P

LfJ< A e(9,<,v)dQ dv where again the integration is over the entire sky, but since e is small outside the antenna beam, the integration is effectively over the antenna beam. Also, we invoked d^ = sin 9 d9 d<.

The denominator of (31) represents the energy-gathering power of the antenna without reference to the specific intensity I(9,<,v) of the sky. It is a quantity that can be calculated precisely for a given antenna and receiver system. An approximate value of Iav may be obtained without integration from an approximation of this quantity. Adopt an average efficiency eav, within a beam of effective solid angle A^eff steradians over an effective frequency bandwidth Aveff. Then (31) becomes P

A eav A^eff Aveff

One can estimate the average specific intensity Iav simply by dividing the observed power by the product, A eav A^eff Aveff.

Antennas with high angular resolution enable us to learn about the smallest features of the specific intensity of the sky. Unfortunately, all positions on the sky can not be observed with high resolution because it could take too long. The number of photons from a small region of sky is small and it takes a long time to accumulate enough for a significant measurement. In addition, there are many more positions on the sky that must be sampled if the same overall solid angle is to be measured.

In contrast, antennas with broad beams will view large regions of the sky and hence collect large numbers of photons, but the angular resolution will be poor. Such measurements provide the broad features of the specific intensity function. Similar statements may be made with regard to frequency resolution. In practice both high and low resolution measurements have their place. These several types of measurements tell us what the sky really looks like as a function of angular position and frequency.

Spectral flux density revisited

Relation to specific intensity

The spectral flux density S(v) used for the study of point sources can be expressed in terms of the specific intensity. Consider the flux from an extended celestial source impinging upon a flat surface of unit area (a very simple antenna) and calculate the amount of flux passing through the surface area (Fig. 4),

S(v) = / I(9, 0, v) cos 9 dQ (Spectral flux density (8.33) J® J0 from specific intensity)

—) ( W ) (sr) m2 Hz m2 Hz sr where the integration is over the angles subtended by the source and 9 = 0 specifies the direction normal to the unit area. The cosine factor represents the reduced effective area presented to the incoming flux at angles off the normal; it is a simple version of e(9, v). The spectral flux density S(v) represents all the power reaching the unit area from the entire range of angles of the source.

If S(v) is measured, the expression (33) yields an expression for I(v)av. Let the source lie near 9 = 0 and let its extent be small so cos 9 * 1 over the source. Again use the definition of an average to obtain, fe¡0I(e,0,v) cos9 dQ s(v) -2

The average specific intensity is merely the spectral flux density divided by the solid angle of the source (or of the antenna if the source fills the beam). This also follows directly from the units (or meanings) of the two quantities.

Figure 8.4. A flat surface of unit area receiving flux from an extended (diffuse) source. The spectral flux density S(v) (W m-2 Hz-1) incident on the surface is obtained from an integration of the specific intensity I(0,0, v) over the angles of the source with the projected area taken into account by the factor cos 0.

Figure 8.4. A flat surface of unit area receiving flux from an extended (diffuse) source. The spectral flux density S(v) (W m-2 Hz-1) incident on the surface is obtained from an integration of the specific intensity I(0,0, v) over the angles of the source with the projected area taken into account by the factor cos 0.

Specific intensity of pulsars

If a source is point-like at celestial position 0,0 we mean only that its angular size is smaller than the resolution of the instrument imaging it. In this case one measures the flux density S. What can one say about the specific intensity? If the resolution (beam size) is AQbeam, one only knows that the solid angle subtended by the source is less than this. This leads to a lower limit to the specific intensity,

It is sometimes possible to limit the size of the solid angle of the source from other evidence. Upon the discovery of isolated neutron stars as radio pulsars, the rapid time variation of the intensity allowed astronomers to deduce upper limits to the size of the radiating regions. For example, the peak width of the repetitive pulse emitted by the pulsar in the Crab nebula is At ~ 2 ms. All parts of a flaring region must physically communicate with each other in this time, so that each part knows that the other parts are flaring. (Otherwise how would one part know when to flare?) Since the communication can occur no faster than the speed of light, this limits the overall size of the flaring region to D = cAt or 600 km in this case. The flaring region must be smaller than this.

At the distance of the Crab, r = 6000 LY ~ 6 x 1019 m, this 600-km size yields an angular size for the emission region of D/r = 10-14 rad and a solid angle AQ ~

(D/r)2 = 10 28 sr. With a measured peak flux density S = 10 23 W m 2 Hz 1 at frequency 430 MHz, one finds that,

S(v) 10 - 23 5 ,ii I(v)av > AO = 77^28 = 105 W m-2 Hz-1 sr-1 (8.36)

^"timing 10

This specific intensity is so high as to be physically implausible for emission from a body or plasma in thermal equilibrium. It can be shown, see (11.24), that this value of I implies a temperature of ~1027 K at 430 MHz. At much lower temperatures, ~1012 K, the particles in a hot gas will have sufficient energy to create mesons, protons and neutrons in profusion. Thus, energy input into a region of limited size will tend to go into particle production rather than into more kinetic energy (i.e., temperature) of the existing particles.

Even for non-equilibrium particle distributions such as those found in synchrotron radiation (Chapter 11), such high intensities are implausible because the high energy electrons collide with, and lose their energies to the very photons they are emitting; this is called inverse Compton scattering. This too occurs at an apparent temperature of ~ 1012 K. This is called the Compton limit. More on these topics will be in Astrophysics Processes; see Preface.

The high specific intensity of radiation from radio pulsars was one piece of evidence that other processes were at work. Coherent radiation by groups of relativistic electrons streaming along magnetic field lines in the direction of the observer can result in high observed specific intensities. Similarly, jets of material emerging from a source at highly relativistic speeds, v ~ c relative to, and in the direction of, the observer, can appear to be extremely bright because of relativistic effects.

Surface brightness

We define here the surface brightness and show that it is numerically equal to the specific intensity measured with a distant antenna.

Power emitted from a surface Heretofore, the specific intensity has been used as a standard way to describe the power received by a telescope. However, the units of specific intensity, W m-2 Hz-1 sr-1, are also appropriate for the description of the power emitted from a surface. Consider a mathematical surface (Fig. 5) of area dA in the upper atmosphere of a star, with its normal in the 9,( direction of the observer. The horizontal "surface" of the star, with normal at 9 = 0, is distinct from this surface. The power radiated from unit area of this surface per unit solid angle in the (9,() direction in unit bandwidth (Av = 1 Hz) at frequency v is traditionally called surface brightness, d«

Figure 8.5. Radiation leaving the surface of a celestial object in the direction 9,< toward an observer into solid angle d«. The surface brightness B(9,<,v) in units of W m-2 Hz-1 sr-1 describes the power emitted per unit solid angle in the direction 9,< and per unit frequency interval at frequency v from a unit area that is perpendicular to the view direction 9,<.

Figure 8.5. Radiation leaving the surface of a celestial object in the direction 9,< toward an observer into solid angle d«. The surface brightness B(9,<,v) in units of W m-2 Hz-1 sr-1 describes the power emitted per unit solid angle in the direction 9,< and per unit frequency interval at frequency v from a unit area that is perpendicular to the view direction 9,<.

B(9, v) = Surface brightness (W m-2 Hz-1 sr-1) (8.37)

(Do not confuse with B magnitudes or with baseline B.)

The surface brightness describes the power coming from the apparent surface of objects such as the sun or the Crab nebula as a function of angle and frequency. The power emitted Pm from a physical surface dA into frequency interval dv at frequency v and into solid angle d« = sin 9 d9 d< is m dpm = B(9,<,v) dA dv d« (W; emitted power) (8.38)

This serves to define B(9,<,v). Note that the units balance.

Equality of emitted and received intensity (B = I)

The units of surface brightness B (37) and of the specific intensity I (26) are identical (W m-2 Hz-1 sr-1). Here we demonstrate that, in a given observation of a diffuse (resolved) source, their magnitudes are equal, B = I. This is an amazing result that tells us that the measurement from earth of a specific intensity gives us directly the actual numerical value of the surface brightness at the surface of the object being observed no matter its distance. This statement is true if one neglects absorption of all kinds and redshifts due to Hubble expansion of the universe.

Consider first the geometry illustrated in Fig. 6a. A very large cloud of atoms, e.g., a distant nebula, emits with a uniform surface brightness B. In Fig. 6b, an antenna at distance r looks out at this sky with a beam that is significantly smaller in solid angle than that of the cloud. The portion of the cloud that is viewed by the

Figure 8.6. Geometry for proof that surface brightness equals the specific intensity, B = I. (a) Radiation from an atom in the cloud is directed toward the telescope of area da which subtends a small solid angle d«. (b) The relatively large beam of the telescope, of solid angle dQ, views a (large) segment dA of the cloud.

beam of the antenna has area dA. This area subtends a solid angle dQ = dA/r2 as viewed from the antenna. The area of the antenna has the smaller value da and faces the source region. Viewed from the cloud (Fig. 6a), the antenna subtends a small solid angle d« = da/r 2. Each atom of the cloud is presumed to radiate isotropically (equally in all directions).

The power received by the antenna is, in terms of the detected specific intensity I and from (29), dA

where care is required in choosing the correct elements of area and solid angle from Fig. 6. This again is simply the definition of specific intensity.

Each atom in the cloud can radiate toward the antenna, into a cone of solid angle d«. The power emitted toward the antenna by all the atoms that are in the antenna beam, within the area dA,is, from (38), dpm = B(0,0,v) dA dv d« = B dA dvdO- (8.40)

The emitted power d Pm is defined here to be exactly that which is emitted toward the antenna, all of it and no more. It must therefore be equal to the received power dpec given in (39), dpm = dpec (Equal powers) (8.41)

Substitute (39) and (40) into (41); the result is B = I, as anticipated,

The import of (42) is that the surface brightness of a diffuse celestial object such as a supernova remnant can be measured directly, without additional calibration factors, knowledge of the distance, etc. The specific intensity at the antenna gives directly the brightness of the celestial surface! Therein lies the importance of the concept of specific intensity; it is the same thing as surface brightness!

Consider an optical CCD image of an optical nebulosity, such as a supernova remnant, the Cygnus Loop. The energy collected by a 1-m2 telescope and deposited on one pixel corresponding to one square arcsecond is a direct measure of the energy emitted into a 1"x 1" solid angle by a 1-m2 segment of the nebulosity which is located in the region that is focused on the pixel. It is not a matter of being proportional; it is an equality that yields the numerical value of the surface brightness (in W m-2 Hz-1 sr-1) of that part of the supernova remnant. The physical conditions of the plasma emitting the radiation can be deduced with the aid of this fundamental quantity.

The ability to measure surface brightness independent of its distance means that a particular nebula will yield the same measured specific intensity regardless of its distance from the antenna, as long as its angular size is larger than the antenna beam. The detected power per unit solid angle does not depend on the distance of the emitting nebula.

This is not as unreasonable as one might first think. Consider two shells of identical emitting material, one twice the distance from the antenna as the other (Fig. 7), r2 = 2r1. Each atom of the farther shell will only provide 1 /4 the radiation to the antenna compared to an atom in the closer shell, because the flux varies as r-2. However, the diverging antenna beam intersects four times more area of the farther shell than it does of the closer because the intercepted area varies as r2. Hence, four times as many emitting atoms in the farther shell are viewed by the antenna. Thus the power received by the antenna is the same in the two cases.

Antenna beam dr

Antenna

Shell #1

Shell #2

Antenna beam dr

Antenna

Shell #1

Shell #2

Figure 8.7. Antenna viewing shells of emitting gas at two distances, one twice the other. The specific intensity I detected from shell #1 equals that from shell #2.

This gives us another way to justify the relation I = B. Consider the specific intensity of a body like the sun, and bring it closer and closer to your detector until the sun's "surface" is right against your detector. We have shown that the measured specific intensity I will not change during this process. When the sun's surface is finally up against the detector, the detector is in fact measuring directly the emitted brightness B. The geometry of Fig. 5 applies; your detector is the surface in the figure with radiation entering its backside. Thus one again finds that I = B , as long as absorption and redshifts are not present.

Liouville's theorem

The origin of this equality lies in a fundamental theorem of physics, known as Liouville's theorem. It makes use of the concept of phase space, a six-dimensional space with three spatial dimensions, x, y, z and three momentum dimensions, px, py , pz . The density in phase space is the number of particles or photons per unit (vol mom)3, with SI units, m-3 (kg m s-1)-3 = (m2kg s-1)-3.

The theorem states that the density of particles in phase space remains constant even as those particles travel through space under the influence of no forces or of a restricted class of "smooth" forces, of which magnetic fields are an example. Thus cosmic rays (charged particles) traversing the Galaxy, spiraling along the interstellar magnetic field lines, will have the same phase-space density at the earth as they do at their origin, to the extent particles are not removed by collisions en route. (We examine this topic further in Astrophysics Processes; see Preface.) The specific intensity (W m-2 Hz-1 sr-1) is uniquely related to the phase-space density, so it too is conserved as the photons travel through space.

Interesting implications of this equality B = I surround us, for example starting a fire with a magnifying glass by focusing the sunlight. Another ramification is that radiation passing through an optical system has the same brightness before and after its passage. Thus the moon has the same brightness when viewed with binoculars as when it is viewed with naked eyes. The image is bigger but the energy received per pixel of the retina, or per square arcmin, is the same. This is true of the sun also. Thus the filter material safe for naked eye viewing of the sun is also safe for viewing the sun with binoculars as long as it is fully covers both objective lenses.

Energy flow - names, symbols, and units

The names and symbols for the specific intensity and related quantities are hardly standardized in practical use. This is less than helpful to the suffering student (and teacher). However, we try to be consistent within this text. A summary of the names and units of the several energy-related quantities we use are given in Table A4 of the Appendix. Each is directly related to the specific intensity I through an appropriate integration. In fact, the entries immediately following the first line are sequential integrals of I .It is clear from the table that other quantities could be defined, e.g., W Hz-1, by appropriate integration of I.

One often sees the specific intensity (W m-2 Hz-1sr-1) designated with a subscript v as Iv. The subscript v is used to designate clearly that the quantity is per unit frequency interval (Hz-1), our usual definition of I. Similarly, the subscripted variable Ix (W m-2 m-1 sr-1 = W m-3 sr-1) is used to designate the specific intensity per unit wavelength interval (m-1). There is a one-to-one conversion from Ix to Iv,

c which follows from the condition Iv dv = - Ix dA; see (11.12).

In this context, the symbol I without subscript is likely to refer to the specific intensity integrated over the entire frequency (or wavelength) band of the measurement. If so, it would have units (W m-2 sr-1) and could be called the integrated intensity, I = / Iv dv. This quantity is not included in Table A4.

In this text, the subscripts are dropped, and the following correspondences are adopted:

Similarly, the spectral flux density, S(v) can have subscripts v or A. Again our practice is S = Sv.

The key to clarity in this sea of definitions is the dimension of the units applied to a given number or symbol. In general, it is dangerous to trust the English language in this matter. On the other hand, the units tell all. One must take care to specify the units of symbols used as well as the units of numerical values.

Volume emissivity

A source that appears extended on the sky in the angular directions 9,$ is undoubtedly also extended along the line of sight, toward and away from the observer. Such diffuse objects are three-dimensional clouds of material, often gases with some degree of ionization. It is convenient and physical to consider the amount of power emitted by unit volume of an emitting cloud of atoms or ions, namely the volume emissivity, j (Wm-3 Hz-1), j(r,v) = Volume emissivity

where the vector r specifies the location of the differential volume element. This is the power emitted from unit volume of the plasma into all directions, per unit frequency interval. This is sometimes designated jv.

The volume emissivity derives from the actual physical processes that give rise to the emission of photons. If the cloud is transparent (optically thin) to its own radiation, one can easily relate j to the surface brightness B of the cloud. We now know that B is equal to the specific intensity I observed by an earth-bound antenna. Thus the observed quantity I can be related to a physical quantity j(v), as we now demonstrate.

Relation to specific intensity

Let the emission from a volume element in the cloud be isotropic; the radiation leaves the source in all directions equally. Since all directions constitute 4n sr, the power emitted into one steradian is

Further assume that this cloud is optically thin; it does not absorb radiation emitted toward the observer by more distant parts of the cloud. It is also possible to consider optically thick clouds that severely absorb their own radiation, or clouds that partially absorb it. (See discussion of radiative transfer in Section 11.5.) The optically thin case is presented here; the surface brightness B of such a cloud may be calculated from the volume emissivity.

Consider a cloud of thickness A (m) viewed by an antenna with a beam of solid angle dfi (Fig. 8). A volume element of the cloud of area d A (facing the antenna) and thickness dr along the line of sight is shown as an inset to the figure. The emitted power per unit volume, and per (Hz sr) is given by (46). Multiply this by the volume of our element dV = d A dr to obtain (j d A dr )/(4n) (WHz-1 sr-1), the power emitted from the volume element per (Hz sr). Next divide by the area of the element d A to obtain the emitted power per unit area of the cloud as well as per (Hz sr). This, by definition, is the incremental emitted surface brightness dB from one slice dr of the cloud, j d A dr j dr dB = ---= ---(Wm-2 Hz-1 sr-1) (8.47)

4n dA 4n

This relates dB and the volume emissivity j.

Now, from (42), the detected specific intensity I is equal to the surface brightness B. Thus, for the first shell of thickness dr, j dr dI = dB = — (8.48)

Figure 8.8. Extended transparent cloud containing atoms that are emitting photons in all directions. The cloud is divided into thin spherical shells of thickness dr along the line of sight. Emission from a volume element dA dr (inset) leads to a relation between the observed specific intensity I (W m-2 Hz-1 sr-1) and the volume emissivity j (W m-3 Hz-1) of the cloud. The latter quantity is power emitted by unit volume of the gas into unit frequency interval.

Figure 8.8. Extended transparent cloud containing atoms that are emitting photons in all directions. The cloud is divided into thin spherical shells of thickness dr along the line of sight. Emission from a volume element dA dr (inset) leads to a relation between the observed specific intensity I (W m-2 Hz-1 sr-1) and the volume emissivity j (W m-3 Hz-1) of the cloud. The latter quantity is power emitted by unit volume of the gas into unit frequency interval.

Note that in (48), the contribution to the specific intensity dI measured by the antenna is independent of the distance r of the shell. This is in accord with our previous discussion about the distance independence of specific intensity.

The radiation from all the shells may now be summed to obtain the specific intensity (brightness) of the entire cloud, of thickness A (Fig. 8). If the volume emissivity j is constant throughout the cloud, each additional thickness dr makes an additional equal contribution to the measured specific intensity I because distance to the antenna is not a factor and because the cloud does not absorb its own radiation Thus, one can sum over the slices of cloud,

i where A is the thickness of the cloud.

If j is not constant throughout the cloud but varies with distance r, integration is required,

1 fA

In this expression, if j is independent of r,

The expression (50) tells us that the detected specific intensity I is a measure of the volume emissivity j summed along the line of sight through the entire thickness of the cloud. It also can be written in terms of the average volume emissivity jav measured through the thickness of the cloud, jav =

IQ' jdr

Use this to eliminate the integral in (50),

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