Prediction of Uncertainties in Atmospheric Properties Measured by Radio Occultation Experiments

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Advances in Space Research 46 (2010) 58–73 www.elsevier.com/locate/asr

Prediction of uncertainties in atmospheric properties measured by radio occultation experiments

Paul Withers

Center for Space Physics, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA

Received 3 September 2008; received in revised form 9 November 2009; accepted 4 March 2010

Abstract

Refraction due to gradients in ionospheric electron density, N e, and neutral number density, nn, can shift the frequency of radio signals propagating through a planetary atmosphere. Radio occultation experiments measure time series of these frequency shifts, from which N e and nn can be determined. Major contributors to uncertainties in frequency shift are phase noise, which is controlled by the Allan Deviation of the experiment, and thermal noise, which is controlled by the signal-to-noise ratio of the experiment. We derive expressions relatingpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uncertainties in atmospheric properties to uncertainties in frequency shift. Uncertainty in N e is approximately 2 ð4prDf fcme0=Ve Þ 2pH p=R where rDf is uncertainty in frequency shift, f is the carrier frequency, c is the speed of light, me is the electron mass, 0 is the permittivity of free space, V isp speed,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie is the elementary charge, H p is a plasma scale height and R is planetary radius. Uncertainty in nn is approximately ðrDf c=Vf jÞ H n=2pR where j and H n are the refractive volume and scale height of the neutral atmosphere. Predictions from these expressions are consistent with the uncertainties of the radio occultation experiment on Mars Global Sur- veyor. These expressions can be used to interpret results from past radio occultation experiments and to perform preliminary design studies of future radio occultation experiments. Ó 2010 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Radio occultation; Measurement uncertainties; Instrument design

1. Introduction 1971; Yakovlev, 2002; Kliore et al., 2004). Sections 2–6 sum- marize the principles behind radio occultation experiments. Radio occultation experiments are commonly used by They are aimed at readers who may be familiar with the spacecraft missions to study planetary atmospheres, partic- atmospheric measurements made by radio occultation ularly to measure vertical proﬁles of both the ionospheric experiments, but who are not familiar with the implementa- electron density and the number density of neutral gases tion of such experiments. These sections provide the intellec- (Tables 1 and 2). In this paper, the term “atmosphere” is tual foundation required by subsequent sections. Section 7 used inclusively to encompass both the “ionosphere” and discusses sources of uncertainties. Section 8 discusses uncer- “neutral atmosphere”. The aim of this paper is to derive tainties for gravity tracking observations. Sections 9 and 10 simple expressions for uncertainties in atmospheric proper- obtain expressions for uncertainties in atmospheric properties that can be used to interpret results from past radio ties measured by radio occultation experiments. Section 11 occultation experiments and to perform preliminary design summarizes some complications of real radio occultation studies of future radio occultation experiments. experiments. Sections 12–14 discuss these results. The application of radio occultation experiments to planetary science has been described previously by several 2. Planetary radio occultations authors (e.g. Phinney and Anderson, 1968; Fjeldbo et al., Planetary radio occultations depend on the interaction E-mail address: [email protected] of a radio signal and a planetary atmosphere. In most

Table 1 Table 1 (continued) List of symbols. Symbol Meaning Location of Symbol Meaning Location of first use first use ln Real part of refractive index of the neutral Section 5 A Amplitude Eq. (22) atmosphere a Impact parameter Section 3 m Refractivity Eq. (5) a a j j Value of for ray path Section 3 me Refractivity of the ionosphere Section 3 B Half of noise bandwidth Eq. (23) mj Value of m for ray path j Section 3 C Signal power Eq. (23) mn Refractivity of the neutral atmosphere Section 3 C1 A coefficient Eq. (21) q Mass density Section 5.1 C 2 A coefficient Eq. (21) rAD Allan Deviation Eq. (24) C3 A coefficient Eq. (21) rNe Uncertainty in N e Eq. (38) c Speed of light Eq. (3) rnn Uncertainty in nn Eq. (39) e Elementary charge Eq. (1) rthermal Uncertainty in frequency due to thermal noise Eq. (23) f Frequency Eq. (1) rV Uncertainty in V Section 8 fmeasured Measured frequency Eq. (25) rDf Uncertainty in Df Eq. (26) fnominal Nominal frequency Eq. (25) rm Uncertainty in m Eq. (37) f p Plasma frequency Eq. (2) sthermal Time interval relevant for thermal noise Section 7.2 fR Received frequency Section 2 sphase Time interval relevant for phase noise Section 7.3 fT Transmitted frequency Section 2 / Phase Section 7.2 G Gravitational constant Section 12 H A scale height Eq. (28) H Neutral scale height Eq. (39) n cases, geometrical optics provides a sufficiently accurate H p Plasma scale height Eq. (38) h A vertical lengthscale Section 7.1 description of the interaction and the radio signal can be K0 A modified Bessel function Eq. (32) treated as a ray. The propagation of a radio signal through kB Boltzmann’s constant Eq. (16) a medium is determined by the medium’s complex refrac- L Distance between tangent point Section 11 tive index. The real part of the refractive index controls and closest radio element M Planetary mass Section 12 phase speed and the imaginary part of the refractive index m Mean molecular mass Section 12 controls extinction (Eshleman, 1973). Variations in the real me Electron mass Eq. (1) part of the refractive index lead to refraction and bending N 0 Noise power density Eq. (23) of the ray. The refractive index of a planetary atmosphere N e Electron density Eq. (1) is determined by the neutral gases and plasma it contains. n Label of a sample period Eq. (24) As shown in Section 6, it is generally possible to consider nn Total neutral number density Eq. (13) nn;i Number density of constituent i Eq. (12) contributions to the refractive index from the neutral atmo- p Pressure Section 5.1 sphere and plasma separately (Eshleman, 1973). ps Standard pressure Section 5.2 In the case that the effects of magnetic fields and colli- pW Partial pressure of water vapour Eq. (21) sions between charged particles and neutrals can be R Planetary radius Section 7.1 r Radius of closest approach of Section 3 neglected, the complex refractive index of ionospheric ray path to planetary center plasma, g,is(Rishbeth and Garriott, 1969): r r j j Value of for ray path Section 3 2 s Signal Eq. (22) 2 N ee g ¼ 1 2 2 ð1Þ T Temperature Section 5.1 4p me0f T s Standard temperature Section 5.2 t Time Eq. (22) where N e is electron density, e is the elementary charge, me V Speed Eq. (3) is the electron mass, 0 is the permittivity of free space and f y Relative deviation in frequency Eq. (24) is frequency. Refraction in ionospheric plasma is almost a Bending angle Section 2 entirely due to electrons, not ions, so only the electron den- a Value of a for ray path j Section 3 j sity features in Eq. (1). The total plasma density approxi- apol Molecular polarizability Section 5.2 b 1 Eq. (15) mately equals the electron density if the number density Relative permittivity or Section 5.2 of negatively charged ions is much less than the electron dielectric constant density. The plasma frequency, fp, is defined as: 0 Permittivity of free space Eq. (1) g Complex refractive index Eq. (1) N e2 f 2 ¼ e ð2Þ h Thermal noise Eq. (22) p 4p2m j Refractive volume Eq. (13) e 0 ji Refractive volume of constituent i Eq. (12) and the complex refractive index of ionospheric plasma can k Wavelength Section 12 be expressed as 1 f 2=f 2. In this case, the real part of the l Real part of refractive index Section 3 p refractive index of ionospheric plasma is always less than le Real part of refractive index Section 4 of the ionosphere unity. If f > fp, then the complex refractive index is real. lj Value of l for ray path j Section 3 If f < fp, then the complex refractive index is imaginary. 60 P. Withers / Advances in Space Research 46 (2010) 58–73

Table 2 experiment by reducing the signal strength. Critical refrac- Atmospheres investigated by radio occultation experiments. tion, defocusing and multipath propagation, which compli- Object Spacecraft cate radio occultation experiments, can also occur as a Mercurya Mariner 10 radio signal propagates through an atmosphere. These Venusb Mariner 5, Mariner 10, Venera 9, are discussed in Section 9. The focus of this paper is on Venera 10, Pioneer Venus Orbiter, the effects of refraction, not extinction. Accordingly, “the Venera 15, Venera 16, Magellan, Venus Express real part of the refractive index” is abbreviated to “the Earthc GPS/MET, Oersted, CHAMP, IOX, SAC-C, GRACE, COSMIC, Metop-A refractive index” in the remainder of this paper. Moond Pioneer 7, Luna 19, Luna 22, SMART-1, SELENE We now introduce some common terms. In planetary sci- Marse Mariner 4, Mariner 6, Mariner 7, ence, a radio occultation experiment involves two widely- Mars 2, Mariner 9, Mars 4, Mars 5, separated radio elements and a planet. One radio element Mars 6, Viking 1, Viking 2, Mars transmits a radio signal that passes close to the planet under Global Surveyor, Mars Express Jupiterf Pioneer 10, Pioneer 11, Voyager 1, Voyager 2, Galileo observation before being received by the second radio ele- Iog Pioneer 10, Galileo ment. In some cases, the second radio element transmits a Europah Galileo radio signal, possibly related to the received signal, back to Ganymedei Galileo j the first radio element. In other cases, the second radio ele- Callisto Galileo ment functions as a receiver only. The former is called a Saturnk Pioneer 11, Voyager 1, Voyager 2, Cassini Titanl Voyager 1, Cassini “two-way” experiment and the latter is called a “one-way” Uranusm Voyager 2 experiment. A two-way experiment requires that both radio Neptunen Voyager 2 elements transmit and receive signals, whereas a one-way Tritono Voyager 2 p experiment does not. A two-way experiment is “coherent” 1P/Halley Vega 1, Vega 2, Giotto if the second signal is derived directly from the first signal a Fjeldbo et al. (1976). b and “non-coherent” if it is not. In a “spacecraft-spacecraft” Yakovlev (2002) and references therein. experiment, each radio element is located on a spacecraft. In c Wu et al. (2009) and references therein. d Stern (1999) and references therein; Yakovlev (2002) and references a “spacecraft-Earth” experiment, one radio element is therein; Maccone and Pluchino (2007), Pluchino et al. (2007), Imamura located on a spacecraft and the other is located on Earth. et al. (2008a,b). An “uplink” signal is transmitted from Earth to a spacecraft e Mendillo et al. (2003) and references therein; Pa¨tzold et al. (2005). f and a “downlink” signal is transmitted from a spacecraft to Yelle and Miller (2004) and references therein. Earth. Note that a single radio element can transmit at one g Kliore et al. (1975), Hinson et al. (1998a). h Kliore et al. (1997). or more frequencies and that the use of multiple frequencies i Kliore et al. (1998), Kliore (1998). can reduce certain sources of error (Section 7). An “ingress” j Kliore et al. (2002). or “entry” occultation occurs when, from the point of view k Gehrels and Matthews (1984) and references therein; Nagy et al. of one radio element, the other radio element disappears (2006). behind the planet. Conversely, an “egress” or “exit” occulta- l Gehrels and Matthews (1984) and references therein; Flasar et al. (2007). tion occurs when the other radio element reappears from m Lindal et al. (1987). behind the planet. n Tyler et al. (1989), Lindal (1992). There are many possible types of radio occultation exper- o Tyler et al. (1989). iments. Examples include one-way spacecraft-spacecraft p Pa¨tzold et al. (1997) and references therein. experiments, such as the COSMIC mission (Rocken et al., 2000; Anthes et al., 2008), that use a GPS satellite as the Radio signals that propagate from a region of low electron transmitter and a satellite in low-Earth orbit as the receiver; density into a region of high electron density are reflected one-way uplink experiments, such as New Horizons (Tyler where f ¼ fp and never propagate into regions where et al., 2008), that use a ground-based transmitter and a f < fp. A radio occultation experiment is generally de- spacecraft receiver; one-way downlink experiments, such signed such that f > fp. Radar sounders, which probe as Mars Global Surveyor (Tyler et al., 2001), that use a ionospheres and magnetospheres, are another common spacecraft transmitter and a ground-based receiver; and type of planetary radio science instrument. They rely on two-way experiments, such as Mars Express (Pa¨tzold the reflection of radio signals by plasma when f < fp, et al., 2004), in which a ground-based transceiver uplinks a and hence typically use lower frequencies than radio occul- signal to a spacecraft transceiver, which then downlinks a tation experiments (Reinisch et al., 2000; Gurnett et al., related signal to the ground-based transceiver. 2005). At any given time, one point along the radio signal’s ray In contrast, the real part of the refractive index of a neu- path is closer to the center of mass of the planet than any of tral atmosphere is always greater than unity and neutral the other points on the ray path. Depending on the altitude atmospheres do not reflect radio signals. Extinction of of closest approach, this ray path may pass through the the radio signal in regions where the imaginary part of atmosphere of the planet. If so, the ray path is refracted the refractive index is large can degrade the quality of the by the atmosphere and it differs from the ray path the P. Withers / Advances in Space Research 46 (2010) 58–73 61 signal would have taken in the absence of refraction. Both exceeding 1 kHz at the 1 bar level in the atmospheres of Ve- the ionosphere and neutral atmosphere contribute to this nus, Jupiter and Titan (Section 13). refraction. Two vectors can be defined, one by the ray path before it entered the atmosphere and one by the ray path after it exited the atmosphere. The “bending angle”, a,is 3. Measurement of atmospheric refractivity from bending of defined as the angle between these two vectors (Fjeldbo ray paths et al., 1971). This is illustrated in Fig. 1, which shows a one-way downlink experiment at Mars. The frequency During an occultation, the received frequency is measured as a function of time. This is then compared to a pre- received by one radio element, fR, is not the same as the fre- diction of the frequency that would have been received in quency transmitted by the other radio element, fT . For instance, the classical Doppler shift expected between two the absence of refraction by the planetary atmosphere to radio elements with a relative speed of 5 kms1 and an find Df as a function of time. At each timestep, a can be X-band frequency of 8.4 GHz is 140 kHz. Refraction of obtained from Df , the known spacecraft trajectory and the ray path by the planetary atmosphere introduces an Eq. (3). The impact parameter, a, is defined as the radial additional frequency shift, so the actual received frequency distance of closest approach that the ray path would have is not identical to the frequency that would have been in the absence of refraction. At each timestep, a can be received in the absence of refraction in the planetary atmo- obtained from the known trajectories of the two radio ele- sphere. For small bending angles, the magnitude of this fre- ments. If the atmosphere’s refractive index, l, is spherically quency difference, Df , approximately satisfies: symmetric, then it can be shown that a and a are related by (Fjeldbo et al., 1971): Z Df V a ÀÁ r¼1 d ln lðrÞ dr ð3Þ aj aj ¼2aj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ f c 2 r¼rj dr 2 ðÞlðrÞr aj where V is the relative speed of the two radio elements and where r is radius and subscripts indicate a specific ray path, c is the speed of light (Hinson et al., 1999, 2000). Eq. (3) which is usually equivalent to a specific time at the receiver. neglects some geometrical and other factors; more accurate The radius of closest approach of the ray path whose im- expressions are given in Appendix A of Fjeldbo et al. pact parameter is aj is rj. Here we follow Ahmad and Tyler (1971) and Eq. (2.2.8) of Kursinski (1997). It is important (1998) in adopting the sign convention that positive bend- to note that this approximate relationship should not be ing is towards the center of planet, whereas Fjeldbo et al. used to derive bending angles from measured frequency (1971) used the opposite convention. It is also convenient shifts. For a typical radio occultation experiment in the to define refractivity, m, which can be positive or negative, Martian atmosphere, the maximum value of Df is about as: 10 Hz, much smaller than the total Doppler shift (Hinson et al., 1999). Values are larger in denser atmospheres, m ¼ l 1 ð5Þ

We do not adopt the convention of some other workers who introduce a factor of 106 on the left hand side of Eq. (5). Typically, jmj1 (Sections 4 and 5). The sign of the bending angle is determined by the sign of d ln lðrÞ=dr, which for weak refraction is the sign of dm=dr. Ionospheres have negative refractivity and neutral atmospheres have positive refractivity, so radio signals entering an ionosphere from vacuum are refracted in the opposite direction from radio signals entering a neutral atmosphere from vacuum (Section 2). As described by Fjeldbo et al. (1971) and Hinson et al. (1999), Eq. (4) can be inverted to give an expression for l in terms of a and a: Z 1 a¼1 aðÞa da American Geophysical Union. Reproduced by ln l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 Ó permission of American Geophysical Union 1999 j ¼ ð Þ p 2 a¼aj a2 a Fig. 1. Schematic illustration of a radio occultation experiment, specif- j ically a one-way downlink experiment conducted at Mars by Mars Global Surveyor (MGS). The bending angle is indicated by a, the impact This inversion is an example of an Abel transform (Phinney parameter by a and the radial distance of closest approach by r. and Anderson, 1968; Bracewell, 1986; Ahmad and Tyler, Reproduced with minor modifications from Fig. 2 of Hinson et al. (1999). Fig. 2 of Hinson et al., Initial results from radio occultation 1998). Eq. (6) can be integrated by parts to eliminate the measurements with Mars Global Surveyor, Journal of Geophysical infinite integrand at a ¼ aj (Yakovlev, 2002; Pa¨tzold Research, 104, 26,997–27,012, 1999. et al., 2004). 62 P. Withers / Advances in Space Research 46 (2010) 58–73 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 Z 2 clouds, can be neglected because they are transparent at 1 a¼1 a a da ln l ¼ ln @ þ 1A da ð7Þ typical radio occultation frequencies. If the chemical com- j p a a da a¼aj j j position of the atmosphere is known, then a mean refractive volume, j, can be defined such that: Or equivalently: Z a¼1 mn ¼ jnn ð13Þ 1 1 a da ln lj ¼ cosh da ð8Þ where nn is the total neutral number density. Hence nnðrÞ p a¼aj aj da can be found from mnðrÞ. For many typical atmospheric 29 3 After measurements of aðaÞ are used to determine lj, the gases, j is on the order of 10 m , similar to their classical radial distance to which lj corresponds, rj, must be found. molecular volumes (Eshleman, 1973). For a 1 bar atmo- 29 3 4 This is accomplished via Snell’s law of refraction for spher- sphere at 300 K and j ¼ 10 m , mn ¼ 2 10 , confirm- ical geometry, also known as Bouguer’s rule, which states ing that jmnj1. The mass density profile, qðrÞ, can be that aj and rj are related by (Born and Wolf, 1959; Kursin- found from nnðrÞ and the atmospheric composition. The ski et al., 2000): pressure profile, pðrÞ, can be found from qðrÞ, the known gravitational field and an upper boundary condition via ljrj ¼ aj ð9Þ the equation of hydrostatic equilibrium. The upper bound- Hence lðrÞ can be determined from aðaÞ. The total refractiv- ary condition can be an assumed temperature or pressure. ity of the atmosphere, m ¼ l 1, is the sum of the refractivity It can also be obtained from the scale height of nnðrÞ under of the ionosphere, me, and the refractivity of the neutral certain reasonable assumptions (Withers et al., 2003; atmosphere, mn (Eshleman, 1973). Tellmann et al., 2009). The temperature profile, T ðrÞ, can be found from p r , the atmospheric composition and m ¼ me þ mn ð10Þ ð Þ either nnðrÞ or qðrÞ via an equation of state, such as the In practice, it is possible to determine both me and mn from a ideal gas law. Errors in the upper boundary condition have single measurement of m for most planetary atmospheres minimal effect on the derived pressure and temperature at and radio frequencies. This is addressed in Section 6. Ver- altitudes more than several scale heights below the upper tical profiles of ionospheric electron density and neutral boundary (Withers et al., 2003). Note that neutral atmo- number density can be determined from meðrÞ and mnðrÞ, spheric refractivity is not frequency-dependent. respectively, as shown in Sections 4 and 5. 5.2. Values of refractive volumes 4. Ionospheric refractivity Refractive volume, which is not typically found in stan- The relationship between the complex refractive index of dard reference works, can be related to relative permittiv- an ionospheric plasma and frequency was stated in Eq. (1) ity, , which is also known as the dielectric constant, and for circumstances in which magnetic fields and electron-neu- molecular polarizability, apol. These are commonly found tral collisions can be neglected. For typical radio occultation in standard reference works (Lide, 1994). experiments in planetary ionospheres, the real part of the When the relative permeability (works on electromagne- ionospheric refractive index, le, satisfies jle 1j1 and, tism frequently use the symbol l for permeability; this following Eq. (1), le is given by: paper does not) of a neutral medium is close to unity, its 2 values of l , the refractive index, and are related by N ee n le 1 ¼ me ¼ 2 2 ð11Þ (e.g. Mo¨ller, 1988; Hecht, 2002): 8p me0f pffiffi 12 3 Hence N eðrÞ can be found from meðrÞ. For N e ¼ 10 m ,a ln ¼ ð14Þ large electron density for solar system ionospheres, and Standard reference works, such as Lide (1994), commonly f = 8.4 GHz, m ¼6 107, confirming that jm j1. e e list values of at standard temperature, T , and pressure, Note that ionospheric refractivity is frequency-dependent. s ps.If can be written as: 5. Atmospheric refractivity ¼ 1 þ b ð15Þ

5.1. Introduction to refractive volume where b 1, then, following Eq. (13) and the ideal gas law, the refractive volume, j, satisﬁes: The refractive index of the neutral atmosphere, ln, bk T satisﬁes: j ¼ B s ð16Þ X 2ps ln 1 ¼ mn ¼ jinn;i ð12Þ where kB is Boltzmann’s constant. The Clausius–Mossotti, where ji is the refractive volume of constituent i and nn;i is or Lorentz–Lorenz, equation, which relates and apol can the number density of constituent i (Eshleman, 1973). The also be used to determine j (Ashcroft and Mermin, 1964; refractivity of aerosols and condensates, such as dust or Robinson, 1973). P. Withers / Advances in Space Research 46 (2010) 58–73 63

1 4pn a Table 4 ¼ n pol ð17Þ þ 2 3 Values of C1 for solar system atmospheres. Tabulated values were found using C1 ¼ j=kB, values of j from Table 3 and stated compositions. Eq. (17) assumes that apol is expressed as a volume (cgs Compositions are from Lodders and Fegley (1998), with the exception of units), not in units of C m2 V1 (SI units). Use of SI units Titan (Niemann et al., 2005). The abundances of some constituents have been rounded up to ensure that percentages total 100%. for apol requires a factor of 4p0 in the denominator of the 1 right hand side of Eq. (17), where 0 is the permittivity of Object Composition C1 (K Pa ) 6 free space (Lide, 1994). For 1 1, Eq. (17) can be rear- Venus 96.5% CO2, 3.5% N2 1:3 10 7 ranged to yield: Earth 78% N2, 21% O2,1%Ar 7:8 10 6 Mars 95.7% CO2, 2.7% N2, 1.6% Ar 1:3 10 7 ¼ 1 þ 4pnnapol ð18Þ Jupiter 86.4% H2, 13.6% He 4:5 10 7 Saturn 96.7% H2, 3.3% He 3:9 10 From Eqs. (14) and (18): 7 Titan 95.1% N2, 4.9% CH4 8:1 10 Uranus 82.5% H , 15.2% He, 2.3% CH 4:7 107 l ¼ 1 þ 2pnnapol ð19Þ 2 4 n 7 Neptune 80% H2, 19% He, 1% CH4 4:9 10 7 From Eqs. (13) and (19): Triton 100% N2 8:0 10 7 Pluto 100% N2 8:0 10 j ¼ 2papol ð20Þ Lide (1994, pp. 6–154) lists relative permittivities at 20 °C (293.15 K) and 101.325 kPa for approximately 40 com- 6. Separation of atmospheric and ionospheric refractivities pounds, including He, Ar, H2,N2,O2,CH4 and CO2. Lide (1994, pp. 10–193 to 10–202) also lists polarizabilities of Section 3 showed that m can be found by a single-fre- hundreds of compounds, again including He, Ar, H2,N2, quency radio occultation experiment. Strictly speaking, it O2,CH4 and CO2. Refractive volumes derived from rela- is not possible to determine both me and mn from a single tive permittivities and polarizabilities using Eqs. (16) and measurement of m. However, several properties of me and (20) are the same for these seven species to two significant mn make it possible to do so in practice. As shown in Sec- figures. Values are listed in Table 3. Refractive volumes tions 4 and 5, me is always negative and mn is always positive. may also be determined by laboratory measurements (e.g. Plasma is much more refractive than neutral gas. For typ- Essen and Froome, 1951; Essen, 1953; Orcutt and Cole, ical frequencies, jmej is ten orders of magnitude greater than 1967; Kołos and Wolniewicz, 1967; Tyler and Howard, mn given identical electron and neutral number densities 1969; Bose and Cole, 1970; Bose et al., 1972). (Eqs. (11) and (13)). Consider an isothermal neutral atmosphere, loosely based on the Martian atmosphere, that has 5.3. Values of atmospheric refractivity a surface pressure of 6 mbar, temperature of 200 K, scale height of 8 km and refractive volume of 1029 m3. Its verti- Total refractivity is often represented as the sum of sev- cal profile of mn, which can be found from Eq. (13),is eral factors. The following representation is particularly shown in Fig. 2. Suppose that the ionosphere in this atmo- common for the terrestrial atmosphere (e.g. Kursinski sphere is a Chapman layer with its maximum electron den- 11 et al., 1997; Kursinski, 1997; Tellmann et al., 2009). sity of 2 10 m3 occurring where the product of the p p N m C C W C e ¼ 1 þ 2 2 3 2 ð21Þ T T f 200

The Ci values are coefficients and pW is the partial pressure of water vapour. C1 is j=kB, where kB is Boltzmann’s con- 2 1 150 stant (Eq. (13)). C2 is 0.373 K mbar (Kursinski et al., 2 2 1997). C3 is e =ð8p me0Þ (Eq. (11)). Table 4 lists values of C1 for solar system atmospheres. 100 Altitude (km) Table 3 50 Refractive volumes, j, of common atmospheric gases. Values obtained from permittivities and polarizabilities as discussed in Section 5.2. 0 Species j (m3) 10 -18 10 -16 10 -14 10 -12 10 -10 10 -8 10 -6 10 -4 29 He 1:3 10 Absolute value of refractivity Ar 1:0 1029 H 5:1 1030 2 Fig. 2. The solid line shows m and the dashed line shows m for the N 1:1 1029 n e 2 idealized neutral atmosphere and ionosphere discussed in Section 6. O 1:0 1029 2 Neutral atmospheric refractivity dominates at low altitudes and iono- CH 1:6 1029 4 spheric refractivity dominates at high altitudes, with an intermediate CO 1:8 1029 2 region where the magnitudes of both refractivities are extremely small. 64 P. Withers / Advances in Space Research 46 (2010) 58–73 absorption cross-section, neutral number density and scale radio elements and, if any radio element is on Earth, the eph- height equals unity (Withers, 2009, and references therein). emerides of Earth. Uncertainties in Df as a function of posi- For a cross-section of 1017 cm2, the altitude of the iono- tion result from uncertainties in the ephemerides of the spheric peak is 115 km. Assuming a frequency of planet being observed, uncertainties in Df as a function of 8.4 GHz, its vertical profile of me, which can be found from time, and uncertainties in the trajectories of the radio ele- Eq. (11), is shown in Fig. 2. In the ionosphere, neutral den- ments (Section 3). The latter includes uncertainties in the tra- sities are low and electron densities are relatively high, so mn jectories of all spacecraft carrying radio elements and, if any is very small and me is dominant. At lower altitudes, neutral radio element is on Earth, the ephemerides of Earth. Uncer- densities are relatively high and electron densities are low, tainties in a as a function of position result from uncertain- so jmej is very small and mn is dominant. If the uncertainty in ties in Df as a function of position, the frequency, the 9 refractivity is 10 (Section 13), then mn is experimentally trajectories of the radio elements and c (Section 3). Changes indistinguishable from zero above 62 km and me is experi- in frequency are usually sufficiently small that it is irrelevant mentally indistinguishable from zero below 95 km. There whether the transmitted frequency or received frequency is is a region spanning four scale heights where both mn and used in Eq. (3) to find a. Uncertainties in m as a function of me are experimentally indistinguishable from zero. Similar position result from uncertainties in a and deviations from vertical separations of the contributions of neutral gases spherical symmetry (Section 3). Evaluation of the effects of and ionospheric plasma occur in other solar system deviations from perfect spherical symmetry is complicated atmospheres. and beyond the scope of this paper (Ahmad and Tyler, In practice, the analysis of data from a single-frequency 1999). However, they are not usually considered critical radio occultation uses the following assumptions. If the mea- for most applications. The horizontal lengthscalepffiffiffiffiffiffiffiffi over which sured value of m is negative, then me is assumed to be identical this assumption is significant is on the order of 2 2hR, where to the measured value of m and mn is assumed to be zero; if the h is the vertical lengthscale of the feature of interest, often a measured value of m is positive, then me is assumed to be zero scale height, and R is the planetary radius (Kursinski et al., and mn is assumed to be identical to the measured value of m. 1997; Hinson et al., 1999; Kursinski et al., 2000). Uncertain- In the intermediate altitude region below the detectable ion- ties in N e as a function of position result from uncertainties osphere, but above the detectable neutral atmosphere, where in m, the frequency, e; me and 0 (Section 4). Changes in fre- m is experimentally indistinguishable from zero, both me and quency are usually sufficiently small that it is irrelevant mn are assumed to be zero. Although the two refractivities are whether the transmitted frequency or received frequency is not actually zero, they have comparable, but opposite, val- used in Eq. (11) to find N e. Uncertainties in nn as a function ues in this region. These assumptions are not necessary if m of position result from uncertainties in m and j (Section 5). is measured simultaneously at multiple frequencies. Since Not all of these uncertainties are equally important for me depends on frequency, but mn does not, it is straightfor- accurate determination of vertical profiles of neutral number ward to separate me and mn in such experiments. density and electron density. The most important are typically those that affect knowledge of Df . These can be divided 7. Sources of uncertainties into two types: thermal noise and phase noise. If the received signal, sðtÞ, can be described as (Lipa and Tyler, 1979): 7.1. Background stðÞ¼AtðÞsinðÞþ 2pftðÞt hðÞt ð22Þ The basic quantities measured in a radio occultation experiment are characteristics of the received radio signal where t is time, A is amplitude and f is frequency, then as a function of time. The received frequency is determined phase noise is responsible for variations in f and thermal as a function of time from these characteristics. Uncertain- noise is responsible for h, which is typically assumed to ties in the received frequency during a given time interval be normally distributed with zero mean. Variations in A result from the finite duration of the time interval, thermal with time are not considered in this paper. noise and the accuracy of the reference oscillator used as a A radio occultation experiment is composed of many clock at the receiving radio element. Uncertainties in Df dur- parts. This is illustrated in Fig. 3, which shows a one-way ing a given time interval result from uncertainties in the downlink experiment conducted by Mars Global Surveyor. transmitted frequency, the received frequency, the vacuum Note the reference oscillators on the spacecraft (ultrastable Doppler shift between the two radio elements and frequency oscillator) and on the ground (frequency timing subsys- shifts caused by refraction in plasma in the interplanetary tem), which contribute to the phase noise, and the many medium (Section 2). If any radio element is on Earth, then parts through which the radio signal passes, which contrib- frequency shifts caused by refraction in the terrestrial neutral ute to the thermal noise. atmosphere and ionosphere are also present. Uncertainties in the vacuum Doppler shift, meaning the Doppler shift that 7.2. Thermal noise would have been observed in the absence of refraction in the planetary atmosphere under observation, are affected by The uncertainty in frequency due to thermal noise, uncertainties in the trajectories of all spacecraft carrying rthermal,is(Simon and Yuen, 1983; Lindsey, 1972): P. Withers / Advances in Space Research 46 (2010) 58–73 65 American Geophysical Union. Reproduced by permission of American Geophysical Ó

Fig. 3. Block diagram of a radio occultation experiment, speciﬁcally a one-way downlink experiment conducted at Mars by Mars Global Surveyor. Reproduced from Fig. 7 of Tyler et al. (2001). Fig. 7 of Tyler et al., Radio science observations with Mars Global Surveyor: Orbit insertion through one Mars year in mapping orbit, Journal of Geophysical Research, 106, 22,327–23,348, 2001.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2BN 0=C 7.3. Phase noise rthermal ¼ ð23Þ 2psthermal There are many sources of phase noise, including instabil- where 2B is the noise bandwidth (Hz), N 0 is the noise power ities of all reference oscillators used in the experiment and 1 density (W Hz ), C is the signal power (W) and sthermal is variations in refractive index in the terrestrial atmosphere, the time interval for this measurement (seconds). terrestrial ionosphere and plasma in the interplanetary med- A qualitative justification for this expression is as follows. ium. If an oscillator’s output frequency is extremely stable, The signal amplitude can be represented as A sin / þ h, then it is called an “ultrastable oscillator” or USO. This label where / is phase. An experimenter examining a time series is often applied to space-based oscillators. In a one-way of amplitudes can only be confident that one cycle has ended experiment, instabilities of the reference oscillators at both and another begun when A sin / > h. For small values of the transmitter and receiver contribute to the phase noise. h=A, this leads to an uncertainty in phase at any time of The reference oscillator at the transmitter affects the stability h=A. Since amplitude is proportional to the square root of of the transmitted frequency. The reference oscillator at the power, this is equivalent to thep reciprocalffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the square root receiver affects the accuracy with which the received fre- of the signal-to-noise ratio or 2BN 0=C. The uncertainty in quency can be measured. The received frequency cannot be thepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi number of cycles in this time series is 1=2p times this or measured to one part in a billion if the clock at the receiver ð 2BN 0=CÞ=ð2pÞ. The uncertainty in the frequency of this is only stable to one part in a thousand, for example. In a timep seriesffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is the ratio of this to the duration of the time series two-way coherent experiment, the most stable radio element or ð 2BN 0=CÞ=ð2psthermalÞ, as stated in Eq. (23). can be used as the transmitter. The least stable radio element 5 A representative value for C=N 0 is 50 dB Hz or 10 Hz can receive the signal, then transmit a signal that is coher- (Nagy et al., 2006). A representative value for B is 100 Hz ently related to the received signal. In this case, the stability (Hinson et al., 1998b). A representative value of sthermal is of the signal transmitted by the least stable radio element and 1s (Hinson et al., 1999). From Eq. (23), a representative received at the end of the experiment is unaffected by the fre- value of rthermal is 10 mHz, consistent with the value reported quency stability of the least stable radio element. Despite in Hinson et al. (1999). Thermal noise is produced at all eliminating one source of phase noise, two-way coherent stages of the experiment, including each transmitter and experiments are not always better than one-way experi- receiver. ments. The time required to establish the two-way coherent 66 P. Withers / Advances in Space Research 46 (2010) 58–73 link at the onset of an egress occultation results in loss of The phase noise in the complete radio occultation exper- data at low altitudes. Two-way coherent links are also diffi- iment is characterized by an Allan Deviation. In addition, cult to maintain when the signal passes through a dense the contribution to the phase noise from an individual com- atmosphere, such as the atmosphere of Venus or Jupiter, ponent within the complete experiment can be characterized and its amplitude varies dynamically. by an Allan Deviation. Typical values of Allan Deviations Refraction along the ray path from regions other than the for space-based oscillators on the 1–1000 s timescales that atmosphere of the planet under observation are sources of are relevant for radio occultations are on the order of phase noise. Refraction due to plasma in the interplanetary 1013 (Tyler et al., 2001; Kliore et al., 2004). Equivalent val- medium is always a source of phase noise. For a radio occul- ues for ground-based oscillators at the NASA Deep Space tation in which one radio element is ground-based, refrac- Network are on the order of 1014 to 1015 (Dick and Wang, tion in the terrestrial neutral atmosphere and ionosphere 1991; Howard et al., 1992; Thornton and Border, 2003; also contributes to the phase noise. A baseline correction Asmar et al., 2004). for these effects can be made using signals recorded when In a typical two-way coherent radio occultation experi- the ray path is far from being occulted by the planet under ment, the signal is unaffected by the Allan Deviation of the observation and is not refracted at all by its atmosphere, space-based oscillator. The Allan Deviation of the complete but the actual contribution of these effects will inevitably radio occultation experiment must be no smaller than the vary somewhat about the baseline. Spatial variations in the Allan Deviation of each reference oscillator that is used in terrestrial neutral atmosphere and ionosphere are not typi- the two radio elements. In practice, rAD exceeds this lower cally a concern, since the location in the sky of the space- limit due to the other sources of phase noise, particularly based radio element does not change significantly over the variations in interplanetary plasma density along the ray duration of the experiment, typically on the order of minutes. path for single-frequency radio occultation experiments. Turbulence in interplanetary plasma (that is, the solar corona and solar wind) is often the most significant source 7.4. Implications of undesirable refraction (Sniffin, 2001). Large interplanetary plasma densities are not in themselves a major prob- The duration of radio occultation experiments is typi- lem as the effects of homogeneous plasma can be cally set so that some of the measurements are made when removed by baseline corrections. Variations in the column the ray path is far above the atmosphere of the planet being density of interplanetary plasma along the ray path due to observed and unaffected by refraction in this atmosphere. true temporal variations or motion of the ray path through These unaffected measurements provide a baseline from spatially varying regions of plasma can produce phase which thermal noise, phase noise and, for a one-way down- noise. In addition, plasma inhomogeneities on a scale small link experiment, transmitted frequency can be determined. by comparison with the signal’s Fresnel zone can produce The frequency produced by a space-based oscillator drifts phase noise as the assumptions of geometrical optics are with time, so its frequency is not identical to the frequency violated. The relevant length scale is the square root of determined during pre-flight testing. the product of the wavelength and the distance between The uncertainty in Df ; rDf , depends on rthermal and rADf the inhomogeneous region and the closest radio element. in the usual manner: This length scale is on the order of 100 km for wavelengths 2 2 2 2 of a few centimetres and a separation of 1 AU. Variations rDf ¼ rthermal þ rADf ð26Þ in antenna pointing can also be a source of phase noise. 13 The phase noise is characterized by the Allan Deviation, If rAD ¼ 3 10 and f ¼ 8:4 GHz, then rADf equals 3 mHz, less than the representative value of rthermal of rAD, which is defined as (Allan, 1966): ÀÁ 10 mHz. Uncertainties in radio occultation experiments are 1 2 r2 ¼ y y ð24Þ not solely determined by the Allan Deviations of reference AD 2 nþ1 n oscillators. For example, the uncertainties in radio occulta- where angle brackets indicate the expectation value of the tion experiments performed by the Galileo orbiter signifi- enclosed quantity, y is the value of y from the nth sample n cantly exceeded those anticipated during mission design period and y is given by: because the signal power was lower than planned. This oc- fmeasured fnominal curred despite the excellent in-flight Allan Deviation of the y ¼ ð25Þ fnominal onboard oscillator. Signals had to be transmitted through Galileo’s low gain antenna because the high gain antenna The frequency fmeasured is the measured frequency over a failed to deploy (Howard et al., 1992; Hinson et al., 1997). sample period or integration time of duration sphase.If fmeasured comes from a normal distribution with mean fnominal, then rADfnominal equals the standard deviation of 8. Velocity uncertainties for gravitational studies the normal distribution. However, although rADfnominal is often treated as if it were a standard deviation, it is not for- If radio tracking is used to measure the speed of one mally identical to a standard deviation. The Allan Devia- radio element with respect to the other radio element, as tion is a function of the integration time, sphase. in gravitational studies (e.g. Rappaport et al., 1997), then P. Withers / Advances in Space Research 46 (2010) 58–73 67 the uncertainty in speed, rV , is simply the product of c=f For neutral pressures less than hundreds of bars and for and the uncertainty in frequency shift. This is valid for a all planetary ionospheres, jmjj1 as shown in Sections 4 one-way experiment. For a two-way coherent experiment, and 5. Hence rj can be replaced by aj (Eq. (9)) and lr c=f is replaced by c=2f because the signal is Doppler- can be replaced by r (Eq. (5)) to give: shifted on both legs of its journey. Z ÀÁ r¼1 2a m r aj dr The observed value of rDf is typically smaller during a j j aj ¼ exp qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð31Þ gravity tracking flyby than during an occultation because H r¼a H 2 2 j r aj the timescale for frequency changes is much longer during a gravity tracking flyby than during an occultation (e.g. The solution of Eq. (31) is (Arfken and Weber, 1995, Eq. Rappaport et al., 1997; Hinson et al., 1999). This permits (11.122)): longer s , which reduces r , and longer s . The thermal thermal phase 2a m a a Allan Deviations of reference oscillators generally decrease a ¼ j j exp j K j ð32Þ j H H 0 H as the integration time sphase increases from seconds to thousands of seconds. (Howard et al., 1992; Tyler et al., where K0ðaj=HÞ is the modified Bessel function of the sec- 2001; Kliore et al., 2004). For example, the Allan Deviation ond kind of order 0. Since aj is on the order of the plane- of the oscillator on the Galileo Orbiter was 11 12 12 12 tary radius, aj=H 1. In this limit (Arfken and Weber, 3 10 ; 4 10 ; 1 10 and 1 10 for integra- 1995, Eq. (11.127)): tion times of 1, 10, 100 and 1000 s (Howard et al., 1992). sffiffiffiffiffiffiffi For a one-way experiment in which thermal noise is dom- aj pH aj inant, rV satisfies: K0 ! exp ð33Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 2aj H c 2BN =C r ¼ 0 ð27Þ V 2pf s Therefore: thermal rffiffiffiffiffiffiffiffiffi This value is halved for a two-way coherent experiment 2paj aj ¼ mj ð34Þ (Pa¨tzold et al., 2004). Hrffiffiffiffiffiffiffiffi 2pa aðÞ¼a mðrÞ ð35Þ 9. Relationship between bending angle and refractivity H A similar result is stated in Eshleman et al. (1979) and In order to relate uncertainties in N e and n to uncertainties in Df , we must simplify the Abel transform relation- Yakovlev (2002). As long as the refractivity increases expo- ship between the bending angle, a, and the refractivity, m nentially as altitude decreases, the bending angle also in- (Eq. (4)). creases exponentially as altitude decreases. The bending angle, a, is a function of the shape of the From Eqs. (3) and (34), the expected frequency shift, vertical refractivity profile. For neutral atmospheres or Df , satisfies: the topsides of ionospheres above their main peaks, it is rffiffiffiffiffiffiffiffi fV mðrÞ 2pa often reasonable to assume that refractivity, m, which is DfaðÞ ð36Þ proportional to electron density or neutral number density, c H decreases exponentially with increasing altitude. An iono- Recall that this is an approximation since the assumed rela- sphere controlled by diffusion will have an exponential ver- tionship between frequency shift and velocity is approxi- tical profile, as will the topside of a Chapman-like mate. In particular, Eq. (3) is only a realistic ionosphere (Rishbeth and Garriott, 1969; Schunk and approximation for small bending angles. It is important Nagy, 2000; Withers, 2009). to note that this relationship should not be used to derive ÀÁ refractivity profiles from measured frequency shifts. Eq. r rj mðrÞ¼m exp ð28Þ j H (36) remains approximately true if the assumption of an exponential dependence of refractivity on altitude is inva- where mj is the refractivity at radius rj and H is a scale lid, as long as mðrÞ decreases with increasing altitude with height. Using Eq. (5) that defines m as l 1; d ln l=dr characteristic lengthscale H. satisfies: ÀÁ 10. Expressions for uncertainties in electron density and d ln l m r rj ¼ j exp ð29Þ neutral number density dr H H Eq. (4) becomes: The vertical extent of most planetary atmospheres and Z ÀÁ ionospheres is much smaller than the planetary radius, R, r¼1 2ajmj r rj dr although Titan and comets are exceptions. Hence we sub- aj ¼ exp qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð30Þ 2 stitute R for a in Eq. (36). From Eq. (36), the uncertainty H r¼rj H 2 ðÞlðrÞr aj in refractivity, rm, is related to rDf via: 68 P. Withers / Advances in Space Research 46 (2010) 58–73 rffiffiffiffiffiffiffiffiffi cr H absorption reduce the signal strength, but critical refrac- r Df ð37Þ m Vf 2pR tion can prevent the radio signal from leaving the atmosphere at all. Critical refraction occurs when the radius of If the refractivity from Eq. (11) is equated to rm in Eq. (37), curvature of the ray path is smaller than the radius of cur- and the resultant equation is rearranged, then the uncer- vature of a surface of uniform refractivity (Kursinski et al., tainty in electron density, rNe, satisfies: rffiffiffiffiffiffiffiffiffiffiffi 1997). The ray path curves down towards the planetary interior, never propagating out of the atmosphere to the 4prDf fcme0 2pH p rNe ð38Þ receiver. Critical refraction occurs when dl=dr þ l=r ¼ 0 Ve2 R or, when Eq. (28) is valid, when mðr HÞ=H ¼ 1(Karayel where H p is the scale height of ionospheric plasma, which is and Hinson, 1997; Yakovlev, 2002). Defocusing, absorp- not necessarily identical to the neutral scale height at iono- tion and critical refraction are most significant in the deep spheric altitudes. If the refractivity from Eq. (13) is equated atmospheres of Venus and the giant planets. to rm in Eq. (37), and the resultant equation is rearranged, Multipath propagation occurs when the receiver simul- then the uncertainty in neutral number density, rnn, taneously receives radio signals that have taken different satisfies: paths through the planetary atmosphere. Formally, a a rffiffiffiffiffiffiffiffiffi ð Þ is multi-valued and ray crossing occurs. In this case, the crDf H n rnn ð39Þ frequency shift, Df , observed at a given time cannot be Vf j 2pR unambiguously assigned to one closest approach distance, where H n is the scale height of the neutral atmosphere. We r, since the multiple radio signals have different tangent do not consider errors in pressure and temperature in this heights. The geometrical optics treatment of the radio paper, which are derived from a vertical integration of occultation fails. Multipath propagation occurs when “an the neutral number density, except to note that their rela- interval of high refractivity gradient overlies a region of tive errors decrease as altitude decreases (Hinson et al., lower gradient” (Kursinski et al., 1997). This is a particular 1999). problem around thin plasma layers on the bottomsides of Again, it is important to note that these relationships giant planet ionospheres (Hinson et al., 1998b). should not be used to derive neutral number density or ionospheric electron density profiles from measured frequency 12. Discussion of results shifts. We assert without proof that these relationships pffiffiffiffiffiffiffiffiffiffi remain valid when the refractivity does not vary exponen- Both rNe and rnn are proportional to rDf =V H=R, tially with altitude, such as around an ionospheric peak. where H ¼ H p for the ionosphere and H ¼ H n for the neu- Published uncertainties are only weakly dependent on alti- tral atmosphere. The value of rNe is also proportional to f. tude (Hinson, 2007; Tyler et al., 2007), which supports this The value of rnn is also inversely proportional to f j. Values assertion. of rDf ; V and f can be affected byp theffiffiffiffiffiffiffiffiffiffiffiffi designpffiffiffiffiffiffiffiffiffiffiffiffi of the radio occultation experiment, whereas H p=R, H n=R and j 11. Effects of defocusing, critical refraction and multipath are planetary properties that are beyond the experimenter’s propagation control. The uncertainty in Df depends on many variables, The preceding sections have discussed somewhat ideal- including integration time, bandwidth, signal power, noise ized radio occultation experiments. The phenomena of power density, oscillator stability and fluctuations in inter- defocusing, critical refraction and multipath propagation planetary plasma. Integration time can be increased, but at can complicate real radio occultation experiments. These the cost of reduced vertical resolution. The smallest vertical are summarized here. resolution consistent with geometrical opticspffiffiffiffiffiffiffiffi is the Fresnel Refraction can cause near-parallel rays to diverge, which zone diameter, which is on the order of k=L where k is reduces the signal intensity. This is called defocusing the wavelength and L is the distance between the closest (Kursinski et al., 1997; Haüsler et al., 2006). The ratio of radio element and the tangent point of the ray path (Kur- the power of the radio beam after exiting the atmosphere sinski et al., 1997; Hinson et al., 1999; Kursinski et al., to its power before entering the atmosphere is 2000). If the received radio signal is recorded in a closed ð1 Lda=daÞ1, where L is the distance between the point loop system, then the bandwidth is pre-determined before of closest approach to the planet and the closest radio ele- the experiment and must be sufficiently wide to allow for ment (Karayel and Hinson, 1997; Kursinski et al., 2000). differences between the predicted and actual frequency Using Eq. (34) and a=H 1, this ratio is ð1 þ La=HÞ1 shifts. If the received radio signal is recorded in an open (Eshleman et al., 1980). Section 10 quantifies how the loop system, then the bandwidth can be reduced during uncertainties in measured atmospheric properties depend post-processing. The bandwidth must be greater than on the strength of the radio signal. The strength of the the thermal noise and the phase noise. The experimenter radio signal can also be reduced by the absorption of the can influence the signal power by choosing which of the radio signal by atmospheric gases, such as ammonia (Lin- available radio elements act as transmitters. Ground- dal et al., 1981; Howard et al., 1992). Defocusing and based transmitters have much greater signal power than P. Withers / Advances in Space Research 46 (2010) 58–73 69 space-based transmitters. The signal power of a given radio cations than lower frequencies, and are less affected by element is usually determined by the mission’s communica- plasma in the interplanetary medium and terrestrial iono- tion requirements, which are beyond the control of the sphere, so planetary spacecraft are migrating towards experimenter. The noise power density can be minimized Ka-band for communications systems. This transition from by careful hardware design. Uncertainties from oscillator the long-established S-band to Ka-band has both positive instabilities can be reduced by using a two-way experiment and negative consequences for ionospheric occultations. instead of a one-way experiment if the net effect is to Multipath effects, which more pronounced at low frequen- replace a less stable oscillator with a more stable oscillator. cies, are reduced (e.g. Hinson et al., 1998b; Nagy et al., The Allan Deviation of oscillators on planetary spacecraft 2006). If uncertainties are dominated by phase (thermal) has steadily improved over time with values at 1 s integra- noise, then they are two (one) orders of magnitude smaller tion times of 1 1010 on Mariner 4, 5 1012 on Voyager, for S-band than for Ka-band. However, this is mitigated and 2 1013 for Mars Global Surveyor, Cassini and for multi-frequency (X-band and Ka-band) occultations Venus Express (Kliore et al., 1970; Eshleman et al., 1977; by the excellent characterization of noise due to interplan- Tyler et al., 2001; Kliore et al., 2004; Haüsler et al., etarypffiffiffiffiffiffiffiffiffiffiffiffi plasma that Ka-band provides. 2006). The typical Allan Deviation of ground-based oscilla- H n=R is closely related to an important metric for tors at the present day is on the order of 1014 to 1015 at atmospheric escape, the ratio of a molecule’s gravitational integration times of 1s–1000 s (Dick and Wang, 1991; potential energy to kinetic energy at the exobase. This ratio Howard et al., 1992; Thornton and Border, 2003; Asmar equals the ratio of the radius of the exobasepffiffiffiffiffiffiffiffiffiffiffiffi to the neutral et al., 2004). scale heightp atffiffiffiffiffiffiffiffiffiffiffiffi the exobasep (Strobel,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2002). H p=R is not as The effects of fluctuations of interplanetary plasma can significant. H n=R equals kBTR=GMm, where kB is Boltz- be minimized by reducing the path length and increasing mann’s constant, T is temperature, R is planetary radius, the distance of closest approach of the ray path to the GM is the product of the gravitational constant and the Sun. This is a major problem for spacecraft-Earth occulta- mass of the body,pffiffiffiffiffiffiffiffiffiffiffiffi and m is mean molecular mass. Solar sys- tions that occur near solar conjunction when the ray path tem values of H n=R vary by almost one order of magni- between the spacecraft and Earth passes very close to the tude, as shown in Table 5. Refractive volume, j, is typically Sun. Useful measurements of planetary atmospheres are within a factor of two of 1029 m3, as shown in Table 3. rarely obtained near solar conjunction. At these times, fluc- Therefore variations in j from planet to planet only modify tuations in plasma density in the solar corona may be stud- rnn by a factor of two. ied from their effects on radio signals (e.g. Howard et al., 1992; Bird et al., 1992, 1996). As noted in Section 7, 13. Case studies multi-frequency measurements can effectively separate refraction in plasma and neutral gases. This technique To provide a concrete illustration of the results of the can be used to eliminate the uncertainties caused by fluctu- preceding sections, we now estimate uncertainties in iono- ations in the interplanetary medium from measurements of spheric electron density and neutral number density for neutral atmospheric number densities. hypothetical radio occultations at Venus, Mars, Jupiter Speed, V, is effectively determined by the top-level mis- and Titan. Bending angle and Df at the surface or 1 bar sion implementation design, which is usually beyond the level are also estimated. Results are listed in Table 6. control of the radio occultation experimenter. That is, Parameters adopted for case studies at Venus, Mars, Jupi- whether the mission consists of single or multiple space- ter and Titan were influenced by the Venus Express, Mars craft and whether each spacecraft is a flyby spacecraft, Global Surveyor, Voyager and Cassini missions, respec- planetary orbiter, or satellite orbiter. For a single spacecraft mission, possible values of V range between about 1.5 km s1 for a Titan orbiter and 25 km s1 for a fast Table 5 pffiffiffiffiffiffiffiffiffiffiffiffi Values of H n=R for solar system atmospheres. Values were obtained as outer solar system flyby such as New Horizons. Such high discussed in Section 12 using data from Lodders and Fegley (1998) and 1 speeds are rare, and V ¼ 5kms is a reasonable default Strobel (2002). Surface conditions were used for solid planets and value. satellites; conditions at the equatorial 1-bar level were used for fluid planets. Frequency, f, can be selected by the radio occultation pffiffiffiffiffiffiffiffiffiffiffiffi experimenter, subject to the availability of suitable trans- Object H n=R mitting and receiving equipment in the two radio elements. Venus 0.05 Only S-band, which at downlink is typically 2.3 GHz or Earth 0.04 13 cm, was available to the first radio occultation experi- Mars 0.05 ments in the 1960s (Kliore et al., 1970). X-band, which at Jupiter 0.02 Saturn 0.09 downlink is typically 8.4 GHz or 3.6 cm, became available Titan 0.09 in the 1970s (Eshleman et al., 1977). Ka-band, which at Uranus 0.03 downlink is typically 32 GHz or 0.9 cm, has been available Neptune 0.03 to recent missions (Kliore et al., 2004). Higher frequencies Triton 0.10 offer more bandwidth and higher data rates for communi- Pluto 0.13 70 P. Withers / Advances in Space Research 46 (2010) 58–73

Table 6 sity of 3 1020 m3 or 1 order of magnitude greater than Representative parameters and resultant uncertainties in electron and estimated here. Since pressure depends on the upper neutral number densities for radio occultations at Venus, Mars, Jupiter boundary condition used to integrate the equation of and Titan. hydrostatic equilibrium, it is likely that the smallest pres- Venus Mars Jupiter Titan sure reported corresponds to an altitude one or two scale rDf (mHz) 10 10 10 10 g k o s heights below the altitude of the smallest detected neutral f (GHz) 8.4 8.4 2.3 2.3 number density. This would improve the accuracy of the V (km s1) 7.0h 3.4l 14p 5.6t i m q u predicted uncertainty. H n (km) 7 10 25 20 j n r v 9 3 H p (km) 10 25 1000 130 The estimated uncertainties at Mars are 6:3 10 m R (km)a 6050 3400 70,000 2575 and 1:3 1020 m3 for electron density and neutral number j (m3)b 1:8 1029 1:8 1029 6:2 1030 1:1 1029 c 2 4 2 2 density, respectively. Mean values of Mars Global Sur- a (rad) 2:7 10 1:8 10 3:6 10 3:6 10 9 3 d 3 1 3 3 veyor’s uncertainties are 4:6 10 m for electron densi- Df (Hz) 5:4 10 1:7 10 3:9 10 1:5 10 20 3 3 e 9 9 8 9 ties and 1:9 10 m for neutral number densities rNe (m ) 1:5 10 6:3 10 5:8 10 2:7 10 3 f 19 20 20 20 rnn (m ) 3:8 10 1:3 10 1:1 10 7:4 10 (Hinson, 2007; Tyler et al., 2007; Withers et al., 2008). a Lodders and Fegley (1998). The estimated uncertainties are within 50% of the mean b Values of j found using j ¼ kBC1 and C1 from Table 4. observed uncertainties. The estimated value of Df at the c Bending angle at the surface for Mars and Titan and at the 1 bar level surface, 17 Hz, is several times greater than the 6 Hz value for Venus and Jupiter. Values found from Eq. (34) using data from this reported by Hinson et al. (1999) for one particular occulta- Table and atmospheric pressure and temperature from Lodders and tion, but is the same order of magnitude. This difference Fegley (1998). d Frequency shift at the surface for Mars and Titan and at the 1 bar level could be caused by the assumption in Eq. (3) that the fre- for Venus and Jupiter. Values found from Eq. (3) using data from this quency shift is proportional to spacecraft speed, when it is Table. more accurate to assume that the frequency shift is propor- e Values of rNe are calculated using Eq. (38). f tional to the component of spacecraft velocity along the Values of rnn are calculated using Eq. (39). Earth-spacecraft line. This component is less than the g X-band, Pa¨tzold et al. (2009). h Intermediate value for Venus Express orbit, also representative of speed. Note that this should also reduce V in Eqs. (38) 600 km circular orbit. and (39), which will increase rNe and rnn. i 8 3 H n at 1 bar level, value from Tellmann et al. (2009). The estimated uncertainties at Jupiter are 5:8 10 m j Pa¨tzold et al. (2009). and 1:1 1020 m3 for electron density and neutral number k X-band, Tyler et al. (2001). l density, respectively. The uncertainty in Voyager electron Circular orbit around Mars at 400 km altitude. 9 3 m density observations is on the order of 10 m (Hinson H n at the surface, value from Lodders and Fegley (1998). n Hanson et al. (1977). et al., 1998b), consistent with the estimate. However, the o S-band, Hinson et al. (1998b). smallest neutral number density reported during the Voy- p Circular orbit around Jupiter at the orbital distance of Europa. ager 1 egress radio occultation is between 1022 m3 and q H n at the 1 bar level, value from Lodders and Fegley (1998). 1023 m3 (Lindal et al., 1981). This difference of 2–3 orders r Hinson et al. (1998b). s S-band, Nagy et al. (2006). of magnitude is too large to be accounted for by inappro- t Circular orbit around Saturn at the orbital distance of Titan. priate choices for V ; f and H. It is possible that the value of u H n at the surface, value from Lodders and Fegley (1998). rDf appropriate for the neutral atmospheric analysis of v Kliore et al. (2008). Lindal et al. (1981) is much greater than 10 mHz, yet 10 mHz seems consistent with the later ionospheric analysis of Hinson et al. (1998b). It is possible that different band- tively. However, thermal noise properties for Venus widths (2B) and integration times (sthermal) were used by Express, Voyager and Cassini have not been reported in Lindal et al. (1981) and Hinson et al. (1998b), which would the literature. Papers on radio occultation observations affect rDf in accordance with Eq. (23), but resolution of this usually state rAD, but rarely provide information sufficient discrepancy is beyond the scope of this paper. 9 3 to calculate rthermal from Eq. (23). Hence the same value of The estimated uncertainties at Titan are 2:7 10 m and 20 3 rDf , 10 mHz, was assumed in all four case studies based on 7:4 10 m for electron density and neutral number den- the Mars Global Surveyor results of Hinson et al. (1999). sity, respectively. Uncertainties in electron density from Cas- This should be considered when the accuracies of the case sini observations are 108 m3 (Kliore et al., 2008), much studies are evaluated. smaller than estimated here. Neutral atmospheric profiles The estimated uncertainties at Venus are 1:5 109 m3 from Cassini have not yet been published (Flasar et al., and 3:8 1019 m3 for electron density and neutral number 2007). Since Kliore et al. (2008) noted that Df at the iono- 9 3 density, respectively. Pa¨tzold et al. (2009) reported that spheric peak (N e 10 m ) at X-band is only 1 mHz, it is uncertainties in electron densities measured by Venus clear that Cassini’s value of rDf is significantly smaller than Express were 2 109 m3, very similar to those estimated the 10 mHz assumed in Table 6. Yet rearrangement of Eq. here. The smallest pressures reported by Tellmann et al. (38), use of most of the Titan-specific parameters from Table (2009) from Venus Express data were 1 Pa. For a temper- 6, replacement of f ¼ 2:3 GHz (S-band) with f ¼ 8:4 GHz ature of 200 K, this corresponds to a neutral number den- (X-band), and assumption of an electron density of P. Withers / Advances in Space Research 46 (2010) 58–73 71

109 m3 (Kliore et al., 2008) yields a predicted Df of The simple expressions developed here can be used for 1 mHz—exactly the value quoted by Kliore et al. (2008). preliminary design studies of future radio occultation Thus the relationships derived in this work appear consistent experiments. Obviously, more realistic simulations should with Kliore et al. (2008), even if the estimated uncertainties be used for detailed design studies. in Table 6 are too large due to incorrect assumptions about rDf . It is not surprising that Cassini has a much smaller Acknowledgements uncertainty in Df than Mars Global Surveyor, since Cassini carries the most sophisticated planetary radio science inves- PW acknowledges productive discussions with Kerri Ca- tigation yet flown, including the first use of Ka-band signals hoy, Martin Pa¨tzold, the radio science teams of Mars Ex- in planetary radio occultations and averaging of results press and Venus Express, and many colleagues involved obtained using multiple frequencies and multiple ground in the Radio Science Experiment, led by Michael Mendillo, stations (Kliore et al., 2004; Nagy et al., 2006). of “The Great Escape” mission that underwent a Phase A study for NASA’s Mars Scout program. PW also thanks 14. Conclusions Bernd Haüsler for a pre-publication copy of “Venus atmospheric, ionospheric, surface, and interplanetary radio We have derived theoretical relationships for uncertain- wave propagation studies with the Venus Express radio sci- ties in ionospheric electron density and neutral number ence experiment VeRa”, which is a VeRa instrument paper density in radio occultation experiments. Uncertaintiespffiffiffiffiffiffiffiffiffiffi in submitted for publication in a forthcoming ESA Special both densities are proportional to rDf =V H=R, where Publication on the Venus Express mission and payload. H ¼ H p for the ionosphere and H ¼ H n for the neutral atmosphere. Uncertainty in electron density is also propor- References tional to f. Uncertainty in neutral number density is also inversely proportional to f j. The value of rDf depends Ahmad, B., Tyler, G.L. The two-dimensional resolution kernel associated on thermal noise and phase noise. 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