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15.7 Lookback time and the age of the universe _{.}
In this section using the four presentday cosmological parameters 𝐻_{0}, 𝛺_{𝑚,0}, 𝛺_{𝑟,0} and 𝛺_{𝛬,0} we consider the lookback time and the age of the universe.
Remembering (14.26) of Chapter 14, if a comoving galaxy emmited a photon at cosmic time 𝑡, 'lookback time' 𝑡_{0}  𝑡 is given as a function of the photon's redshift by
(15.37) 𝑡_{0}  𝑡 = ¡ò_{𝑡}^{𝑡0} 𝑑𝑡 = ¡ò_{0}^{𝑧} 𝑑𝑧/[(1 + 𝑧)𝐻(𝑧)].
From the cosmological field equation (15.13), on noting that 𝑎 = 𝑅/𝑅_{0} = (1 + 𝑧)^{1} we obtain the useful result
(15.38) 𝐻^{2}(𝑧) = 𝐻_{0}^{2}[𝛺_{𝑚,0} (1 + 𝑧)^{3} + 𝛺_{𝑟,0} (1 + 𝑧)^{4} + 𝛺_{𝛬} + 𝛺_{𝑘,0} (1 + 𝑧)^{2}].
𝑡_{0}  𝑡 = 1/𝐻_{0} ¡ò_{0}^{𝑧} 𝑑𝑧/[(1 + 𝑧)¡î{𝛺_{𝑚,0} (1 + 𝑧)^{3} + 𝛺_{𝑟,0} (1 + 𝑧)^{4} + 𝛺_{𝛬} + 𝛺_{𝑘,0} (1 + 𝑧)^{2}}].
A more convenient integral is obtained by substituting 𝑥 = (1 + 𝑧)^{1} and we have
(15.39) 𝑡_{0}  𝑡 = 1/𝐻_{0} ¡ò^{1}_{(1 + 𝑧)1 } 𝑥/¡î(𝛺_{𝑚,0}𝑥 + 𝛺_{𝑟,0} + 𝛺_{𝛬}𝑥^{4} + 𝛺_{𝑘,0}𝑥^{2}) 𝑑𝑥.
Assuming 𝛺_{𝑟,0} = 0 reasonably, in Figure 15.17 we see 𝐻_{0}(𝑡  𝑡_{0}), the lookback time in units of the Hubble time, as a function of redshift for several values of 𝛺_{𝑚,0} and 𝛺_{𝛬}.
In cosmological model, the age of the universe, i.e. the cosmic time interval between the point of 𝑎(𝑡) = 0 and the present epoch 𝑡 = 𝑡_{0}. Since 𝑧 ¡æ ¡Ä at the big bang, we may obtain an expression of the age of the universe by letting 𝑧 ¡æ ¡Ä in (15.39), so that the lower limit of the integral equals zero. So we can write
𝑡_{0} = 1/𝐻_{0} 𝑓(𝛺_{𝑚,0}, 𝛺_{𝑟,0}, 𝛺_{𝛬,0}),
where 𝑓 is the value of the integral, which is typically a number of order unity. But it is not possible to perform the integral analytically and so one has to resort to numerical integration. Table 15.1*^{} lists the age of the universe 𝑡_{0} for the same values as considered in Figure 15.7. It is interesting to compare these values with estimates of the ages of the oldest stars in globular clusters,
𝑡_{stars} ≈ 11.5 ∓ 1.3 Gyr.
Clearly, one requires 𝑡_{0} > 𝑡_{stars} for a viable cosmology!
Simply by setting 𝑡 = 𝑡_{0} and 𝑎 = 1, we can obtain the age of the universe for each model with bigbang origin. Foe example. from (15.28) it is 𝑡_{0} = 2/(3𝐻_{0}) and from the (15.36) in case of 𝛺_{𝛬,0}) > 0 is given by
𝑡_{0} = 2/(3𝐻_{0})¡î𝛺_{𝛬,0}) sinh^{1} ¡î[𝛺_{𝛬,0}/(1  𝛺_{𝛬,0})] = 2/3𝐻_{0} (tanh^{1}¡î𝛺_{𝛬,0})/¡î𝛺_{𝛬,0}.
15.8 The distanceredshift relation _{.}
We may also obtain a general expression for the comoving 𝜒coordinate of a galaxy emitting a photon at time 𝑡 that is recieved at time 𝑡_{0} with redshift 𝑧. This is given by from (14.27)
𝜒 = ¡ò_{𝑡}^{𝑡0} 𝑐𝑑𝑡/𝑅(𝑡) = 𝑐/𝑅_{0} ¡ò_{0}^{𝑧} 𝑑𝑧/𝐻(𝑧).
We may substitute for 𝐻(𝑧) using the expression (15.38). Thus the 𝜒coordinate of a comoving object with redshift is given by
(15.40) 𝜒(𝑧) = 𝑐/𝑅_{0}𝐻_{0} ¡ò_{0}^{𝑧} 𝑑𝑧/¡î{𝛺_{𝑚,0} (1 + 𝑧)^{3} + 𝛺_{𝑟,0} (1 + 𝑧)^{4} + 𝛺_{𝛬,0} + 𝛺_{𝑘,0} (1 + 𝑧)^{2}}.
As before a simpler form for the integral is obtained by making the substitution 𝑥 = (1 + 𝑧)^{1}, which yields
(15.41) 𝜒(𝑧) = 𝑐/𝑅_{0}𝐻_{0} ¡ò_{(1 + 𝑧)1}^{1} 𝑑𝑥/¡î(𝛺_{𝑚,0}𝑥 + 𝛺_{𝑟,0} + 𝛺_{𝛬,0}𝑥^{4} + 𝛺_{𝑘,0}𝑥^{2}).
From (14.29) and (14.31), the corresponding luminosity distance 𝑑_{𝐿}(𝑧) and angular diameter distance 𝑑_{𝐴}(𝑧) to the object are given by
𝑑_{𝐿}(𝑧) = 𝑅_{0}(1+ 𝑧)𝑆(𝜒(𝑧)) and 𝑑_{𝐴}(𝑧) = 𝑅(𝑡_{0})/(1+ 𝑧) 𝑆(𝜒(𝑧)),
where 𝑆(𝜒) is given by (14.12), whereas the proper distance to the object is simply 𝑑(𝑧) = 𝑅_{0}𝑆(𝜒(𝑧)). It is useful to introduce the notation 𝜒(𝑧) = 𝑐𝐸(𝑧)/𝑅_{0}𝐻_{0}, so that 𝐸(𝑧) denotes the integral in (15.41). Using the expression (15.10) to obtain 𝛺_{𝑘,0}, one can then write
𝑅_{0}𝑆(𝜒(𝑧)) = 𝑐/𝐻_{0} ∣𝛺_{𝑘,0}∣^{1}𝑆(¡î∣𝛺_{𝑘,0}∣ 𝐸(𝑧)) for 𝛺_{𝑘,0} ¡Á 0,
𝑅_{0}𝑆(𝜒(𝑧)) = 𝑐/𝐻_{0} 𝐸(𝑧) for 𝛺_{𝑘,0} = 0,
which allows simple direct evaluation of 𝑑_{𝐿}(𝑧) and 𝑑_{𝐴}(𝑧) in each case.
As was the case in the previous section, for general value of 𝛺_{𝑚,0}, 𝛺_{𝑟,0} and 𝛺_{𝛬,0} it is not possible to perform the integral (15.41) analytically an so one has to resort to numerical integration. Figure 15.8 shows plots of dimensionless luminosity distance (𝑐/𝐻_{0})^{1}𝑑_{𝐿}(𝑧) (top panel) and dimensionless angular distance (𝑐/𝐻_{0})^{1}𝑑_{𝐴}(𝑧) (bottom panel) for various of 𝛺_{𝑚,0} and 𝛺_{𝛬,0}, assuming that 𝛺_{𝑟,0} is negligible: the solid, broken and dotted lines corresponds to spatially flat, open and closed models respectively.
The integral (15.41) can be evaluated in some simple cases. As an example, consider Einsteindesitter(EdS) model (𝛺_{𝑚,0} = 1, 𝛺_{𝑟,0} = 0, 𝛺_{𝛬,0} = 0). In this case we find
𝜒(𝑧) = 𝑐/𝑅_{0}𝐻_{0} ¡ò_{(1 + 𝑧)1}^{1} 𝑑𝑥/¡î𝑥 = 2𝑐/𝑅_{0}𝐻_{0} [1 (1 + 𝑧)^{1/2}].
Thus, the luminosity distance in EdS model is given by
𝑑_{𝐿}(𝑧) = 2𝑐/𝐻_{0} (1 + 𝑧)[1 (1 + 𝑧)^{1/2}],
and the angluardiameter distance by
𝑑_{𝐴}(𝑧) = 2𝑐/𝐻_{0} 1/(1 + 𝑧) [1 (1 + 𝑧)^{1/2}].
We note that 𝑑_{𝐴}(𝑧) has a maximum at redshift 𝑧 = 5/4^{**} and we can find it in Figure 15.8.
What we need to use the above relation are a standard candle and a standard ruler for them to fix the values of 𝛺_{𝑚,0} and 𝛺_{𝛬,0}. Though they arei very hard to find in the universe, in recent years (of 2005), using distant Type Ia supernovae as standard candles and anisotropies in the cosmic microwave background radiation as a standard ruler. The results of these observation suggest tha we live in a spatially flat universe with 𝛺_{𝑚,0} ≈ 0.3 and 𝛺_{𝛬,0} ≈ 0.7.
15.9 The volumeredshift relation _{.}
We found in Section 14.12 that the proper volume in the range 𝜒 ¡æ 𝜒 + 𝑑𝜒 and subtending an infinitesimal solid angle 𝑑𝛺 = sin 𝜃 𝑑𝜃 𝑑𝜙 at the observer is
(15.42) 𝑑𝑉_{0} = 𝑐𝑅_{0}^{2}𝑆^{2}(𝜒(𝑧))/𝐻(𝑧) 𝑑𝑧 𝑑𝛺,
where a redshift 𝑧 is given by 𝑑𝑉(𝑧) = 𝑑𝑉_{0}/(1 + 𝑧)^{3}. We may now express 𝑑𝑉_{0} in terms of 𝐻_{0}, 𝛺_{𝑚,0}, 𝛺_{𝑟,0} and 𝛺_{𝛬,0}. Using (15.40), (15.38) and (15.10) for 𝜒(𝑧), 𝐻(𝑧) and 𝛺_{𝑘} respectlvely, we find that
(15.43) 𝑑𝑉_{0} = (𝑐𝐻_{0}^{1})^{3}/𝘩(𝑧) ∣𝛺_{𝑘,0}∣^{1}𝑆^{2}¡î ∣𝛺_{𝑘,0}∣ 𝐸(𝑧) for 𝛺_{𝑘,0} ¡Á 0,
𝑑𝑉_{0} = (𝑐𝐻_{0}^{1})^{3}/𝘩(𝑧) 𝐸^{2}(𝑧) for 𝛺_{𝑘,0} = 0,
where we define
𝘩(𝑧) ¡Õ 𝐻(𝑧)/𝐻_{0} = ¡î{𝛺_{𝑚,0}(1 + 𝑧)^{3} + 𝛺_{𝑟,0}(1 + 𝑧)^{4} + 𝛺_{𝛬,0} + 𝛺_{𝑘,0}(1 + 𝑧)^{2}}, 𝐸(𝑧) ¡Õ ¡ò_{0}^{z} 𝑑𝑧/𝘩(𝑧).
One must once again resort to numerical integration to obtain 𝑑𝑉_{0}. In Figure 15.9 we plot the dimensionless differential comoving element (𝑐𝐻_{0})^{3}𝑑𝑉_{0}/(𝑑𝑧 𝑑𝛺) as a function of redshift, assuming that 𝛺_{𝑟,0} = 0. In particular, we note that we may explore a large comoving volume in the range 𝑧 = 23 for the currently favored case (𝛺_{𝑚,0}, 𝛺_{𝛬,0}) = (0.3, 0.7).
15.10 Evolution of the density parameters _{.}
It is possible to investigate of the evolution of these densities 𝛺_{𝑚}, 𝛺_{𝑟} and 𝛺_{𝛬}) as the universe expands.
From (15.5) we have the equation of 𝛺_{𝑖}(𝑡) and using Quotient rule we obtain
(15.44) 𝛺_{𝑖}(𝑡) = 8¥ð𝐺/3𝐻^{2}(𝑡) 𝜌_{𝑖}(𝑡) ¢¡ 𝑑𝛺_{𝑖}/𝑑𝑡 = 8¥ð𝐺/3𝐻^{2}[ῤ_{𝑖}  (2Ḣ/𝐻)𝜌_{𝑖}],
where the label 𝑖 denotes '𝑚', '𝑟' or '𝛬' and dots denotes differentiation with respect to cosmic time 𝑡 as usual. From the equation of motion for the cosmological fluid (14.39) we have
ῤ_{𝑖} = 3(1 + 𝑤_{𝑖})𝐻𝜌_{𝑖},
where 𝐻 = Ṙ/𝑅 and the equationof state parameter 𝑤_{𝑖} = 𝑝_{𝑖}/(𝜌_{𝑖}𝑐^{2}). Thus {15.44) becomes
(15.44) 𝑑𝛺_{𝑖}/𝑑𝑡 = 𝛺_{𝑖}𝐻[3(1 + 𝑤_{𝑖}) + 2Ḣ/𝐻^{2}].
We now find an expression for Ḣ in 𝑅, Ṙ or Ȑ and the deceleration parameter 𝑞
Ḣ = 𝑑/𝑑𝑡 (Ṙ/𝑅) = Ȑ/𝑅  (Ṙ/𝑅)^{2} = Ȑ/𝑅  𝐻^{2}, Ḣ/𝐻^{2} = 𝑅Ȑ/Ṙ^{2}  1 = (𝑞 + 1).
Substituting this result into (14.45) and using (15.14), we finally obtain the neat relation
𝑑𝛺_{𝑖}/𝑑𝑡 = 𝛺_{𝑖}𝐻(𝛺_{𝑚} + 2𝛺_{𝑟}  2𝛺_{𝛬}  1  3𝑤_{𝑖}),
Setting 𝑤_{𝑖} = 0, 1/3 and 1 respectively for matter(dust), radiation and the vacuum, we thus obtain
(15.46) 𝑑𝛺_{𝑚}/𝑑𝑡 = 𝛺_{𝑚}𝐻[(𝛺_{𝑚} 1) + 2𝛺_{𝑟}  2𝛺_{𝛬}],
𝑑𝛺_{𝑟}/𝑑𝑡 = 𝛺_{𝑟}𝐻[𝛺_{𝑚} + 2(𝛺_{𝑟}  1)  2𝛺_{𝛬}],
𝑑𝛺_{𝛬}/𝑑𝑡 = 𝛺_{𝛬}𝐻[𝛺_{𝑚} + 2𝛺_{𝑟}  2(𝛺_{𝛬} 1)].
By dividing these equations by one another, we may remove the dependence on 𝐻 and 𝑡 and hence obtain a set of copled firstorder differentiable equations. Therefore, these equations define a unique trajectory that passes through some points. As an illustration, let us consider the case 𝛺_{𝑟} = 0. Dividing two equations then gives
𝑑𝛺_{𝛬}/𝑑𝛺_{𝑚} = [𝛺_{𝛬}{𝛺_{𝑚}  2(𝛺_{𝛬}  1)}]/[𝛺_{𝑚}{(𝛺_{𝑚}  1)  2𝛺_{𝛬}}],
which define a set of trajectories or 'flow line' in the (𝛺_{𝑚}, 𝛺_{𝛬}) plane. The equation highlights the significance of the points (1, 0) and (0, 1) which act as 'attractors'. This is illustrated in Figure 15.10.
It is worth noting that the profound effect of a nonzero cosmological constant on the evolution of the density parameters. In case of 𝛬 = 0, any slight deviation from 𝛺_{𝑚} = 1 in the early universe result in a rapid evolution away from the point (1, 0) along the 𝛺_{𝑚} axis, tending to (0, 0) for an open universe and to (¡Ä, 0) for a closed one. If 𝛬 > 0, however, the trajectory is 'refocussed' and tends to the spatially flat de Sitter case (0, 1). Indded, by the time 𝛺_{𝑚} ≈ 0.3 the universe is close to spatially flat.
15.11 Evolution of the spatial curvature _{.}
We may investigate directly the behavior of the spatial curvature from (15.10) as follows
(15.47) 𝛺_{𝑘} = 1  𝛺_{𝑚}  𝛺_{𝑟}  𝛺_{𝛬} =  𝑐^{2}𝑘/(𝐻^{2}𝑅^{2}).
Differentiating the righthand side with respect to 𝑡 and combining the derivatives (15.46), one finds that
(15.48) 𝑑𝛺_{𝑘}/𝑑𝑡 = 2𝛺_{𝑘}𝐻𝑞 = 𝛺_{𝑘}𝐻(𝛺_{𝑚} + 2𝛺_{𝑟}  2𝛺_{𝛬}),
where 𝑞 is the deceleration parameter. If 𝛺_{𝛬}) = 0, then at some early cosmic time 𝑘 rapidly evolves away from the spatially flat case. In the open case of Figure 15.5 𝛺_{𝑘} ¡æ 1 and in the closed case 𝛺_{𝑘} ¡æ ¡Ä. However if 𝛺_{𝛬}) > 0, at some finite cosmic time the 2𝛺_{𝛬} term in (15.48) will dominate the others, then 𝛺_{𝑘} is 'refocussed' back to 𝛺_{𝑘} = 0.
We may obtain an analytical expression for the 𝑘 as a function of redshift 𝑧. Substituting for 𝑐^{2}𝑘 from (15.47) evaluated at 𝑡 = 𝑡_{0} and noting that 𝑅_{0}/𝑅 = 1 + 𝑧, we obtain the general formula
𝛺_{𝑘}(𝑧) = [𝐻_{0}(1 + 𝑧)/𝐻(𝑧)]^{2}𝛺_{𝑘,0}.
Using (15.38) for 𝐻(𝑧) then gives
𝛺_{𝑘}(𝑧) = 𝛺_{𝑘,0}/{𝛺_{𝑚,0}(1 + 𝑧) + 𝛺_{𝑟,0}(1 + 𝑧)^{2} + 𝛺_{𝛬,0}(1 + 𝑧)^{2} + 𝛺_{𝑘,0}}
From the above expression we find that even if 𝛺_{𝑘,0} differs greatly from zero, at very high redshift i.e. in the distant past 𝛺_{𝑘}(𝑧) must have been very close to zero. Since today we measure the value 𝛺_{𝑘,0} in the range 0.5 to 0.5, this means that at very early epochs 𝛺_{𝑘} must have been very finely tuned to near zero. This tuning of the initial conditions of expansion is called the flatnedd problem and has no solution within standard cosmological models. However, from our above discussion, the presence of a positive 𝛬 explains why the present universe is close to spatially flat.
15.12 The particle horizon, event horizon and Hubble distance _{.}
It is interesting to consider the extent of the region 'accessible' to some comoving observer at a given cosmic time 𝑡.
[Particle horizon]
Let us consider a comoving observer 𝑂 at comoving coordinates 𝜒 and a emitter 𝐸 at 𝜒_{1} which emits a photon at 𝑡_{1} that 𝑂 receives by the time 𝑡 in the condition that 𝜒 < 𝜒_{1}.
Since along the photon path 𝑑𝑠 = 𝑑𝜃 = 𝑑𝜙 = 0, from (14.11), the comoving coordinates 𝜒_{1} of the emitter 𝐸 is determined by
(15.49) 𝜒_{1} = 𝑐 ¡ò_{t1}^{t} 𝑑𝑡'/𝑅(𝑡').
If the integral on the righthand side diverges as 𝑡_{1} ¡æ 0, then 𝜒_{1} can be made as large as we please by taking 𝑡_{1} sufficiently small. In this case, it is possible to receive signals emitted at sufficiently early epochs from any comoving particle. But if the integral converges as 𝑡_{1} ¡æ 0, then 𝜒_{1} can never exceed a certain value for given 𝑡. In this case our vision of the universe is limited by a particle horizon. The 𝜒coordinate of the particle horizon is given by
(15.50) 𝜒_{𝑝𝘩}(𝑡) = 𝑐 ¡ò_{0}^{t} 𝑑𝑡'/𝑅(𝑡')
the corresponding proper distance to the particle horizon is 𝑑_{𝑝} = 𝑅(𝑡)𝜒_{𝑝𝘩}(𝑡).
On differentiating (15.50) with respect to 𝑡, we have 𝑑𝜒_{𝑝𝘩}/𝑑𝑡 = 𝑐/𝑅(𝑡), which is always greater than zero. Thus the particle horizon of a comoving observer grows as 𝑡 increases, and so the parts of the universe that were not in view previously must gradually come into view. Thus if the universe has a bigbang origin then 𝑅(𝑡_{1}) ¡æ 0 as 𝑡_{1} ¡æ 0 and so 𝑧 ¡æ ¡Ä. Thus the particle horizon at any given cosmic time is the surface of infinite redshift, beyond which we cannot see.^{***}
We can obtain explicit expression for the paricle horizon in some cosmological models. For example, a matterdominated model at early epoch obeys 𝑅(𝑡)/𝑅_{0} = (𝑡/𝑡_{0})^{2/3} and a radiationdominated model obeys 𝑅(𝑡)/𝑅_{0} = (𝑡/𝑡_{0})^{1/2}. Substituting these into (15.50) gives the proper distance to the particle horizon at 𝑡 as
𝑑_{𝑝}(𝑡) = 3𝑐𝑡 (matterdominated), 𝑑_{𝑝}(𝑡) = 2𝑐𝑡 (radiationdominated),
These proper distances are karger than 𝑐𝑡 because the universe has expanded while the photon has been travelling. Alternatively, if one has an analytical expression for 𝜒(𝑧) for some cosmological model, then 𝜒_{𝑝𝘩} may be obtained simply letting 𝑧 ¡æ ¡Ä.
The particle horizon for common cosmological models illustrates the horizon problem, i.e. how do vastly separated regions display the sam physical characteristic (e.g. the nearly uniform temperature of the cosmic microwave background). This problem is a serous challenge to standard cosmology.
[event horizon]
Although our particle horizon grows as the cosmic time increase, in some cosmological models there could be events that we may never see. Similarly to the particle horizon we use (15.49) and have following expression for the event horizon
𝜒_{𝑒𝘩}(𝑡) = 𝑐 ¡ò_{t1}^{tmax} 𝑑𝑡/𝑅(𝑡),
So 𝜒_{𝑒𝘩}(𝑡_{0}) is the maximum 𝜒coordinate that can be reached by a light signal sent by us today.
[Hubble distance]
In a similar way to define the Hubble time 𝐻^{1}(𝑡) one can define the Hubble distance
𝑑_{𝐻}(𝑡) = 𝑐𝐻^{1}(𝑡),
which provide a characteristic length scale for the universe. We may also define the comoving Hubble distance
(15.51) 𝜒_{𝐻}(𝑡) = 𝑑_{𝐻}(𝑡)/𝑅(𝑡) = 𝑐/{𝐻(𝑡)𝑅(𝑡)} = 𝑐/Ṙ(𝑡).
The Hubble distance is the length scale at which generalrelativistic effects become important; indeed, on length scales much less than 𝑑_{𝐻}(𝑡), Newtonian theory is often sufficient to describe the effect of gravitation. We further note that the proper distance to the particle horizon for standard cosmological model is typically
𝑑_{𝑝}(𝑡) ¡ 𝑐𝑡 ¡ 𝑐𝐻^{1}(𝑡).
Thus, the Hubble distance is of same order as the particle horizon and so is often described simply as the 'horizon'. But in inflationary cosmologies the particle horizon and the Hubble distance may differ by many order of magnitude. In particular, we note that the partle horizon at time 𝑡 depends on the entire expansion history of the universe to that point, whereas the Hubble distance is defined instantaneously at 𝑡. Moreover, once an object lies within an observer's particle horizon it remains so, but an object can lie within an observer's Hubble distance or outside it according to the situation.
^{} _{}
* According to this table we should take the value 13.5 Gyr which is very close to the known value 13.8 Gyr (as of 2015).
** We can solve max{1/(1 + 𝑧) [1 (1 + 𝑧)^{1/2}]} = 4/27 at 𝑧 = 5/4, (e.g. by using WolframAlpha with iPad).
*** In practice, our view of the universe is not by our particle horizon but by the epoch of recombination occured at 𝑧 ≈ 1500.
^{¡Ø} attention: some rigorous derivation might be required
p.s. Áß°£¿¡ D. Fleisch & J. Kregenow A Student's Guide to the Mathematics of Astronomy (Cambridge University Press 2013)°Ãß!ÀÇ
'spectrum fundamental' Wien's law & Stefan's law¸¦ ºñ·ÔÇØ, ÀÚ»óÇÑ ÇØ¼³ÀÇ parallax, angular size, angular resolution, luminosity, magnitude¿Í
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