2.6 Brightness of images

For calculating the brightness of images we need the definitions and results of Section 2.4. In particular we need the luminosity distance Dlum and its relation to other distance measures. We begin by considering a point source (worldline) that emits isotropically with (bolometric, i.e., integrated over all frequencies) luminosity L. By definition of Dlum, in this case the energy flux at the observer is
F = ---L----. (52 ) 4πDlum2
F is a measure for the brightness of the image on the observer’s sky. The magnitude m used by astronomers is essentially the negative logarithm of F,
( ) m = 2.5 log10 Dlum2 − 2.5log10(L) + m0, (53 )
with m0 being a universal constant. In Equation (52View Equation), Dlum can be expresed in terms of the area distance Darea and the redshift z with the help of the general relation (48View Equation). This demonstrates that the magnification factor μ, which is defined by Equation (42View Equation), admits the following reinterpretation. |μ(s)| relates the flux from a point source at affine distance s to the flux from a point source with the same luminosity at the same affine distance and at the same redshift in Minkowski spacetime.

Dlum can be explicitly calculated in spacetimes where the Jacobi fields along lightlike geodesics can be explicitly determined. This is true, e.g., in spherically symmetric and static spacetimes where the extremal angular diameter distances D+ and D − can be calculated in terms of integrals over the metric coefficients. The resulting formulas are given in Section 4.3 below. Knowledge of D+ and D − immediately gives the area distance D area via Equation (41View Equation). D area together with the redshift determines Dlum via Equation (48View Equation). Such an explicit calculation is, of course, possible only for spacetimes with many symmetries.

By Equation (48View Equation), the zeros of Dlum coincide with the zeros of Darea, i.e., with the caustic points. Hence, in the ray-optical treatment a point source is infinitely bright (magnitude m = − ∞) if it passes through the caustic of the observer’s past light cone. A wave-optical treatment shows that the energy flux at the observer is actually bounded by diffraction. In the quasi-Newtonian approximation formalism, this was demonstrated by an explicit calculation for light rays deflected by a spheroidal mass by Ohanian [245] (cf. [298Jump To The Next Citation Point], p. 220). Quite generally, the ray-optical calculation of the energy flux gives incorrect results if, for two different light paths from the source worldline to the observation event, the time delay is smaller than or approximately equal to the coherence time. Then interference effects give rise to frequency-dependent corrections to the energy flux that have to be calculated with the help of wave optics. In multiple-imaging situations, the time delay decreases with decreasing mass of the deflector. If the deflector is a cluster of galaxies, a galaxy, or a star, interference effects can be ignored. Gould [144] suggested that they could be observable if a deflector of about − 15 10 Solar masses happens to be close to the line of sight to a gamma-ray burster. In this case, the angle-separation between the (unresolvable) images would be of the order 10 −15 arcseconds (“femtolensing”). Interference effects could make a frequency-dependent imprint on the total intensity. Ulmer and Goodman [327] discussed related effects for deflectors of up to 10− 11 Solar masses. Femtolensing has not been observed so far. However, it is an interesting future perspective for lensing effects where wave optics has to be taken into account. This would give practical relevance to the theoretical work of Herlt and Stephani [155156] who calculated gravitational lensing on the basis of wave optics in the Schwarzschild spacetime.

We now turn to the case of an extended source, whose surface makes up a 3-dimensional timelike submanifold 𝒯 of the spacetime. In this case the radiation is characterized by the surface brightness B (= luminosity L per area) at the source and by the intensity I (= energy flux F per solid angle) at the observer. For each past-oriented light ray from an observation event pO and to an event pS on 𝒯, we can relate B and I in the following way. By definition, the area distance Darea relates the area at the source to the solid angle at the observer, so we get from Equation (52View Equation) 2 2 I = BDarea ∕Dlum. As area distance and luminosity distance are related by a redshift factor, according to the general law (48View Equation), this gives the relation

B I = ----------. (54 ) 4π(1 + z)4
This result is, of course, valid only if the radiation from different parts of the emitting surface is incoherent; otherwise interference effects have to be taken into account. The most remarkable feature of Equation (54View Equation) is that all distance measures have dropped out. Save for a redshift factor, the (observed) intensity of a radiating surface is the same for all observers.

The law for point sources (52View Equation) and the law for extended sources (54View Equation) refer to bolometric quantities, i.e., to integration over all frequencies. As every astronomical observation is restricted to a certain frequency range, it is actually necessary to consider frequency-specific quantities. For a point source, one writes ∫∞ L = 0 ℓ(ωS )d ωS and ∫ ∞ F = 0 f(ωO )dωO, where the specific luminosity ℓ is a function of the emitted frequency ωS and the specific flux f is a function of the received frequency ωO. As ωS and ωO are related by a redshift factor, the frequency-specific version of Equation (52View Equation) reads

ℓ (ω (1 + z))(1 + z) f (ωO ) = ----O---------------. (55 ) 4πDlum2
Similarly, for an extended source one introduces a specific surface brightness b and a specific intensity i such that B = ∫∞ b(ω )dω 0 S S and I = ∫∞ i(ω )dω 0 O O. Then one gets the following frequency-specific version of Equation (54View Equation).
b(ωO (1 + z)) i(ωO) = -4π(1-+-z)3-. (56 )
The results summarized in this section can also be derived from the kinetic theory of photons (see, e.g., [89]). In the photon picture, the three redshift factors in Equation (56View Equation) are easily understood: The first reflects the fact that each photon undergoes a redshift; the second relates the rate of emission (with respect to proper time at the source) to the rate of reception (with respect to proper time at the obsever); the third reflects the aberration effect on the angular size of the source in dependence of the motion of the observer.

As an example for the calculation of the brightness of images we consider the Schwarzschild spactime (see Figure 17View Image).

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