolute magnitude, M1, is +4.83, and the quasar’s visual absolute magnitude, M2 = -32.62. We can use equation (26) to compare brightnesses B2 and B1:
Mq - Ms = -2.5 log (Bq / Bs)
Ms - Mq = 2.5 log (Bq / Bs)
Bq / Bs = antilog [ (Ms - Mq) / 2.5 ] = antilog [ (4.83 + 32.62) / 2.5 ] = antilog (14.98)
Bq / Bs = 9.55 x 10^14
In the optical band, quasar APM 8279 is 9.55x10^14, or 955 trillion times brighter than the Sun. This number argees with professional estimates for the quasar's brightness: between 10^14 and 10^15 Suns.
[ https://arxiv.org/abs/astro-ph/9908052 ]
We will again use equation (26) to compare the brightness of the Milky Way, with absolute visual magnitude M1 = -20.8, to the brightness of the quasar, with absolute magnitude M2 = -32.62.
B2 / B1 = antilog ( (M1 - M2) / 2.5 ) = antilog [ (-20.8+32.62) / 2.5 ] = antilog (4.73)
B2 / B1 = 53,703
The quasar is nearly 54,000 times brighter than the Milky Way.
Using the same method to compare the quasar's brightness to that of the Andromeda Galaxy, absolute visual magnitude M1 = -21.5, we calculate that the quasar is about 28,054 times brighter.
We can now use equation (26) to calculate the apparent magnitude APM 8279 would have if it were located at the distance of the Andromeda Galaxy. Using subscript 2 for the quasar, and 3.44 for the apparent visual magnitude of Andromeda, we have:
m2 - m1 = -2.5 log (B2 / B1)
m2 = m1 - 2.5 log (B2 / B1)
m2 = 3.44 - 2.5 log(28054) = 3.44 – 11.12
m2 = -7.68
At the distance of the Andromeda Galaxy, APM 8279 would have the apparent visual magnitude of –7.68, approximately as bright as the crescent Moon, and about 20.5 times brighter than Venus at its maximum apparent magnitude of -4.4.
Absolute visual magnitude of APM 8279, M2 = -32.62, indicates the apparent brightness of the quasar at the distance of 10 pc, or 32.6 ly. In comparison, the apparent magnitude of the Sun seen from Earth is m1 = -26.74. Using equation (26) to compare brightnesses:
m1 - M2 = 2.5 log (B2 / B1)
B2 / B1 = antilog ( (m1 - M2) / 2.5 ) = antilog [ (-26.74 + 32.62) / 2.5 ] = antilog (2.352)
B2 / B1 = 224.9
At the distance of 32.6 light years, the quasar would appear 225 times brighter than the noonday sun.
20) BOLOMETRIC MAGNITUDES, BOLOMETRIC CORRECTION, and LUMINOSITIES
Visual magnitude of an object describes emitted energy, or brightness, in the visible band of the spectrum. Bolometric magnitudes, absolute, Mb, and apparent, mb, describe an object's total energy output, or luminosity, across the entire electromagnetic spectrum.
Bolometric correction, BC, is applied to the visual absolute magnitude, Mv, in order to convert it to bolometric absolute magnitude:
Mb = Mv + BC (29)
Bolometric corrections have negative values because objects emit more energy across the entire spectrum than over a limited range. The bolometric correction is small for objects like the Sun (BC = -0.08) which radiate most of their energy in the visible range, and quite high for very hot or cool objects which emit mostly in the ultraviolet and infrared bands respectively. For example, BC for a hot, spectral type O3 star is -4.3, and for a cool, type M0 star -1.21.
Bolometric correction for quasars is significantly dependent on the inclination angle of the accretion disk relative to the observer. Since this value is presently not known for APM 8279, we will resort to the mean bolometric correction for luminous, broad-lined quasars, which is -2.75 +/-0.40 according to Krawczyk et al. (2013).
[ https://arxiv.org/abs/1304.5573 ].
Mb = Mv + BC
Mb = -32.62 –2.75 = -35.37
The quasar’s approximate bolometric absolute magnitude, Mbq, is –35.37.
The Sun's bolometric absolute magnitude, Mbs = 4.83 - 0.08 = 4.75.
We use equation (26) to compare the luminosity of the quasar, Lq, to that of the Sun, Ls:
Mbq - Mbs = -2.5 log (Lq / Ls)
Mbs - Mbq = 2.5 log (Lq / Ls)
Lq / Ls = antilog [ (Mbs - Mbq) / 2.5 ] = antilog [ (4.75 + 35.37) / 2.5 ] = antilog (16.05)
Lq / Ls = 1.12 x 10^16
Over the entire electromagnetic spectrum, quasar APM 8279 releases 1.12x10^16 times more energy than the Sun.
21) MASS TO LUMINOSITY CONVERSION IN INACTIVE GALAXIES AND IN QUASARS
To estimate the quantity of matter which needs to be converted to energy in order to produce the luminosity of distant galaxies and quasars, Lq, we will use the luminosity of the Sun, Ls = 3.83x10^26 J/s, as a reference.
Einstein's mass-energy equation relates the change in solar mass, Ws, required to generate energy, E = Ls, where the speed of light, C, is 3x10^8 m/s:
Ls = E = Ws x C^2 (30)
Ws = E / C^2 = Ls / C^2 = 3.83x10^26 / (3x10^8)^2
Ws = 4.26x10^9 kg/s = 4.26x10^6 t/s
In a main sequence star, like the Sun, 4.2 billion kilograms, or 4.2 million metric tons of matter is converted to energy to produce one solar luminosity (Ls). In inactive galaxies, luminosity is generated mostly by stars. We can approximate total mass consumption of an ordinary galaxy by comparing its absolute magnitude to that of the Sun. For example, a galaxy with a brightness of 100 million Suns consumes around 100 million times more mass per second than the Sun.
Recall from section 14) that luminosity is the energy per unit time emitted by an object throughout the entire electromagnetic spectrum, while brightness is defined as luminosity within a specified spectral region, usually the visible light. In section 19) we compared the brightness of APM 8279 to that of the Sun using visual absolute magnitudes. But mass-energy reactions generate radiation throughout the electromagnetic spectrum and, to compare mass consumptions, we need to calculate luminosities based on bolometric absolute magnitudes.
In section 20) we calculated that, over the entire spectrum, the luminosity of quasar APM 8279, Lq, is 1.12x10^16 times greater than the luminosity of the Sun, or:
Lq / Ls = 1.12x10^16
Mass-energy conversion efficiency in main sequence stars, like the Sun, is about 0.7%. However, quasar luminosity is generated by accretion disks around supermassive black holes, where pressures and temperatures are much higher than within stars. Consequently, mass-energy conversion efficiency in quasars is approximately 10 times higher, and mass consumption 10 times lower to produce equivalent luminosity.
Compared to the Sun's mass consumpton, Ws = 4.26x10^9 kg/s, mass converted to energy every second by the quasar's accretion disk, Wq, is then given by the following proportion:
Wq / Ws = 0.1 x Lq / Ls
Wq / Ws = 0.1 x 1.12x10^16 = 1.12x10^15
Wq = Ws x 1.12x10^15 = 4.2x10^9 kg/s x 1.12x10^15
Wq = 4.70x10^24 kg/s = 4.70x10^21 t/s
The quasar's accretion disk converts 4.70x10^21 (4,700 billion billion) metric tons of matter into radiation every second. Since the mass of the Earth is about 6x10^21 metric tons, the quasar converts an "Earth" to radiation every 1.3 seconds.
Extreme pressure necessary for the mass to energy reaction in the accretion disk is created by gravitational compression. Extreme temperatures result from the friction between the particles of infalling matter whereby their kinetic energy is converted to heat.
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