Impedance and voltage power spectra of a monopole antenna in a warm plasma -- derivation and application to CubeSats
aa r X i v : . [ phy s i c s . s p ace - ph ] S e p Confidential manuscript submitted to
Radio Science
Impedance and voltage power spectra of a monopole antenna ina warm plasma - derivation and application to CubeSats
Ronald Maj , , Iver H. Cairns and M.M. Martinović , , School of Physics, The University of Sydney, NSW 2006, Australia SPACE Research Centre, RMIT University, Melbourne, Victoria, Australia Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85719, USA LESIA, Observatoire de Paris, Meudon, France Department of Astronomy, Faculty of Mathematics, University of Belgrade, Serbia
Key Points: • An antenna response function is derived for monopole antennas F m ( x ) but is onlyuseful in certain circumstances (e.g. modelling the QTN) • Integration over F m ( x ) to model the impedance of the antenna is well-behaved for thereal part but does not converge for the imaginary part of the warm plasma impedanceintegral. The shot noise and capacitance predictions do not converge as a result. How-ever, this is also true for the well-known double-sphere antenna response function • Restricting the integration on physical grounds did not lead to a useful general crite-rion to limit the integration. Further work is necessary to find the source of the issueswith shot noise and capacitance or determine a new monopole response function
Corresponding author: Ronald Maj, [email protected] –1–onfidential manuscript submitted to
Radio Science
Abstract
The impedance for a monopole antenna is derived and compared with the cases forwire dipole and double-sphere antennas. This derivation produces a new expression for themonopole antenna response function, F m ( x ) . The monopole, wire dipole, and double-sphereresponse functions are compared by modeling an antenna in Earth’s ionospheric plasma (i.e.electrostatic and collisionless) and predicting the antenna capacitance and voltage powerspectra for quasi-thermal noise (QTN) and shot noise. The monopole antenna current dis-tribution is assumed to be a half-triangular current distribution (considering only the positivehalf of the triangular distribution). The predictions for the shot noise and capacitance pre-sented problems, as the integral over wavenumber-space or k -space did not converge for largevalues of k . The derived expression therefore remains a current problem and necessitates fu-ture work to determine a more general expression. In this paper we bring the problem of anappropriate analytic monopole antenna response function to the attention of the communityand outline a number of tests that can be used to verify any future expression. Antennas play a vital role in communication, for example allowing signals to be sentfrom mobile devices to receiving towers or even between Earth-orbiting satellites and groundstations. Antennas are also important in researching the natural environment through studyof the electromagnetic and electrostatic waves and other signals that may be present. Tomodel an antenna, parameters such as the gain, radiation pattern, and impedance may needto be known. This is also true for antennas used to diagnose space plasmas through quasi-thermal noise (QTN) and shot noise spectroscopy assuming Maxwellian or Kappa velocitydistribution functions [
Couturier et al. , 1981;
Kellogg , 1981;
Meyer-Vernet , 1983;
Meyer-Vernet and Perche , 1989;
Maksimovic et al. , 1995;
Le Chat et al. , 2009;
Maj and Cairns ,2017]. Quasi-thermal noise is due to the thermal motions of plasma particles producing elec-trostatic Langmuir waves which can be detected with sensitive receivers. Shot noise is due tothe impact of plasma particles with the antenna producing voltage peaks/troughs which canbe approximated by a step function at low frequencies and are again detectable with a sen-sitive receiver. In these cases, the impedance of the antenna in a warm plasma needs to beknown to predict the power spectrum and therefore determine the plasma properties. –2–onfidential manuscript submitted to
Radio Science
Authors such as
Balmain [1965],
Kuehl [1966],
Meyer-Vernet [1979], and
Couturieret al. [1981] have outlined derivations for the impedance of a dipole antenna assuming a tri-angular current distribution. However, the literature does not appear to contain a derivationfor a monopole antenna. We derive the monopole impedance following the general methodoutlined in
Kuehl [1966] and
Couturier et al. [1981], expanding on intermediate steps andusing a different expression for the current distribution, namely one for a monopole antenna.Through this process a new expression for the antenna response function is derived and com-pared to the wire dipole response function F ( x ) introduced in Kuehl [1966] as well as to theresponse function for a double-sphere antenna [
Meyer-Vernet and Perche , 1989]. This re-sponse function is the main difference between the monopole, wire dipole and double-spherecalculations of the impedance.This paper proceeds by presenting the theoretical background for the impedance inSection 2. The derivations for the dipole and monopole impedance are then detailed in Sec-tions 3 and 4, respectively. The ensuing predictions for the voltage power spectrum due toquasi-thermal noise (QTN) and shot noise are outlined in Section 5. The capacitances fordipole and monopole antennas calculated using the derived impedances, plus their compar-isons with standard expressions in the low frequency approximation, are presented in Section5. These results are discussed and the possibility of restricting the maximum wavenumber k value based on physical grounds is investigated in Section 6. Finally, Section 7 concludes thepaper. Determining the impedance of an antenna in a plasma requires a model representationof the plasma. The most common ways of doing this involve using either the magnetohy-drodynamic equations (which are well suited to describe the plasma macroscopically) or theVlasov equations (which take a kinetic approach and reveal microscopic details). The deriva-tion here will use the kinetic approach and follow the work of
Kuehl [1966] and
Couturieret al. [1981]. In this section we derive the resistance of a wire dipole antenna of radius a andlength L for each arm. Firstly, we will assume the following [ Kuehl , 1966]: • the plasma is unmagnetized • the plasma is at thermal equilibrium (Maxwellian and non-zero temperature, i.e. warm), • collisions are neglected, –3–onfidential manuscript submitted to Radio Science • ion motions are neglected, • the effects of electron and ion sheaths around the antenna are neglected [ Meyer-Vernet ,1993], • the field is weak enough to allow linearized equations (any external electromagneticfield is also neglected), • the antenna is long with length L ≫ λ D where λ D = p ǫ k B T e / e n e is the electronDebye length (electron density and temperature n e and T e , respectively), ǫ the per-mittivity of free space, k B the Boltzmann constant, and e the electric charge, and • the antenna radius a is finite with a ≪ L and a ≪ λ D .Following Couturier et al. [1981] the antenna impedance Z a is given by Z a = R − iX = − I ∫ ® E (® r ) · ® J (® r ) d ® r (1)where R is the resistance, X the reactance, i the imaginary unit, I is the peak current flowinginto/out of the antenna, ® E (® r ) the electric field of the source, and ® J (® r ) the current distribution.Fourier transforming and using Parseval’s theorem allows us to write [ Couturier et al. , 1981] Z a = iI ( π ) ωǫ ∫ J ∗ i (® k ) Λ − ij (® k , ω ) J j (® k ) d ® k (2)where the summation over dummy indices is implied, J ∗ i is the complex conjugate of J i and Λ − ij (® k , ω ) is defined with respect to the plasma dielectric permittivity tensor ǫ ij (® k , ω ) as[ Sitenko , 1967] Λ ij (® k , ω ) = k c ω (cid:18) k i k j k − δ ij (cid:19) + ǫ ij (® k , ω ) . (3)To simplify, we consider only the longitudinal component and so (2) yields [ Meyer-Vernetand Perche , 1989] Z a = iI ( π ) ωǫ ∫ | ® k · ® J (® k )| k ǫ L d ® k (4)as the impedance for the antenna. The current distribution ® J (® k ) used will determine the finalform that (4) takes. The longitudinal dielectric permittivity ǫ L is defined as ǫ L = + k λ D + i ω √ π W (cid:16) ω kv T (cid:17) k λ D (cid:18) ω p k v T (cid:19) (5)where ω p = π f p is the angular electron plasma frequency, v T = q k B Tm e is the thermalspeed of electrons with mass m e , W ( z ) is the Faddeeva function defined as W ( z ) = e − z erfc (− iz ) , (6) –4–onfidential manuscript submitted to Radio Science and erfc ( z ) is the complementary error function.The dielectric permittivity tenor ǫ ij defines how waves will travel in the given plasmaenvironment. We have decided to use the Earth’s ionospheric environment at low latitudesand low altitudes ( ≈
300 km) as our test case to derive the expressions for the antenna re-sponse functions for a dipole and monopole antenna. This is to say that the plasma is as-sumed to be at thermal equilibrium and unmagnetized, allowing the dielectric permittivitytensor to be separated into independent longitudinal and transverse components, which iswhy we only consider ǫ L in our equations.The unmagnetized condition is not entirely accurate as the geomagnetic field strengthis on the order of 20 µT to 60 µT at 300 km altitude, according to the International Geo-magnetic Reference Field model [ Thébault et al. , 2015]. This places the electron cyclotronfrequency between 0 . . f p ranges from1 . . ω p and having ad-ditional thermal Bernstein and upper hybrid waves modes detectable. However, in this paperwe will concentrate on the unmagnetized case as a first order approximation. Also, these re-sults could be extended down to altitudes of 100 km to 140 km where the plasma is mostlyunmagnetized and approximately collisionless (although the effect of collisions are notable ataround 120 km [ Martinović et al. , 2017]).Relaxing the thermal equilibrium condition requires a different expression for the ve-locity distribution function (VDF) of the plasma particles. In this paper we will compare thepredictions for the Maxwellian VDF against those for a Kappa VDF in order to explore anydifferences between thermal and non-thermal conditions. In effect this involves substitutingthe longitudinal dielectric permittivity ǫ L in (5) with [ Chateau and Meyer-Vernet , 1991] ǫ L = + z r © « κ − + (− ) κ + ( κ − ) !! iz κ Õ p = ( κ + p ) ! p ! 1 ( i ) κ + + p ( z + i ) κ + − p ª®¬ (7)where z = ω /( k v √ κ ) , r = ω / ω p , and v = v T p ( κ − )/ κ . It should be noted that theformalism in Chateau and Meyer-Vernet [1991] and expressed above is for integer values of κ only, and a more general expression can be found in [ Le Chat et al. , 2009]. Using a Kappadistribution also alters the definition of the Debye length as the value of κ affects the distance –5–onfidential manuscript submitted to Radio Science over which plasma particles are shielded. We will call this modified Debye length λ D − κ with[ Chateau and Meyer-Vernet , 1991] λ D − κ = v ω p r κ κ − . (8)Other distributions such as a Flat-Top or Heaviside VDF could also be used to revealnon-thermal effects. Kappa distributions with values of κ ≈ − Vasyliunas , 1968] and κ ≈ − Chateau and Meyer-Vernet ,1991;
Maksimovic et al. , 1997a,b] provide a good fit to measured energy spectra and otherobservations. Therefore we will use κ = Meyer-Vernet and Perche , 1989] V QT N = k B T e ℜ( Z a ) (9)where ℜ( Z a ) denotes the real part of the impedance Z a , that is, the resistance of the antenna.This can be expressed as R = R L + R T , which are the longitudinal ( L ) and transverse ( T )components of the resistance, which correspond to electrostatic and electromagnetic waves,respectively. In the non-thermal Kappa-distribution case, the QTN expression we will use is[ Chateau and Meyer-Vernet , 1991] V QT N − κ = κ + π ǫ κ ! ( κ − ) !! m v r ∫ + ∞ zF (cid:18) ruz √ κ − (cid:19) [( + z ) κ | ǫ L | ] − dz (10)where u = L / λ D − κ and the integral is over z as defined previously. The shot noise has theform [ Meyer-Vernet and Perche , 1989] V S = e N e | Z a | (11)where | Z a | = Z ∗ a × Z a with Z ∗ a the complex conjugate of Z a , and N e = /√ π n e v T S is theimpact rate of plasma electrons with S the antenna (or satellite) surface area. Equation (11)is only valid below the electron plasma frequency [ Meyer-Vernet , 1983].For the capacitance we use C a = ω Im ( Z a ) (12)which is the case for an ideal capacitor and involves the antenna impedance Z a for an arbi-trary antenna. We can also calculate the capacitance by using the analytic approximation for –6–onfidential manuscript submitted to Radio Science the reactance [
Meyer-Vernet and Perche , 1989] of a long dipole ( L ≫ λ D ), which gives C a = πǫ L log ( λ D / a ) (13)for a wire dipole and C a = πǫ a (14)for a double-sphere antenna.The expressions (13) and (14) are only valid for low frequencies f ≪ f p . Thereforein the dipole case, we are able to compare the approximations (13) and (14) with the capac-itance calculated from the dipole impedance and (12). For a monopole placed a small dis-tance perpendicularly from a reference plane we expect that C monopole = C dipole . In thiscase the monopole creates an image resembling the dipole configuration but with only onehalf the voltage input/output and therefore double the capacitance [ Balanis , 2016].The impedance is therefore a critical part of both the QTN and shot noise expres-sions. The expression for the impedance that we use in this paper is based on Equation (15)in
Meyer-Vernet and Perche [1989]. Specifically we will aim to derive an expression of theform Z a = i π ωǫ ∫ ∞ F ( k ) ǫ L dk (15)where F ( x ) is the appropriate antenna response function. Depending on the type of antennaand the current distribution over the antenna, the form of F ( x ) will vary. For a wire dipole F ( k ) is given by [ Meyer-Vernet and Perche , 1989] F ( k ) = F ( k L ) J ( ka ) (16)where J ( x ) is the zeroth Bessel function of the first kind and F ( x ) takes the form F ( x ) = x (cid:16) Si ( x ) − Si ( x ) (cid:17) − (cid:0) x (cid:1) x (17)where Si ( x ) = ∫ x ( t ) t dt is the sine integral. Equation (16) assumes that a is finite and in thecase of an infinitely thin antenna (16) simplifies to F ( k ) = F ( k L ) .The function F ( x ) was derived by Kuehl [1966], and has been used by many authorssince then, including
Couturier et al. [1981],
Kellogg [1981],
Meyer-Vernet and Perche [1989],
Le Chat et al. [2009],
Martinović et al. [2016] and
Maj and Cairns [2017]. Section 3 derivesthe expression (17), while in Section 4 two new expressions are derived for the monopoleresponse function to be used as F ( x ) in (15). –7–onfidential manuscript submitted to Radio Science
To derive the dipole response function, we begin by assuming a triangular current dis-tribution along the antenna [
Balmain , 1965]. This is expressed as ® J (® r ) = I π a (cid:16) − | z | L (cid:17) δ ( r − a ) ˆ z , for | z | < L . for | z | > L (18)Here cylindrical coordinates are used, so that the delta function means that all the current ison surface of the antenna ( r = a ), the antenna is aligned along the z -axis, and I is the peakcurrent flowing into/out of the antenna.If a function is separable, i.e. f ( x , x , ... ) = f ( x ) f ( x ) ... , then Fourier transformingallows us to write F [ f ( x , x , ... )] = F [ f ( x )]F [ f ( x )] ... where F represents the Fouriertransformation operation F [ f ( x )] = ∫ ∞−∞ f ( x ) e − i ® k ·® r d ® r . (19)Taking the Fourier transform of the current distribution (18) gives F [ ® J (® r )] = ® J (® k ) = I J ( k ⊥ a ) ∫ ∞−∞ (cid:18) − | z | L (cid:19) e − ik z z dz ˆ z = I J ( k ⊥ a ) (cid:20)∫ − L (cid:16) + zL (cid:17) e − ik z z dz + ∫ L (cid:16) − zL (cid:17) e − ik z z dz (cid:21) ˆ z = I J ( k ⊥ a ) (cid:20)∫ L − L e − ik z z dz + ∫ − L ze − ik z z L dz − ∫ L ze − ik z z L dz (cid:21) ˆ z (20)Integration by parts (cid:16)∫ uv ′ = [ uv ] − ∫ u ′ v (cid:17) can be used to solve the second and third inte-grals above with u = z and v ′ = e − ik z z . This then gives ® J (® k ) = I J ( k ⊥ a ) (cid:20) ie − ik z L − ie ik z L k z + k z L (cid:18) iLe ik z L − iLe − ik z L − ( e ik z L + e − ik z L − ) k z (cid:19) (cid:21) ˆ z = I J ( k ⊥ a ) (cid:20) − cos ( k z L ) k z L (cid:21) ˆ z = I k z L sin (cid:18) k z L (cid:19) J ( k ⊥ a ) ˆ z (21)using the trigonometric identity cos ( θ ) = − ( θ ) .The expression for the current distribution ® J (® k ) in (21) can now be substituted into (4)to give Z a = i ( π ) ωǫ ∫ sin (cid:16) k z L (cid:17) J ( k ⊥ a ) k k z L ǫ L d ® k (22) –8–onfidential manuscript submitted to Radio Science which if we use spherical coordinates in ® k -space and carry out the φ integration can be sim-plified to Z a = i ( π ) ωǫ ∫ π ∫ π ∫ ∞ sin (cid:16) k z L (cid:17) J ( k ⊥ a ) k k z L ǫ L k sin ( θ ) dkd θ d φ = i π ωǫ ∫ π ∫ ∞ sin (cid:16) k cos ( θ ) L (cid:17) J ( k sin ( θ ) a ) k cos ( θ ) L ǫ L sin ( θ ) dkd θ (23)where the θ dependence has been made explicit in the second line. As we have assumed that a ≪ L we can approximate J ( k ⊥ a ) with unity since k ⊥ a ≪ Couturier et al. , 1981].Concentrating only on the θ integral and making the substitution u = cos ( θ ) gives ∫ π sin (cid:16) k cos ( θ ) L (cid:17) k cos ( θ ) L sin ( θ ) d θ = ∫ − sin (cid:16) kLu (cid:17) k L u du . (24)Note that as the plasma is isotropic ǫ L has only radial k dependence, remaining constant overthe θ integration. Using integration by parts with u = sin ( k Lu / ) and v ′ = / k L u in(24) gives ∫ − sin (cid:16) kLu (cid:17) k L u du = − sin (cid:16) kLu (cid:17) k L u − + ∫ − k Lu ( ( k Lu ) − sin ( k Lu )) du = − (cid:16) kL (cid:17) k L + ∫ sin ( k Lu ) k Lu du − ∫ sin ( k Lu ) k Lu du = Si ( k Lu ) k Lu − Si ( k Lu ) k Lu − (cid:16) kL (cid:17) k L = F ( k L ) (25)where F ( x ) , with x = k L , is the dipole antenna response function defined in (17) and wehave used the fact that integrals over even functions have the property that ∫ a − a f ( x ) dx = ∫ a f ( x ) dx . Placing this back into (23) gives Z a = i π ωǫ ∫ ∞ F ( k L ) J ( ka ) ǫ L dk , (26)which is Equation (15) in Meyer-Vernet and Perche [1989] or (15) in this text with F ( x ) inplace of F ( x ) .In summary, we have shown in this section how the antenna response function in (17)and the antenna impedance expression in (26) are derived for the dipole case. We will use asimilar procedure in the next section to derive a monopole antenna response function F m ( x ) in order to retain the general expression (15) as the antenna impedance, with F ( x ) replacedby the monopole function F m ( x ) . –9–onfidential manuscript submitted to Radio Science
To derive the monopole response function, we need to assume a certain current dis-tribution across the antenna. A monopole antenna fed with current from one end will havea peak of I at one end and zero current at the other, so we choose a half-triangular currentdistribution. However the end points of the antenna can be arbitrarily placed on the axisalong which the antenna lies. In this paper, we define the end points of the antenna to be at z = − L / z = L /
2; however, the method below produces the same results using z = z = L . The current distribution is defined as ® J (® r ) = I π a (cid:16) − zL (cid:17) δ (® r − a ) ˆ z , for | z | < L , for | z | > L (27)for an antenna aligned along the z -axis. Taking the Fourier transform of (27) gives ® J (® k ) = I J ( k ⊥ a ) ∫ L / − L / (cid:18) − zL (cid:19) e − ik z z dz ˆ z = I J ( k ⊥ a ) " ∫ L / − L / e − ik z z dz − L ∫ L / − L / ze − ik z z dz ˆ z . (28)Carrying out the z integration (28) becomes ® J (® k ) = I J ( k ⊥ a ) (cid:20) ie − ik z L / − ie ik z L / k z − k z L (cid:18) iL ( e − ik z L / − e ik z L / ) − ( e ik z L / − e − ik z L / ) k z (cid:19) (cid:21) ˆ z = I J ( k ⊥ a ) i sin (cid:16) k z L (cid:17) k z L − ie ik z L / k z ˆ z (29)where integration by parts is used for the ze − ik z z integral (with u = z and v ′ = e − ik z z ).Now calculating | ® k · ® J (® k )| for use in (4) we obtain | ® k · ® J (® k )| = I k z J ( k ⊥ a ) © « i sin (cid:16) k z L (cid:17) k z L − ie ik z L / k z ª®®¬ © « − i sin (cid:16) k z L (cid:17) k z L + ie − ik z L / k z ª®®¬ = I J ( k ⊥ a ) © « (cid:16) k z L (cid:17) k z L − ( k z L ) k z L + ª®®¬ . (30)After substituting (30) into (4), using spherical coordinates, and carrying out the φ integra-tion we obtain Z a = i π ωǫ ∫ π ∫ ∞ © « (cid:16) kL cos ( θ ) (cid:17) k L cos ( θ ) − ( k L cos ( θ )) k L cos ( θ ) + ª®®¬ J ( ka sin ( θ )) sin ( θ ) ǫ L dkd θ (31) –10–onfidential manuscript submitted to Radio Science which can be simplified to create a new antenna response function F m ( x ) by concentratingon the θ integral. Using the substitution u = cos ( θ ) we write the θ integral as ∫ π © « (cid:16) kL cos ( θ ) (cid:17) k L cos ( θ ) − ( k L cos ( θ )) k L cos ( θ ) + ª®®¬ sin ( θ ) d θ = ∫ − © « (cid:16) kLu (cid:17) k L u ª®®¬ du − ∫ − sin ( k Lu ) k Lu du + ∫ − du = − (cid:16) kLu (cid:17) k L u − + ∫ − k Lu sin (cid:18) k Lu (cid:19) cos (cid:18) k Lu (cid:19) du − ( k L ) k L + = − (cid:16) kL (cid:17) k L − ( k L ) k L + ∫ − sin ( k Lu ) k Lu du = © « k L − (cid:16) kL (cid:17) k L ª®®¬ (32)where integration by parts was used in the first integral on the second line with u = sin ( k Lu / ) and v ′ = / k L u . Placing (32) back into (31) yields Z a = i π ωǫ ∫ ∞ © « k L − (cid:16) kL (cid:17) k L ª®®¬ J ( ka ) ǫ L dk . (33)We now define F m ( x ) , the monopole response function, as F m ( x ) = − sin (cid:0) x (cid:1) x ! (34)where x = k L and the constant 1 / Z a = i π ωǫ ∫ ∞ F m ( k L ) J ( ka ) ǫ L dk . (35)This can now be used in the same way as (26) to determine the resistance and capacitance ofthe antenna, and the spectra for QTN and shot noise. For large values of x , (34) convergestowards 1 / / × ( / + x / ) .The result for the monopole antenna response function (34) is very similar to that ofa spherical (rather than wire) dipole antenna. The antenna response function for a double-sphere dipole antenna is [ Meyer-Vernet and Perche , 1989;
Le Chat et al. , 2009] F s ( x ) = (cid:18) − sin ( x ) x (cid:19) (36) –11–onfidential manuscript submitted to Radio Science -4 -2 x -0.0500.050.10.150.20.250.3 F ( x ) F (x) - Dipole (wire)F m1 (x) - MonopoleF s1 (x) - Dipole (spheres) Figure 1.
Comparison of the antenna response function for a wire dipole antenna F ( x ) , as in Equation (17)of Kuehl [1966] or (17) in this text, a spherical dipole antenna F s using (36), and the monopole expression F m ( x ) in (34). which in the limit of large x converges to 1 / x /
24. For a double-sphere dipole of finite a , the F ( k ) function in (15) would be F ( k ) = F s ( k L ) sin ( ka ) k a . (37)In the case of an infinitely small a , this simplifies to F ( k ) = F s ( k L ) . Here we compare the three antenna response functions F ( x ) and F s ( x ) for dipolesand F m ( x ) for a monopole and the effects they have on generic spectra. We derived F ( x ) and F m ( x ) in (17) and (34), respectively, and showed the final expression for F s in (36). Wecompare the functions in isolation as a function of x first and then within expressions thatrepresent antennas immersed in an isotropic, unmagnetized thermal (or non-thermal) plasmalike the ionosphere. The second step allows us to investigate the voltage power spectrum forQTN and shot noise, as well as the predicted capacitance and resistance of the antennas.Figure 1 compares the wire and double-sphere dipole response functions F ( x ) and F s ( x ) , with the monopole response function F m ( x ) . The wire dipole function has a Gaussian- –12–onfidential manuscript submitted to Radio Science Wavenumber (k) -0.006-0.004-0.00200.0020.0040.0060.0080.01 I n t e g r a nd k D Dipole-wire Z Integrand - RealMonopole Z Integrand - RealDipole-wire Z Integrand - ImagMonopole Z Integrand - Imag Wavenumber (k) -12 -10 -8 -6 -4 -2 I n t e g r a nd k D Dipole-wire Z Integrand - RealMonopole Z Integrand - Real(k D ) -1 (k D ) -2 Figure 2.
Real and imaginary parts of the integrand, F n ( k L ) J ( ka )/ ǫ L , plotted against k (bottom axis) and k λ D (top axis) for (26) and (35), corresponding to a wire dipole antenna and a monopole antenna for n = m
1, respectively. The log-linear panel (left) reveals the form of the mostly negative imaginary parts whilethe log-log panel (right) includes additional ( k λ D ) − and ( k λ D ) − lines to show the rate of decrease for thereal parts of the integrand at large k . In both panels L = . a = × − m, ω = . × rad s − , andaverage ionospheric conditions at 300 km altitude are used, i.e. n e = . × m − and T e = like form, peaks near x = . x → x → ∞ . The monopolefunction F m ( x ) is approximately zero for small x but begins to rise near x &
1, peaks at0.125 at x ≈ .
5, and then has an oscillatory form. At larger x this oscillation dampensand F m ( x ) → .
125 as x → ∞ . The double-sphere dipole function has a similar risingand then oscillatory behavior albeit with F s ( x ) → .
25 as x → ∞ instead. However, thepeak of the function significantly overshoots its asymptotic value and dampens while it os-cillates to the value 0.25. Also, the peak value is reached sooner for the two dipole responsefunctions compared to the monopole function. Overall, the similar functional form meansthe monopole response function is expected to produce similar results to the double-spheredipole case for QTN and shot noise. This will be seen in the figures that follow.Figure 2 compares how the integrands in the antenna impedances Z a given by (26)and (35) vary with the antenna response function used. To calculate the integrand, average –13–onfidential manuscript submitted to Radio Science ionospheric conditions at 300 km altitude were used, with n e = . × m − and T e = ω = . × rad s − was chosen as a sample value for ω , the antennalength L = . a = × − m. Note that L / λ D = . ω / ω p = .
08. Figure 2 shows that the real parts of both integrands have a peak near k λ D ≈ − k λ D = k , but this happens much faster for thedipole case. The monopole function oscillates towards zero, decreasing approximately as k − , which is clear in the log-log plot with the ( k λ D ) − relationship overlaid. The dipole casealso oscillates towards zero at large k (due to the Bessel function), decreasing approximatelyas k − .The imaginary parts of the integrands in Figure 2 are also similar in their behavior butwith some differences. Both become negative near k ≈
10 m − ( k λ D ≈ .
1) and have anegative peak near k ≃
50 m − to 200 m − ( k λ D ≃ . − . k λ D = κ = λ D − κ is now defined by (8). The prop-erties of the functions are very much the same in Figures 2 and 3, showing that changingto a kappa distribution intorduces no qualitative differences. In detail, the real parts show asmooth rise to positive values and then an oscillatory decrease towards zero, while the imag-inary parts show smooth non-oscillatory behavior with a peak near k λ D − κ ≈ −
3. Onedifference for the Maxwellian case is that the peaks are shifted to slightly higher k , which isimportant only around the plasma peak where the pole of integration is located. A promi-nent and sharp peak is reached for the real part of the dipole integrand and imaginary part ofthe monopole integrand, albeit reaching a smaller positive and larger negative height than inFigure 2, respectively.Figure 4 uses the double-sphere antenna response function in place of the function forthe wire dipole case in Figure 2, so that we can clearly see the very strong similarities be- –14–onfidential manuscript submitted to Radio Science Wavenumber (k) -0.025-0.02-0.015-0.01-0.00500.0050.010.0150.02 I n t e g r a nd k D- Dipole-wire Z Integrand - RealMonopole Z Integrand - RealDipole-wire Z Integrand - ImagMonopole Z Integrand - Imag Wavenumber (k) -12 -10 -8 -6 -4 -2 I n t e g r a nd k D- Dipole-wire Z Integrand - RealMonopole Z Integrand - Real(k D- ) -1 (k D- ) -2 Figure 3.
Real and imaginary parts of the integrand, F n ( k L ) J ( ka )/ ǫ L , plotted against k (bottom axis)and k λ D (top axis), assuming a kappa distribution. This means that (7) was used for ǫ L , rather than (5) as inFigure 2, and (8) is used to define the Debye length, with the notation also changing to λ D − κ . Otherwise thesame variables, conditions and format are used as in Figure 2. Wavenumber (k) I n t e g r a nd k D Dipole-spheres Z Integrand - RealMonopole Z Integrand - RealDipole-spheres Z Integrand - ImagMonopole Z Integrand - Imag Wavenumber (k) -12 -10 -8 -6 -4 -2 I n t e g r a nd k D Dipole-spheres Z Integrand - RealMonopole Z Integrand - Real(k D ) -1 Figure 4.
Same as Figure 2 but comparing the monopole response with a double-sphere dipole antennarather than a wire dipole. –15–onfidential manuscript submitted to
Radio Science -2 -1 Normalized Frequency (f/f p ) -18 -17 -16 -15 -14 -13 -12 V o l t a g e P o w e r S p ec t r a l D e n s i t y ( V . H z - ) Dipole-wire QTN Spectrum - Maxwellian VDFMonopole QTN Spectrum - Maxwellian VDFDipole-spheres QTN Spectrum - Maxwellian VDF -2 -1 Normalized Frequency (f/f p ) -18 -17 -16 -15 -14 -13 -12 V o l t a g e P o w e r S p ec t r a l D e n s i t y ( V . H z - ) Dipole-wire QTN Spectrum - VDFMonopole QTN Spectrum - VDFDipole-spheres QTN Spectrum - VDF
Figure 5.
Comparison of the quasi-thermal noise (QTN) voltage power spectrum predicted for dipole andmonopole antennas using (9) and (15) assuming a (left) Maxwellian VDF and (right) kappa VDF. The wiredipole case uses (26), the monopole uses (35) and the double-sphere dipole uses (37) in (15). Parameters usedare L = . a = × − m, n e = . × m − , T e = κ =
4. Based on these conditions f p = . tween the integrals in (34) and (36) for the monopole and double-sphere dipole cases. Thefactor of 2 difference in the leading constant for the expressions is clearly seen in Figure 4,with the double-sphere values roughly double their monopole counterparts. Most impor-tantly, the general shapes of the functions are effectively identical - for the real part a largebroad peak followed by the oscillatory decrease to zero as ≈ k − while the imaginary partsshow a single smooth negative peak or trough. The small wiggles for k values below thepeak in the double-sphere dipole integrand is one feature that differs between the two cases.Figure 5 shows the predicted QTN spectra for the dipole and monopole antennas im-mersed in an ionospheric plasma with either a (left) Maxwellian or (right) kappa VDF with κ =
4. The antenna response functions in (17), (34) and (36) are used for the wire dipole,monopole and double-sphere impedance Z a , respectively, in (9) and (15). The plasma condi-tions used are averages for the ionosphere at 300 km above Earth’s surface obtained from theIRI model as used in Figures 2 - 3. –16–onfidential manuscript submitted to Radio Science -2 -1 Normalized Frequency (f/f p ) -4 -3 -2 -1 N o r m a li z e d U n i t s Dipole-wire QTN Spectrum - Maxwellian VDFMonopole QTN Spectrum - Maxwellian VDFDipole-spheres QTN Spectrum - Maxwellian VDF -2 -1 Normalized Frequency (f/f p ) -4 -3 -2 -1 N o r m a li z e d U n i t s Dipole-wire QTN Spectrum - VDFMonopole QTN Spectrum - VDFDipole-spheres QTN Spectrum - VDF
Figure 6.
Same spectra and conditions as in Figure 5 but normalized to the QTN amplitude at f = × Hz for f p = . The wire dipole spectrum (blue) is relatively flat at about 2 . × − V Hz − for fre-quencies below the plasma frequency f p = . × Hz and then rises to a peak just above f p . Comparing this with the approximation in Table 1 of Meyer-Vernet and Perche [1989],for f ≪ f p the voltage power should be V = × − × √ T e × ( λ D / L ) V Hz − . Forthe conditions used in this paper, V = . × − V Hz − , which is within 8% of the cal-culated value and therefore our calculation agrees well with past literature. For frequenciesgreater than f p the spectrum has a f − . relationship, which was calculated using the curvefitting tool in Matlab assuming a power law ( ax b ) and fitted for values from f / f p = . ≈ f − relationshipabove f p is well known in the literature for wire dipole antennas [ Meyer-Vernet and Perche ,1989]. For the kappa distribution (gold), the wire dipole spectrum also starts from a similarlevel for f ≪ f p but the peak is not as prominent. The fall off for large f follows a very sim-ilar relationship as for the Maxwellian, f − . , with 95% confidence the exponent is between-3.14 and -3.16.The monopole spectrum (red) is also relatively flat for low frequencies, but with amagnitude of 2 . × − V Hz − that is a factor of 10 higher than for the wire dipolecase, and rises to a slightly higher peak just above f p . However, the calculated fall-off for –17–onfidential manuscript submitted to Radio Science f ≫ f p is different and the spectrum is proportional to f − . (with 95% confidence theexponent is between -2.18 and -2.17). Interestingly, a fall-off of ≈ f − is well known fordouble-sphere antennas [ Meyer-Vernet and Perche , 1989]. In the kappa case, the monopolefall-off is f − . , lying between -2.08 and -2.09 with 95% confidence. The double-spheredipole spectrum (gold) is very similar to the monopole, although a factor of 2 higher, asseen in both the left and right panels of Figure 5. The value of the flat spectrum for f < f p (4 . × − V Hz − ) also agrees within 1% of the approximation from Meyer-Vernet andPerche [1989] of V = m e v T /( π / ǫ ) = . × − V Hz − for f ≪ f p . The relationshipfor large values of f is effectively the same as the monopole case with the same confidencelevels.The similarities between the monopole and double-sphere spectra are even more obvi-ous in Figure 6, which shows the spectra normalized to their value at f = × Hz. Boththe double-sphere and monopole spectra overlap almost exactly for the Maxwellian case atall frequencies. For the kappa distribution there is a small ( ≃ − f p - the double-spherespectrum is slightly larger, which is expected due to the suprathermal electrons of a kappadistribution [ Chateau and Meyer-Vernet , 1991;
Le Chat et al. , 2009]. However, the shape andvariations of the QTN spectra for the monopole and double-sphere antennas are extremelysimilar.Figure 6 also shows the large prominence of QTN peak above f p of the wire dipolespectrum compared to the double-sphere dipole and monopole cases. For the MaxwellianVDF, the QTN peak rises to ≈
70 times the flat spectrum value (at f ≪ f p ) for the wiredipole whereas the ratio is only ≃
10 for the monopole and double-sphere dipole spectra.In the kappa distribution case, the wire dipole peak only rises to about 20 times the flat spec-trum level compared to a ratio ≃ − f ≫ f p for the wire dipole spectrum is also clear in Figure 6.The height and width of the QTN peaks would affect the expected voltage signal at thereceiver as the measured spectra involve some integration over frequency - the higher andwider the peak the larger the expected voltage signals. The peak prominence is generallydetermined by L / λ D , as described in Couturier et al. [1981]; for L / λ D ≤ L / λ D becomes larger, the peak just above f p becomes –18–onfidential manuscript submitted to Radio Science -3 -2 -1 Normalized Frequency (f/f p ) -14 -12 -10 -8 -6 -4 V o l t a g e P o w e r S p ec t r a l D e n s i t y ( V . H z - ) Distribution - Maxwellian
Dipole-wire - k range: 10 -4 - 10 Dipole-wire - k range: 10 -5 - 10 Dipole-wire - k range: 10 -6 - 10 Monopole - k range: 10 -4 - 10 Monopole - k range: 10 -5 - 10 Monopole - k range: 10 -6 - 10 Dipole-sph - k range: 10 -4 - 10 Dipole-sph - k range: 10 -5 - 10 Dipole-sph - k range: 10 -6 - 10 -3 -2 -1 Normalized Frequency (f/f p ) -14 -12 -10 -8 -6 -4 V o l t a g e P o w e r S p ec t r a l D e n s i t y ( V . H z - ) Distribution - = 4
Dipole-wire - k range: 10 -4 - 10 Dipole-wire - k range: 10 -5 - 10 Dipole-wire - k range: 10 -6 - 10 Monopole - k range: 10 -4 - 10 Monopole - k range: 10 -5 - 10 Monopole - k range: 10 -6 - 10 Dipole-sph - k range: 10 -4 - 10 Dipole-sph - k range: 10 -5 - 10 Dipole-sph - k range: 10 -6 - 10 Figure 7.
Comparison of the shot noise voltage power spectra for wire dipole, double-sphere (dipole-sph)and monopole antenna configurations predicted using either (17), (36) or (34), respectively, and (11) and (15)for various ranges of k and (left) Maxwellian and (right) kappa ( κ =
4) VDFs for L = . a = × − m, n e = . × m − , and T e = k ranges identified in the inset over which (15) is integrated. Values of k are in m − . higher and sharper. In Figures 5 and 6 L / λ D = . ≫
1, explaining the prominence of thepeaks seen (especially for the wire dipole).Figure 7 shows a series of shot noise spectra for dipole and monopole antenna con-figurations, predicted using (11) with different k integration limits for both Maxwellian andkappa VDFs. Again (17), (34) and (36) are used for the antenna response functions in (11)but the surface area S is introduced as an additional parameter through N e = /√ π n e v T S . (38)For the dipole antennas this is the surface area of the two antenna arms while for the monopoleantenna this is the surface area of the spacecraft body. This is because a monopole measuresthe potential difference between the antenna arm and the spacecraft body and the spacecraftarea is assumed to be much larger than the antenna. In this paper we assume that a 1U Cube-Sat (10 cm ×
10 cm ×
10 cm sized spacecraft) is connected to the monopole antenna. For theantenna parameters we are considering, a = × − m and L = . S –19–onfidential manuscript submitted to Radio Science is about 160 times larger for the monopole than for the dipole case. However, the fact that amonopole rather than a dipole antenna is used affects the theoretical shot noise level as well -the total noise for a dipole should be twice that for one arm [
Meyer-Vernet , 1983]. Therefore,from this difference in S we expect for the monopole shot noise spectra to be larger by abouta factor of 80, all else being equal, than the dipole spectra. In fact, many space missions thatcarry on-board both a dipole and monopole antenna (or option for switching between thetwo configurations) have seen increased noise levels in the monopole case. This effect hasbeen seen in comparisons of dust impacts of the PRA (monopole) and PWS (dipole) instru-ments on board the Voyager 2 spacecraft [ Mann et al. , 2011] and for the Wind and STEREOspacecraft antennas [
Meyer-Vernet et al. , 2009]. However, there is still debate in the literatureon the exact explanation for these variations [
Gurnett et al. , 1983;
Tsintikidis et al. , 1994;
Meyer-Vernet et al. , 2014;
Ye et al. , 2016;
Maj and Cairns , 2018].Figure 7 uses the same ionospheric conditions as Figures 5 and 6. For each antennaconfiguration spectra are shown for three different ranges of k in the integrals in (15). It isclear at a glance of Figure 7 that the predicted shot noise spectra are convergent for the wiredipole case but not for the monopole or double-sphere dipole cases. In detail, for the wiredipole case increasing the integration range from k = − - 10 m − to 10 − - 10 m − brings the shot noise level only very slightly higher but with clear convergence to the finalspectrum. For the monopole case, however, increasing the k integration range increases theshot noise level considerably and there is no sign of convergence occurring; from k = − - 10 m − to 10 − - 10 m − the level increases by almost an order of magnitude. This is ex-pected from Figure 2 because the real part of the integrand does not converge quickly enoughtowards zero as it has a k − functional form, this leading to only logarithmic convergence.This behavior also occurs for the double-sphere antenna, as seens in Figure 4 and 7.We can also see in Figure 7 that the effects of changing the VDF to a kappa distribu-tion from a Maxwellian are minimal. In detail, for these parameters the shot noise spectrumlevel decreases by less than a factor of 2 for the wire-dipole case while the monopole anddouble-sphere spectra vary even less. The similarities in the double-sphere and monopoleantenna response functions in (34) and (36) and Figures 4 - 6 thus predictably carry over toproduce very similar shot noise spectra in Figure 7 when the integration limits are changed,albeit separated by five orders of magnitude. Therefore increasing k merely increases the fi-nal integral and does not allow the use of the monopole antenna response function (34) or thewell established double-sphere expression (36) to predict the shot noise level. –20–onfidential manuscript submitted to Radio Science
Table 1.
Effect of changing the integration limits on the calculated capacitance (12) which uses theimpedance to calculate capacitance k Integration Limits Monopole Capacitance Double-sphere Capacitance Wire Dipole10 − - 10 m − . × − F 1 . × − F 2 . × − F10 − - 10 m − . × − F 1 . × − F 2 . × − F10 − - 10 m − . × − F 7 . × − F 2 . × − F10 − - 10 m − . × − F 5 . × − F 2 . × − FThis may be caused by the implementation used in this paper and a more reasonableresult could be achieved by considering the ∆ V ω term (known as the correction term) de-scribed in Meyer-Vernet [1983]. In
Meyer-Vernet [1983] the author thoroughly derives thevoltage power spectrum for finite width dipole antennas in a plasma which is separated intoQTN and shot noise terms plus an additional ∆ V ω term. This additional term is ignorable if ω × min ( L , λ D )/ v T < ω , ω p and the frequency range used here may be too closeto f p . However, calculations with f = . × − F for the k range 10 − - 10 m − as a ‘final’ result (computationally fasterthan the 10 − - 10 m − integration limit considered). However, for both the monopole anddouble-sphere cases changing the range of k integration changes the capacitances signifi-cantly, by a factor of ≈ . × − F, which is within 4%of the value calculated from (12). The same approximation for the double-sphere antennayields 1 . × − F, which is within 10% of the value calculated above with k limits of10 − - 10 m − but further increases in the k range lead to a clear reduction in the calculatedcapacitance. Finally, the approximation C monopole = C dipole = . × − F for themonopole capacitance is over 2 orders of magnitude larger than the values calculated from(12) and the discrepancy becomes larger the greater the range of k that is used. Thus the –21–onfidential manuscript submitted to Radio Science values predicted for the capacitances yield several problems associated with the use of theantenna response function (34).
The predicted QTN spectrum, as presented in Figure 5, is one case where our newlyderived monopole response function provides a reasonable, converged prediction which canbe tested and used for measurement in the real-world. The reason is that unlike the predic-tions for the shot noise and capacitance made using this new function, increasing the range of k in the impedance integral for the monopole case does not impact on the final result. This isdue to the fact that in for the QTN only the real part of Z a matters if we disregard gain effectsand the factor of i outside the integral in the impedance (35) implies that it is the imaginarycomponent of the integrand that contributes to the real part of Z a . In Figure 2 the imaginarycomponent for the monopole case is shown to be well behaved and although it has a largermagnitude and sharper peak than the wire dipole case it follows the same functional formand crucially approaches zero quickly enough at large k ( ∝ k − ), being visually close tozero by k = − . This ensures that the integral is finite when integrating up to higher k values. Therefore (34) may be useful for predicting the monopole QTN spectrum, both forMaxwellian and kappa VDFs; however, comparison with observational data is required toverify this.The other applications of the wire dipole, double-sphere and monopole antenna re-sponse functions in Section 5, however, reveal a number of issues with the newly derivedmonopole response function. Although the same procedure was used to derive both thewire dipole and monopole functions, there are a number of features that are problematic forthe monopole case. Firstly, Figure 1 reveals the non-zero and almost constant nature of themonopole function for large x = k L , which in Figure 2 is seen to affect the real part of the in-tegrand used in the impedance integral (35). At large k & m − , the real component of themonopole integrand oscillation rises to a much larger peak and does not fall off as quicklyas the dipole integrand, in fact falling off ∝ k − instead of ∝ k − . This is problematic whenintegrating from k = k = ∞ (or choosing numerical limits close to these values) as thefinal result only converges logarithmically. This is evident in Figure 7 where we have usedincreasingly larger k ranges for the impedance integral in order to determine the level of shotnoise. Importantly, all these features are in common with the double-sphere antenna, whichhas a widely cited antenna response function very similar in functional form to the monopole –22–onfidential manuscript submitted to Radio Science -3 -2 -1 Normalized Frequency (f/f p ) -20 -15 -10 -5 V o l t a g e P o w e r S p ec t r a l D e n s i t y ( V . H z - ) Maxwellian VDF -3 -2 -1 Normalized Frequency (f/f p ) -22 -20 -18 -16 -14 -12 -10 -8 -6 Kappa VDF = 4
Dipole-wire - k max : 2 /LDipole-wire - k max : 2 / D Monopole - k max : 2 /LMonopole - k max : 2 / D Dipole-wire - k max : 10 Figure 8.
Comparison of shot noise spectra for a wire dipole and monopole antenna using the same equa-tions and conditions as Figure 7 but varying the maximum k reached in the integrals (26) and (35) between k max = π / L =
21 m − , k max = π / λ D = − , and k max = m − . The minimum k used in theintegral in all cases is the same: k min = − m − . Two VDFs are presented: (left) Maxwellian and (right)kappa. For the kappa case, λ D − κ rather than λ D is used for the Debye length. response function derived in this paper. The reason for these problems is the k − fall-off ofthe integrand’s real part, so that the integral does not converge as k → ∞ .Restricting k on physical grounds is a possible solution to this infinite integral prob-lem. One way is to use the Debye length as a limiting factor, i.e. setting k max = π / λ D ,which would imply that the wavelength cannot be smaller than the Debye length. Anotherway is to use the antenna length itself and set k min = π / L , implying that the length of theantenna constrains the wavelength that is detected to the case of a single wavelength fittingacross the length of one antenna arm. Figure 8 compares the wire dipole and monopole pre-dictions for each of k max = π / L and 2 π / λ D , as well as comparing these against the dipoleshot noise spectrum for the k range 10 − - 10 m − from Figure 7, represented in Figure 8 asthe dashed curve. We have also performed this comparison for both Maxwellian and kappaVDFs, showing that the results are similar and that the VDF does not affect the final result.For the dipole antenna case, it is clear that restricting k max does not recreate the same result –23–onfidential manuscript submitted to Radio Science as seen in Figure 7. With k max = π / λ D = . × m − the spectrum is 2.8 times lowerthan the dashed curve while with k max = π / L = . − the spectrum is not only 2 × times lower but the simple power law relationship for V ( f ) breaks down at higher frequen-cies. This shows that the majority of the response comes from the range 2 π / L − π / λ D forthe wire dipole case. For the monopole case with k max = π / L , the spectrum has the samefunctional form but is 2 orders of magnitude closer to the dashed line (the wire dipole spec-trum). For k max = π / λ D the monopole spectrum is over 4 orders of magnitude larger thanthe dashed dipole spectrum, and as seen in Figure 7 this behavior continues as k max is in-creased further. Again, a major contribution comes from the range 2 π / L − π / λ D .The Debye length is the distance over which the electric potential decreases by a factor1 / e and is used to approximate when a charged object is shielded from the effect of chargesfurther away. Therefore, plasma waves with wavelengths shorter than about the Debye lengthwill be damped by the charges in the plasma. However, restricting the wavelength λ to onlyone Debye length (or k max = π / λ D ) may not be enough to integrate over all physicallysignificant wavelengths - a factor 1 / e is still 37% of the initial charge. In Figures 9 and 10 weshow the predictions for increasing k max to 2 × , × and 8 × π / λ D , equivalent to decreasingthe wavelength to 1 / , / / λ D .In the dipole case, as expected increasing k max brings it closer to its final convergentshot noise level - at k max = π / λ D the shot noise spectrum is 92% (76%) of the result for k max = for the Maxwellian VDF (kappa VDF). Increasing k max from 4 π / λ D to 8 π / λ D increases the shot noise level by 37% (150%) but from 8 π / λ D to 16 π / λ D the increase is only5 .
6% (71%) for the Maxwellian VDF (kappa VDF). A further 8 .
5% (31%) increase on the k max = π / λ D spectrum would bring it to the Maxwellian (kappa) spectrum for k max = . For the monopole case, we initially see apparent convergence as in the dipole casebut divergence at the highest k max calculated. Based on our calculations, increasing k max from 4 π / λ D to 8 π / λ D increases the shot noise level by 190% (580%) while from 8 π / λ D to16 π / λ D the increase is 28% (300%) for the Maxwellian (kappa) VDF. A value of k max = π / λ D implies that the wavelength is on a scale which is shielded to less than 5% of theelectric potential of undressed charges. However, increasing from 16 π / λ D to 32 π / λ D thereis an increase in the spectrum level by 53%, greater than the 28% increase from 8 π / λ D to16 π / λ D . Therefore, there may be a physically significant effect between these two values but –24–onfidential manuscript submitted to Radio Science -3 -2 -1 Normalized Frequency (f/f p ) -16 -14 -12 -10 -8 -6 -4 V o l t a g e P o w e r S p ec t r a l D e n s i t y ( V . H z - ) Maxwellian VDF -3 -2 -1 Normalized Frequency (f/f p ) -16 -14 -12 -10 -8 -6 -4 Kappa VDF = 4
Dipole-wire - k max : 4 / D Dipole-wire - k max : 8 / D Dipole-wire - k max : 16 / D Monopole - k max : 4 / D Monopole - k max : 8 / D Monopole - k max : 16 / D Monopole - k max : 32 / D Dipole-wire - k max : 10 Figure 9.
Comparison of shot noise spectra as in Figure 8 but using k max = n × ( π / λ D ) with n = , , k max = is also included. k is in m − and λ D − κ rather than λ D is used in the kappa VDF case. ultimately the spectrum is not converging. A crucial point here is that with no obvious refer-ence it is difficult to ascertain how close these values are to the "true" shot noise level. In theMaxwellian case, 16 π / λ D may be a sufficiently large value of k max to account for most ofthe physically significant effects, especially based on the results for the dipole case (reaching92% of the final value) but further verification is necessary, experimental tests being best.For the kappa VDF, it is clear in Figure 9 that convergence is slower than for the MaxwellianVDF. The dipole spectrum at k max = π / λ D is only 76% of the reference spectrum at k max = while the monopole spectrum increases by integer multiples of the previous lev-els when increasing k max from 4 π / λ D to 8 π / λ D and then from 8 π / λ D to 16 π / λ D . It may bedifficult to achieve a specific kappa VDF in a laboratory setting to experimentally discover orverify the necessary k max limit, however this would again be the best approach.We may also choose the capacitance as a criteria for restricting k max . Using the monopoleimpedance (35) calculated for k max = π / λ D and k max = π / λ D we find using (12) that C mono π / L = . × − F and C mono π / λ D = . × − F, respectively. These areover an order of magnitude away from the value calculated in Section 5’s last paragraph,i.e. C monopole = C dipole = . × − F. If we use the low frequency approximation –25–onfidential manuscript submitted to
Radio Science
Normalized Frequency (f/f p ) -3 V o l t a g e P o w e r S p ec t r a l D e n s i t y ( V . H z - ) -11 Dipole
Normalized Frequency (f/f p ) -3 -5 Monopole
Dipole-wire - k max : 4 / D Dipole-wire - k max : 8 / D Dipole-wire - k max : 16 / D Monopole - k max : 4 / D Monopole - k max : 8 / D Monopole - k max : 16 / D Dipole-wire - k max : 10 Figure 10.
Zoomed-in version of the left, or Maxwellian, panel of Figure 9 showing the (left) wire dipoleand (right) monopole predictions for shot noise.
Table 2.
Effect of restricting k max on capacitance results under plasma different conditions. Columns 4 and5 are calculated using (12) and (13), respectively. Altitude Density n e Temperature T e Capacitance with k max = . × πλ D Capacitance for f ≪ f p
300 km 5 . × m − . × K 5 . × − F 5 . × − F800 km 6 . × m − . × K 1 . × − F 3 . × − F1500 km 1 . × m − . × K 1 . × − F 3 . × − F C monopole as the necessary criteria for determining the k max cut-off then we find that a valueof k max =
207 m − = . × ( π / λ D ) brings the calculated capacitance within 3% of C monopole .To test this criterion for restricting k max under different plasma conditions we calcu-late the impedance with k max = . × ( π / λ D ) under average ionospheric conditions at800 km and 1500 km altitude from the IRI, as summarisized in Table 2. The table also in-cludes the IRI model values for n e and T e at 800 km and 1500 km, and a comparison againstthe low frequency approximation (13). Although these values are closer than the orders ofmagnitude difference seen previously for the restrictions based on the length of the antenna –26–onfidential manuscript submitted to Radio Science and Debye length, these capacitances are still about a factor of 3 different from the predic-tions using the low frequency approximation. Also, using k max = . × ( π / λ D ) forthe wire dipole case does not lead to reasonable capacitances either. For 300 km altitude, acapacitance of 3 . × − F is obtained, which is a factor of 13 larger than the dipole ap-proximation, while at 800 km and 1500 km the capacitance is larger by a factor of 3.9 and2.8, respectively, from the low frequency approximation.Therefore the main problem with restricting k max is that a choice must be made onwhat this value should be. Basing it on physical grounds, such as restricting k max to 2 π / λ D ,does not recreate the same results as in Figure 7 for the dipole antenna. Increasing this limitto 4 or 8 × π / λ D will bring the spectrum closer but what this multiplicative factor should beis ultimately an arbitrary choice and would need to be determined experimentally to be valid.Restricting k max based on the expected capacitance for the monopole antenna also did notproduce accurate results for other plasma conditions or the dipole case.Therefore the issues with using the monopole antenna response function (34) anddouble-sphere response function (36) to predict shot noise and the antenna impedance whileproducing reasonable QTN predictions reveal that some physics is missing and needs to beresolved in order to use and predict these quantities more generally. The specific issue is withthe k − fall-off of the integrand for the real part of the impedance, which produces results forshot noise and capacitance that are non-convergent for both the monopole and double-sphereantennas. Future work must be carried out to solve the issues. One approach may be to limit k max on some other physical grounds. Another may be to find a more accurate representa-tion of the current distribution for the monopole and double-sphere antennas. In addition,experimental measurement of the shot noise spectra, impedances, and response functionsmay clear up some of the issues. Empirical results could also be used to find an appropriatevalue for k max , if it exists.In future, we will also look at using other theoretical assumptions to derive an appro-priate analytical antenna response function for monopole antennas. In particular, we plan onusing the work of Kellogg [1981] as a starting point to derive a new function. We provide thecode used to obtain the plots and results in this paper as a repository online (listed in the Ac-knowledgements). We hope this will prove useful for others in the community in exploringtheir own antenna response functions. –27–onfidential manuscript submitted to
Radio Science
The wire dipole antenna response function was re-derived in this paper following stepssimilar to
Kuehl [1966] and
Couturier et al. [1981]. This procedure was then used to derivea new function (34) for the monopole antenna response function, which produces reasonablepredictions for the QTN spectrum.The wire dipole response function was shown to be a well-behaved function that ap-proached zero sufficiently rapidly ( ∝ k − ) for small and large values of k L to produce reason-able predictions for the QTN and shot noise spectra in chosen plasma environments consid-ered, specifically the lower ionosphere at altitudes of 300 km, 800 km and 1500 km above theEarth’s surface [ Maj and Cairns , 2017]. Specifically, the QTN and shot noise spectra con-verged as the k range was increased for the integral in the impedance expression (26) whilethe antenna capacitance that was calculated using this impedance was within 4% of the ap-proximation determined by Meyer-Vernet and Perche [1989].The derived monopole response function, on the other hand, approached a constantfor large k L , very similar to the double-sphere antenna response function which was ex-plored as a point of comparison. Both the monopole and double-sphere function producedill-defined predictions for the shot noise and capacitance. The imaginary component of (15)does not converge to zero as quickly as the wire dipole function for large values of the inte-gration variable k ( ∝ k − versus ∝ k − , respectively). This produced non-convergent resultsfor the shot noise spectrum level and capacitance - as the integration domain increases theshot noise level increases while the capacitance decreases. However, the monopole QTNspectrum does converge since this is determined by the real part of the impedance, which hasa different functional form that does allow convergence at large k .Restricting the upper limit (maximum k ) of the k integrals based on physical groundswas investigated as one way to create finite results for the shot noise and capacitance. How-ever, using the Debye length, antenna length or the capacitance as criterion for restrictiondoes not produce acceptable results. Specifically, the wire dipole results for shot noise andcapacitance are not reproduced well using a wavenumber restriction, revealing a lack of gen-erality for these criterion. Experimental testing would be one way to find such a k restrictionor to verify the validity of a yet unknown general monopole response function. We providean online repository of the code used to produce our results so that others may explore theirown expressions. –28–onfidential manuscript submitted to Radio Science
Acknowledgments
R. Maj was financially supported under a University of Sydney Postgraduate Awards (SC0649)scholarship. M. M. Martinović was financially supported by the Ministry of Education, Sci-ence and Technological Development of Republic of Serbia through financing the projectON176002 and by NASA grant 80NSSC19K0521. The code used for calculations and fig-ures in this paper can be found at: https://github.com/ronaldmaj/Monopole_Paper_Calc_Fig_Script - the code for calculating kappa distributions within this was adaptedfrom
Odelstad [2013]. The International Reference Ionosphere (IRI) was used as a source ofmodel data for the electron density and temperature values in this paper: https://ccmc.gsfc.nasa.gov/modelweb/models/iri2016_vitmo.php . References
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