TRP results from a directed non-symmetric emission of energy. A suitable model for this effect is the emission of photons by a flat two-dimensional plate with surface area dA. The well known dispersion relation:
E=p
(m0c2)2+ (|~pPh|c)2, (3.1) yields the relation EPh =|~pPh|c for photons with a zero photon rest mass (m0 = 0) [63]. HereEPhis the energy carried by the photon,|~pPh|is the photon momentum and c is the speed of light. Using the photon model derived by Einstein, the frequency of the photonνPh is proportional to the photons energy:
EPh=h νPh, (3.2)
where h is Planck’s number. Thus the photon momentum can be expressed in terms of frequency as:
|p~Ph|= h νPh c = h
λ, (3.3)
where λ is the wavelength, at which the photon is emitted. Figure 3.2 shows the resulting change of momentum for an emission of photons (A) and an absorption of photons (B).
3.2 Analytical model for TRP 17
Figure 3.2: Photon and interacting surface momentum for photon emission (A) and photon absorption (B).
For photon emission the effective momentum acting on the emitting source~popposes the direction of the emitted photons momentum. In a closed system this follows directly from the postulation of the conservation of momentum. Thus the momentum of the emitted photon corresponds to the momentum loss of the emitter (~pPh+~p = 0). In case of absorption, which will later be needed for the development of a SRP model, the directions of the photon momentum and the resulting momentum acting on the absorber are aligned. If N Photons are emitted within the time dt this leads to a source momentum derivative of:
d|~p|
dt = h νPhdN
c dt , (3.4)
which is equivalent to the force acting on the emittor directed against the moving direction of the photons. Note that Pe = h ν dNdt is the total power carried by the emitted photons. Thus the resulting force acting on the emitter can be expressed as:
|F~TRP|= d|~p| dt = Pe
c . (3.5)
Equation (3.5) is also valid for the absorption of photons. Here the resulting momentum gained by the absorbing surface is aligned with the moving direction of the impacting photons. In case of emission this resulting force acting on the emittor is calledThermal Recoil Force. If the source of emission posesses a mass m, a corresponding thermal recoil pressurePTRP can be defined as:
PTRP = |F~TRP| m = Pe
m c. (3.6)
Note that for this definition the TRP obtains the unit m/s2 which differs from the typical definition of a pressure in units of N/m2. This results from the fact that it is sensible to specify orbit perturbations as specific values with respect to spaceraft dimensions. If needed, the TRP can be translated into pressure units by multiplication by the ratio of spacecraft surface area to spacecraft mass. For a complete description of PTRP, the magnitude of the emitted power has to be determined. Assuming a flat, two-dimensional grey radiating surface of area A, the energy flux emitted with the emissivity ελ for a specific wavelength λ can be expressed in terms of the spectral radianceLλ as [64]:
Lλ =ελ·2h c2 λ5
1
ek λ Th c −1, (3.7)
where h is Planck’s number, c is the speed of light, k is the Boltzmann constant and T is the homogeneous surface temperature. Integration over all wavelengths of the spectrum leads to the radianceL:
L= Z ∞
0
Lλdλ=εσ
π T4 with σ= 2π4k4
15h3c3 , (3.8) which can be interpreted as an energy density emitted by the radiating surface. Here εis the effective emissivity value for the respective wavelength band. The fraction of energy emitted in the direction of the elevation angle θ is defined as the intensity of radiation I and can be described with Lambert’s cosine law, assuming the radiation pattern to be hemispheric:
I =L A cosθ . (3.9)
By rewriting equation (3.9), the radiation intensity in a given direction specified by the elevation angleθ can be related to the intensity emitted in normal directionIn:
I =In cosθ . (3.10)
The directions of emission can be characterised by discrete solid angle surface elements forming a virtual unit hemisphere above the emitting surface. These fractions of the surface of the unit hemisphere are determined by the azimuth angleφand the elevation angleθ with 0≤φ <2π and 0≤θ < π/2. Figure 3.3 shows the definition of the solid angle elements dΩ for an emitting differential surfacedAwith normal direction ~n.
Figure 3.3: Definition of the solid angle elementdΩ, [64].
The total energy flux EA emitted bydA can now be calculated by integrating the energy fluxes received by each solid angle element surface over the complete hemispher-ical surface:
EA= Z
Ω
L cosθ dΩ. (3.11)
Here the solid angles can be expressed as dΩ = sinθ dθ dφ, giving:
EA(φ, θ) = Z φ2
φ1
Z θ2
θ1
Lcosθsinθ dθ dφ . (3.12)
3.2 Analytical model for TRP 19 For φ1 = θ1 = 0, φ2 = 2π and θ2 = π2 the total energy flux Ptot received by the hemispherical surface evolves to
Ptot = Z
A
EAdA
= Z
A
Z 2π
0
Z π
2
0
ε σ T4
π cosθ sinθ dθ dφ dA
= ε A σ T4, (3.13)
which is the well known Stefan Boltzmann law for radiating grey bodies. Using equation (3.8) and equation (3.10) and the relation Ptot = Pe, the total emitted power Pe can be formulated in terms of the intensity as:
Pe=Inπ . (3.14)
With equations (3.2) - (3.5) the the energy flux dE received by a specific solid angle element:
dE=Lcosθsinθ dθ dφ dA , (3.15) can be rewritten as a solid angle specific thermal recoil force:
|F~TRP,Ω|= 1
cdE , (3.16)
for eachdΩ [63]. Due to the symmetric character of the hemisphere only force compo-nents normal to the emitting surface plane will contribute to a resulting recoil. This introduces an additional cosθterm into the force integral. As a result of the vanishing of all parallel components, the direction of the resulting force can be determined by the normal vector of the emitting surface~n, where|~n|= 1. Thus the effective thermal recoil force component resulting from emission into a specific solid angle element evolves to:
F~TRP,Ω,ef f =−~n1
cL cos2θ sinθ dθ dφ dA , (3.17) where the negative sign is demanded by the balance of energy between emitted photons and emitter. Integration over the hemispheric surface withφ= 2π and θ= π2 leads to the total (effective) thermal recoil forceF~TRP with:
F~TRP =−~n 2
3cπ L A=−~n 2 3cπ εσ
πA T4 =−~n 2
3cPtot. (3.18) With this the power fraction contributing to a resulting force in normal direction of an emitting surface evolves to a ratio of 23 of the total emitted power and the TRP can be characterised by:
P~TRP =−~n2 3
Ptot
m c . (3.19)