Principles of Planetary Photometry

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1 Daft Nov. 4, 24 Chapte 1. Pinciples of Planetay Photomety 1. Intoduction. The subject of planetay photomety is, in substantial pat, a subset of that banch of mathematical physics known as adiative tansfe, fo which the classical and definitive wok is that of Chandasekha (196). Hee we pesent this aspect of the subject in a moden context and although we have adheed as much as possible to the symbols, nomenclatue and notation of Chandasekha, the following changes and additions have been made. (i) The quantity called by Chandasekha intensity I is hee called adiance L. I make no apology fo this since it confoms with moden intenational adiometic standads. (ii) (iii) (iv) A plane paallel beam of adiation is specified by its adiant flux density F athe than its net flux π F, the latte being a moe geneally defined quantity. Shothands fo incident, eflected and tansmitted adiation, with subscipts i, and t have been intoduced. Reflectance functions in addition to Chandasekha s fomulations ae pesented. 2. Radiance and the Equation of Tansfe. Radiance may be egaded as the fundamental quantity of adiative tansfe. d ω dp Obseve θ n da da cosθ Fig.1. Conside, as shown in figue 1, an emitting (o eflecting) su face of aea da which emits dp watts of powe into solid angle d ω about the diection of an obseve at angle θ to the suface nomal vecto n of da, the latte pesenting a pojected aea da cosθ to the obseve. The adiance detected by the obseve is then M.B. Faibain, 24.

2 Daft Nov. 4, 24 Radiance then has the following popeties: dp L = (1) dω dacosθ It is defined at a point and in a specified diection. It is independent of the distance fom which it is obseved Its SI units ae watts pe squae mete pe steadian (W m -2 s -1 ). This can be intepeted in eithe of two ways. Eithe, it is the powe pojected into unit solid angle fom unit pojected aea of an extended suface (i.e. pojected on a plane at ight angles to the line of sight fom the obseve); o it is the powe aiving pe unit aea at the obseve fom unit solid angle (subtended at the obseve) of the extended souce. That these ae equivalent is shown in efeence (2). If adiance is the fundamental quantity of adiative tansfe, then the fundamental law is the equation of tansfe dl κρds = L I. (2) dl Hee is the ate of change of adiance with, and in the diection of, position s in a ds given medium, ρ is the density of the medium (kg m -3 ) and κ is the mass attenuation coefficient (m 2 kg -1 ). Hee attenuation efes to any pocess which educes the bightness of a beam of adiation, and so includes absoption and scatteing. Some authos use the wod extinction fo attenuation, and some (paticulaly in the field of stella atmosphees) use the wod opacity to efe to the mass attenuation coefficient. Equation (2) could be ead as follows: as a beam of a diance L taveses the distance δ s it will be diminished in adiance by the amount κρ Lδ s and enhanced by the amount κρ I δ s. The quantity I is called the souce function and, as we shall see, a typical poblem of planetay photomety is to find a solution fo this quantity befoe solving the equation of tansfe as a whole. 3. Diffuse Reflection and Tansmission. The fundamental poblem of planetay photomety is the diffuse eflection and tansmission of a plane paallel beam of adiation by a scatteing medium, which we would undestand as a planetay atmosphee and/o suface o a planetay laye such as the ings of Satun. Such media may be idealised as locally plane paallel stata in which physical popeties ae unifom thoughout a given laye. In such cases we may use a hybid Catesian and spheical fame of efeence in which the Oxy plane is the suface and z-axis points in the diection of the suface nomal. Diections ae then M.B. Faibain, 24.

3 Daft Nov. 4, 24 specified by the pola and azimuthal angles ϑ and ϕ ( culy theta and culy phi ) espectively. Futhe, with poblems of this kind, athe than woking in actual physical distances it is pefeable to wok in tems of nomal optical thickness τ, measued downwads fom z =, such that dτ = κρ dz. Radiation that has tavesed a path of optical thickness t is attenuated by a facto of Using the diection cosine plane paallel media is t e. µ = cosϑ the standad fom of the equation of tansfe fo dl( τ, µ, ϕ) µ = L( τ, µ, ϕ ) I( τ, µ, ϕ ) dτ (3) Fo a scatteing medium, the only contibution to the souce function is the scatteing of that adiation which has been incident on the medium fom extenal souces, so that, by totalling the contibutions impinging on level τ fom all diections, the souce function is I( τ, µ, ϕ ) = π p ( µ, ϕ; µ ', ϕ ') L( τ, µ ', ϕ' ) dϕ' dµ ' (4) whee p is the nomalised phase function which detemines the angula distibution p of the scatteing. A convenient way to think of p is that ω d is the pobability that a photon tavelling in the diection ( µ ', ϕ' ) would be scatteed into an elemental solid angle dω in the diection ( µ, ϕ). Radiation tavesing a nomal optical thickness δτ in the diection ( µ, ϕ) will be attenuated by the amount δ L = Lδτ / µ. Of this amount, a faction can be attibuted to that caused by scatteing alone this faction is called the single scatteing albedo. It then follows that the phase function p must be nomalised accoding to p dω = 1 (5) and if p is a constant, then p = and the scatteing is isotopic. 4. Diections and Notation. The stength of a plane paallel beam of adiation is specified by the adiant flux density F watts pe squae mete such that F = dp/da, whee A is the aea pependicula to the diection of popagation. Thus F is equal to the net flux πf used by Chandasekha, with the impotant exception that F is used only fo a plane paallel beam. M.B. Faibain, 24.

4 Daft Nov. 4, 24 Since the equation of tansfe deals only in adiances, we will now addess the athe intiguing question, What is the adiance of a plane paallel beam? Figue 2 shows a ay of a plane paallel beam of flux density F incident on the suface of a scatteing medium. We shall let the esulting incident adiance be L i, which, using Chandasekha s notation would be specified in position and diection as M.B. Faibain, 24.

5 Daft Nov. 4, 24 L i = L, µ, ϕ ), µ = cos (6) ( ϑ z F ϑ θ i ϕ y x τ Fig. 2. Many authos specify the pola diection of this adiation in tems of an angle of incidence, say i o θ i, as the angle between the suface nomal and the incident ay, such that i = π ϑ and define µ as µ = cos i. The angula distibution of F (o, athe, its lack of it) may be specified analytically by making use of the Diac delta function, which has the popety that f ( x) δ ( x a) dx = f ( a) and, pehaps moe impotantly, fo any ε > a + ε a ε f ( x) δ ( x a) dx = f ( a). The incident adiance on the s uface in the diection µ, ϕ ) is then 1 ( L = Fδ µ µ ) δ ( ϕ ). (7) i ( ϕ Figue 3 shows a eflected beam of adiance L whee L = L(, + µ, ϕ), µ = cosϑ (8) 1 If equation (7) bothes you in that its ight hand side does not appea to have units of adiance, then do not woy; in chapte 2 we will demonstate that indeed it does have such units. M.B. Faibain, 24.

6 Daft Nov. 4, 24 z ϑ L ϕ y x τ Fig. 3. Again, many authos define the pola diection of eflection in tems of an angle of eflection, the angle between the suface nomal and the eflected ay, say θ (which is the same as ϑ ), and define µ as cos θ. Figue 4 shows a tansmitted ay of adiance L t whee L t = L( τ, µ, ϕ), µ = cosϑ (9) z ϑ y x ϕ τ L t Fig. 4. The question then aises: ae the values of µ and µ, intoduced by Chandasekha and taken up by othes, albeit with diffeent definitions, always positive? The answe to this is, mostly, yes, but cae needs to be taken depending on the context in which they occu, as, fo example, in the following cases. Conside the poblem of detemining the phase angle α π between incident and eflected o tansmitted diections µ and µ, whee M.B. Faibain, 24.

7 Daft Nov. 4, cosα = µ µ + (1 µ )(1 µ ) cos( ϕ ϕ). (1) Fo eflection both µ and µ ae positive, but fo tansmitted ays µ must be negative. The phase function p is often expessed in tems of α o cos α. O, conside, as in figue 5, the optical path lengths of eflected and tansmitted adiation fom a depth within a medium of optical thickness t. The total optical path fo the eflected ay is τ / µ + τ / µ and fo the tansmitted ay τ / µ + ( t τ ) / µ t in which µ, µ and µ t ae all positive. L i L τ = τ τ = t L t Fig Reflectance Functions. In the most geneal case of diffuse eflection, the eflectance of a suface will depend on both the diection of the incident adiation and that of the eflected adiation. The bidiectional eflectance distibution function, f, links the iadiance E to the eflected adiance, such that L = f ( µ, ϕ; µ, ϕ ) E( µ, ϕ ). (11) Fo a suface iadiated with flux density F, the iadiance is simply the component of the flux density pependicula to the suface so that, abbeviated, we can wite E = µ F, (12) L = f µ F (13) One of the simplest examples of a eflectance ule is that of a Lambetian eflecting suface fo which the adiance is isotopic, so that λ L = µ F, (14) π M.B. Faibain, 24.

8 Daft Nov. 4, 24 whee λ is sometimes efeed to as the Lambetian albedo. Although it is not stictly physically coect, it is convenient (Chandasekha, p147) to identify λ with the single scatteing albedo, so fo Lambet s law the BRDF is f =. (15) π Fo the most pat, we shall efe all eflectance ules used to a BRDF; altenative eflectance functions will be discussed in Diffuse Reflection the Lommel-Seelige Law. The Lommel-Seelige eflectance ule is a time-honoued law which is still vey much in use today. It is based on a model which is possibly the simplest fom which a solution may be eadily obtained fo the souce function and the equation of tansfe. As such it is a single scatteing model in which the scatteing is isotopic, i.e. p =. Conside a suface iadiated by flux density as shown in figue 3, so that the incident adiance is given by equation (7). Of this incident adiation, only a faction will penetate to optical depth τ without being scatteed o absobed and the souce function is τ,, ) / µ L τ µ ϕ = F e δ ( µ µ ) δ ( ϕ ϕ ) (16) ( 1 I( τ, µ, ϕ ) = = Fe 1 2π 1 τ / µ Fe τ / µ δ ( µ ' µ ) δ ( ϕ ' ϕ ) dϕ ' dµ ' (17) Thus the contibution to the adiance fom isotopic scatteing in the diection ( + µ, ϕ) fom a laye of thic kness d τ at a depth τ is dl = Fe µ τ / µ dτ (18) so that the adiance emeging fom the suface, without incuing any futhe absoption o scatteing, is τ / µ Fe τ / µ dl (, µ, ϕ) = e dτ (19) µ Note that dl is the contibution to the total adiance fom the laye esulting fom single scatteing. The Lommel-Seelige model consides only the scatteing of the collimated incident light. It does not take into account scatte ing of diffuse light which M.B. Faibain, 24.

9 Daft Nov. 4, 24 has made its way indiectly to the same position by being scatteed one o moe times, i.e. it does not conside multiple scatteing. Fo a planetay suface, the laye is semi-infinite ( t = ) and the totaladiance in the diection µ is esulting in L F = µ. (2) exp[ τ ( )] dτ µ µ F µ µ L = (21) 4 πµ µ + µ and since the iadiance is E = Fµ and L = f E it follows that the bidiectional eflectance distibution function (BRDF) which defines the Lommel-Seelige eflectance ule is 1 f =. (22) 4 π µ + µ It is inteesting to note that Chandasekha neve quite deives the Lommel-Seelige law fomally and explicitly; indeed the name is not even mentioned. Howeve, he does come vey close on at least two occasions see Chandasekha p146 and p Othe Reflectance Functions. It is impotant to distinguish between a eflectance function and the eflectance law it epesents. So fa we have only consideed one such function, the BRDF, so that the Lommel-Seelige law expessed in tems of the BRDF is given by equation (23) and the specific equation fo the adiance is given by 1 L µ F = (23) µ + µ Chandasekha takes a quite diffeent appoach, linking the adiance to the incident flux density though a facto 1 / 4µ, poviding a consistent set of scatteing functions S and tansmission functions T, so that in the case of eflection fom a semi-infinite suface we have L F F = S( µ, ϕ ; µ, ϕ ) = S( µ, ϕ; µ, ϕ), (24) 4µ µ M.B. Faibain, 24.

10 Daft Nov. 4, 24 whee Chandasekha always uses π F fo incident adiant flux density F. Although, at least at fist sight, this fomulation may seem stange, even counteintuitive, thee is a eason fo it; the µ in the denominato is used to satisfy the Helmholtz pinciple of ecipocity (Chandasekha, p171), so that S µ, ϕ; µ, ϕ ) = S( µ, ϕ ; µ, ). (25) ( ϕ Compaing equations (23) and (24), it follows that fo the Lommel-Seelige law the Chandasekha scatteing function is S ( µ µ µ, µ ) = µ + µ, (26) whee it can be seen that the ecipocity pinciple does indeed hold. Anothe function to be found in the liteatue is the bidiectional eflectance, which links the adiance to the incident flux density, so that the Lommel-Seelige law is then µ ( µ, ) =, L = µ F (27) µ + µ So, which, if any, of the above functions is the best fo planetay applications? Thee does not appea to be any standad in use in the liteatue, indeed the situation would seem to be quite the opposite, many authos making up thei own ad hoc eflectance o scatteing functions to suit the poblem at hand. (This can make fo vey fustating eading, especially when wods such as flux, intensity and bightness ae used loosely, as, sadly, is often the case). The autho can see no compelling eason to pefe one function ove anothe. What is impotant is fo authos to state clealy and without ambiguity the popeties of the eflectance func tion and ule(s) which they ae using. 8. Diffuse Reflection and Tansmission. A scatteing laye of finite optical thickness t may be used to model e.g. a planetay ing. If we use the Lommel-Seelige model, then the eflected adiance of such a laye may be detemined by changing the uppe limit of the integal in equation (2) so that L F = µ t exp[ τ ( )] dτ µ µ (28) esulting in M.B. Faibain, 24.

11 Daft Nov. 4, 24 L 1 = µ + µ [1 exp{ t( 1 1 µ + µ )}] µ F (29) Fo the tansmitted adiance, it is eadily shown that dl t = τ / µ Fe ( t τ) / µ µ e dτ (3) and in the special case µ = µ, integation esults in L t Ft = e µ t / µ (31) and othewise L t F µ = µ µ [ e t / µ t / µ e ] (32) In all cases the values of µ and µ ae positive; some authos even explicitly put in absolute value symbols to emphasise this point! 9. Radiances of Planetay Sphees. We conclude this chapte by applying some of the wok coveed so fa to a planetay situation. We will conside two hypothetical planets idealised as smooth sphees. One planet will have a suface eflecting accoding to Lambet s law, the othe the Lommel-Seelige law. The obseve is able to esolve both planets equally well, so that we may compae and contast the distibution of adiance acoss the pojected discs at vaious phase angles. Conside a unit sphee cented at the oigin of an Oxyz coodinate system iadiated with flux density F fom the x-diection. A distant obseve in the xy plane detects the adiance at phase angle α (the angle Sun-planet-Eath). Using the spheical coodinates ( 1, Θ, Φ) fo the suface of the sphee, it can be shown fo the angles of incidence and eflection µ = sin Θ cosφ µ = sin Θ cos( Φ α) (33) The pojected sphee, the disc seen by the obseve, will have pojected coodinates y' = sin Θ sin( Φ α) z = cos Θ (34) M.B. Faibain, 24.

Daft Nov. 4, 24 2 2 such that y ' + z 1.

iadiated and not obscued fom the obseve. Defining a quantity elative adiance, π L / F, we can diectly compae the adiances of the two sphees, as shown in the table.

12 Daft Nov. 4, such that y ' + z 1. Except at zeo phase, not all the illuminated suface will be visible, since fo each point on the disc both the condition µ > and µ > must be satisfied in ode that the point be both iadiated and not obscued fom the obseve. Defining a quantity elative adiance, π L / F, we can diectly compae the adiances of the two sphees, as shown in the table. Relative Radiances of Sphees Lambetian µ Lommel-Seelige 1 µ 4 µ + µ The esulting images ae eadily pogammed. In those that follow (le ftmost Lambetian, middle Lommel-Seelige), each planet has been adjusted so that the maximum elative adiance is white. The ightmost image shows the outline of the lune visible to the obseve. α = α = 3 o α = 6 o M.B. Faibain, 24.

13 Daft Nov. 4, 24 α = 9 o α = 15 o α = 12 o At opposition the Lambetian sphee is limb-dakened, wheeas the Lommel-Seelige sphee is unifomly bight. As the phase angle inceases fom zeo the Lommel- Seelige sphee becomes dakened towads the teminato and bightened at the limb. Fo phases geate than ninety degees, the cusps of the Lommel-Seelige sphee ae moe pesistant than the Lambetian. Without commenting any futhe, the images do make inteesting compaisons with the phases of the Moon. Refeence Notes. Sections 2, 4, 5, 7, 8 and 9 ae based on the autho s intepetation of Chandasekha s book, chaptes I, III, VI and IX. 1. Chandasekha, S., 196, Radiative Tansfe, Dove, New Yok. The ideas of a quantity F defined only fo a plane paallel beam and the use of the BRDF (bi-diectional eflectance distibution function) fo astonomical applications ae taken fom M.B. Faibain, 24.

14 Daft Nov. 4, Leste. P. L., McCall, M. L. & Tatum, J. B., 1979, J. Roy. Aston. Soc. Can., 73, 233 who use F fo flux density, which clashes with the F of the π F used by Chandasekha this is the eason fo using F. (See also Nicodemus, F.E., Applied Optics, 4, 767 (1965) and 9, 1474 (197) JBT) Section 1 is a evised and coected adaptation of an aticle by the autho 3. Faibain, M. B., 22, J. Roy. Aston. Soc. Can., 96, 18. Depending on the hadwae used, the images shown may display some spuious contouing. M.B. Faibain, 24.

Black Body Radiation and Radiometric Parameters:

Black Body Radiation and Radiometic Paametes: All mateials absob and emit adiation to some extent. A blackbody is an idealization of how mateials emit and absob adiation. It can be used as a efeence fo