3 MHz space antenna

ABSTRACT

Little is known about the radio astronomical universe at frequencies below 10 MHz because such radiation does not penetrate the ionosphere. A Cubesat-based antenna for the 1–10 MHz band could be rapidly and economically deployed in low Earth orbit. When shielded by the Earth from Solar emission, it could observe weak extra-Solar System and outer Solar System sources. We suggest possibly observable sources, and application to study of the topside ionosphere.

instrumentation: miscellaneous, MF/HF orbital antenna, topside ionosphere, Lloyd's mirror

1 INTRODUCTION

Low frequency (⁠|$\lesssim 25\,$| MHz) radio astronomy has been comparatively little studied because it is difficult to observe these frequencies from the ground. Science motivations for its study include very high redshift cosmology, low frequency sky surveys, solar/space weather, possible transient emissions from several classes of astronomical sources including fast radio bursts (FRB), soft gamma repeaters (SGR), gamma-ray bursts (GRB) and pulsars (PSR), and perhaps serendipitous discoveries.

The ionosphere reflects, at normal incidence, radiation at frequencies below a varying cutoff of 5–10 MHz, and at higher frequencies at grazing angles. Scintillation may be prohibitively strong even at frequencies at which the waves propagate. These values depend on the Solar cycle and activity, season and time of day, but largely preclude ground-based astronomy below 10 MHz. The lowest frequency successful ground-based observations appear to have been those of Bridle & Purton (1968) at 10.03 MHz, of Caswell (1976) at 10 MHz, of Cane (1979) at 5.2 MHz and of Ellis (1962) at 4.8 MHz, although some results at lower frequencies have been reported (Reber & Ellis 1956; Ellis 1957, 1965; Getmantsev et al. 1969). The abandonment of such low frequency ground-based observations more than 40 yr ago reflected a consensus that they are infeasible or not scientifically promising.

The ionosphere is not an obstacle to space-based observation. In fact, its existence is advantageous for observations at low frequencies because it shields space-based instruments from terrestrial electromagnetic interference while reflecting the sky. Bassett et al. (2020) observed extensive terrestrial interference above the ionosphere at frequencies 5–10 MHz, but very little at 3 MHz where the ionosphere is an effective shield. A few receivers of low frequency radiation have been flown: Alouette (Hartz 1964) in low Earth orbit (LEO), at a similar altitude to that proposed here but in a high-inclination rather than equatorial orbit, Radio Astronomy Explorer-1 (RAE-1) in a 5850 km orbit, higher than the LEO orbit contemplated here (Alexander et al. 1969; Weber, Alexander & Stone 1971) and Radio Astronomy Explorer-2 (RAE-2) in Lunar orbit (Alexander et al. 1975). The prospects of space-based low frequency radio astronomy were reviewed in a conference (Kassim & Weiler 1990). All observations of emission from outside the Solar System must contend with absorption and emission by interstellar plasma (Cong et al. 2021), and better characterization of this environment would be one of the scientific goals of such observations.

Several space instruments have observed Solar emissions at these low frequencies, including WAVES on the deep space Wind and STEREO spacecraft (Kaiser 2005; WAVES 2020; STEREO 2022). The Sun Radio Interferometer Space Experiment (SunRISE) (Kasper et al. 2019, 2022) is planned for geosynchronous orbit. Solar emission, including Type III bursts, might be a source of confusion or interference for observations of extra-Solar System sources. There can be no such confusion or interference when an antenna is shielded from the Sun by the Earth, which occurs for satellites in LEO with a duty cycle of about 40 per cent, but never, or rarely, in deep space or geosynchronous orbits (GEO).

After a long period of somnolence, technical and scientific progress argue for reviving space-based low frequency radio astronomy. The technical progress consists of the development of ‘Cubesats’, satellites consisting of one or more 10 cm cubes (Shkolnik 2018) and of microelectronics capable of sophisticated on-board data analysis within a small spatial envelope and with minimal power. Cubesats are simple enough that they are built as student projects. Launch into LEO may be free, piggybacking on other launches. The scientific progress includes the discovery (Lorimer et al. 2007) of FRB with durations |${\cal O} (\text{1 ms})$|⁠; the study of transients is a rapidly developing branch of radio astronomy. The behaviour of steady extra-Solar System radio sources at frequencies |$\lesssim 10\,$| MHz is also unknown; strong steady sources might be localized by a dipole antenna in LEO by occultation by the Earth’s limb, or by the Moon for favourably located sources (Andrew, Branson & Wills 1964).

Uncharacterized but spatially smooth deviations of the ionosphere from spherical symmetry might slightly limit the accuracy of source localization by this method, but would not greatly affect the strength of the reflected glint and of the peak of the autocorrelation. This peak would still distinguish a transient or rapidly varying source from the background of steady sources as well as from terrestrial interference, that may be significant even for low frequency exo-ionospheric observation of steady sources (Alexander et al. 1975).

Rajan et al. (2016) proposed a deep-space antenna array for low frequency radio astronomy, Sundkvist et al. (2016) proposed the CUbesat Radio Interferometry Experiment (CURIE) to observe Solar radio bursts, Yan, Wu, Gurvits et al. (2023) described the Low Frequency Interferometer and Spectrometer (LFIS) in Lunar orbit and Bentum et al. (2020) proposed the OLFAR (Orbiting Low Frequency Antennas for Radio Astronomy) system involving hundreds or thousands of satellites, linked to synthesize a large number of apertures. Burns (2020) proposed a Lunar-orbiting DAPPER and Lunar-surface based FARSIDE for low frequency observations, and Chen et al. (2020) proposed a low frequency interferometer array in Lunar orbit. Li et al. (2021) placed a lander, including a low frequency radio spectrometer, on the far side of the Moon and the necessary data relay satellite in orbit. These or similar projects offer the prospect of great scientific return some time in the future, but at high cost.

We propose a modest instrument that might, at less cost and sooner, perform a preliminary survey of |$\lesssim 10\,$| MHz radio astronomy. It would be based on a Cubesat in LEO with two centre-fed orthogonal half-wave dipole antennae at a nominal frequency of 3 MHz |$\lambda = 100\,$|m) these would be extended to lengths |$L = \lambda /4 = 25\,$| m by centrifugal force in each of four coplanar orthogonal directions. This nominal frequency is suggested because it is below the frequencies used by short-wave radio (Wikipedia 2022a) and likely also by over-the-horizon radar (Wikipedia 2022b); cf. the signals observed by RAE-2 in Lunar orbit (Alexander et al. 1975).

The orbital altitude is chosen above the peak of ionospheric electron density, where this density is low enough that it does not preclude transmission of extra-terrestrial radiation at the frequency of observation. A plasma frequency of 3 MHz (electron density |$n_e = 1.15 \times 10^5\,$| cm|$^{-3}$|⁠) in the topside ionosphere typically occurs at altitudes of 600–800 km (the peak electron density is several times higher and occurs at |$\approx$| 300–400 km) (International Reference Ionosphere 2022), depending on latitude, longitude, season, phase in the Solar cycle, time of day, and Solar activity. It is desirable to be above this critical altitude most of the time, so a nominal altitude |$h = 1000\,$| km is assumed. Refraction by the ionospheric plasma is significant at that altitude, but does not degrade the signal received by an antenna with little or no angular resolution, such as the dipole antennae considered.

This paper presents a theoretical analysis of the capabilities of the proposed instrument, based on the laws of physics, known properties of the ionosphere and possible but hypothetical extrapolated properties of astronomical radio sources. It does not contain the engineering design that would be required for a funding proposal. Most attention is paid to the instrument and its prospect of observing steady sources, such as supernova remnants, Galactic electron cosmic rays and active galactic nuclei. Interstellar and intergalactic dispersion and scatter broadening may preclude the detection of transients at frequencies |$\lesssim 10\,$| MHz, as discussed in an Appendix.

2 THE ANTENNA

A minimal system is shown in Fig. 1. Two orthogonal centre-fed half-wave (at the nominal frequency of 3 MHz) dipole antennae are extended from a Cubesat that contains amplifiers, data handling and storage electronics, and a higher frequency antenna for transmitting data to a ground station. Power is provided by Solar cells on the Cubesat. Larger telescopes with multiple half-wave antennae (separated by distances |$\sim \lambda /4$| to minimize capacitive coupling) could provide some angular resolution. The configuration is similar to that of Alouette (Hartz 1964), but the construction and deployment made compatible with a Cubesat.

$Sketch of orbiting exoatmospheric radio telescope for observations at 1–10 MHz (numerical values in text apply for $\nu = 3\,$ MHz). Knudsen cell thrusters set the structure rotating, extending the wire antennae. The thrusters are aligned tangentially by ties to insulating diagonal aramid fibres, tightened by centrifugal force but that have no electromagnetic effects.$

Figure 1.

Sketch of orbiting exoatmospheric radio telescope for observations at 1–10 MHz (numerical values in text apply for |$\nu = 3\,$| MHz). Knudsen cell thrusters set the structure rotating, extending the wire antennae. The thrusters are aligned tangentially by ties to insulating diagonal aramid fibres, tightened by centrifugal force but that have no electromagnetic effects.

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2.1 Parameters

A minimum antenna wire radius is set by the requirement that resistive losses in the wire be small compared to its radiation resistance. For a resistance |$\Omega$| in an aluminum wire of length |$\lambda /2$| the wire radius

$$\begin{eqnarray} r = \sqrt{\lambda /2 \over \pi \sigma _{\rm Al}\Omega } \approx 0.21 \sqrt{{\text{10 Ohm} \over \Omega }{\lambda \over \text{50 m}}}\ \text{mm}, \end{eqnarray}$$

(1)

where the conductivity of aluminum |$\sigma _{\rm Al} = 3.6 \times 10^7\,$| mho m⁻¹ (⁠|$\text{1 mho} = 1/\text{ohm}$|⁠). This corresponds to 26 AWG (American Wire Gauge). The mass of the two half-wave antennae is modest:

$$\begin{eqnarray} M &=& 2\pi r^2 \rho _{\rm Al} \lambda /2 \approx {\lambda ^2 \rho _{\rm Al} \over 2 \sigma _{\rm Al} \Omega } \\&\approx & 40 \left({\text{3 MHz} \over \nu } \right)^2 \left({\text{10 Ohm} \over \Omega }\right)\ \text{g}. \end{eqnarray}$$

(2)

Past low frequency space antennae have been partial cylinders or tubes, extended, like a carpenter’s rule, by mechanical stiffness. This requires a comparatively thick, stiff, and massive antenna. The advantage of a centrifugally deployed wire antenna is that no mechanical thickness or stiffness is required; it is extended by tension. This permits a very thin and low mass antenna, limited only the requirement (equation 1) that it have sufficient electrical conductivity.

2.2 Sensitivity

$$\begin{eqnarray} F_{\rm thresh} &=& {S \over N}{4 \pi \over \lambda ^2} {k_B T_{\rm rec} \over G_{a}\sqrt{Bt_{\rm int}}} \\ &\approx & 5 \times 10^4{S/N \over 10} {T_{\rm rec}/3 \times 10^6\, \text{K} \over \sqrt{(Bt_{\rm int}/10^6)}}\ \text{Jy}, \end{eqnarray}$$

(3)

where |$S/N$| is the required signal-to-noise ratio, |$\lambda = 100\,$| m the radio wavelength, |$T_{\rm rec}$|⁠, the noise temperature of the receiver, is scaled to the Galactic synchrotron sky brightness at 3 MHz (Alexander et al. 1969), |$G_{a}$| the telescope’s antenna gain (taken as unity for a dipole), B the receiving bandwidth, and |$t_{\rm int}$| the integration time.

2.3 Data download

A satellite in equatorial orbit passes over a near-equatorial ground station once per orbit. Data can be stored and downloaded with each passage over the ground station. The satellite would be within 2000 km for about 300 s each orbit, implying a minimum data transmission rate of |$\sim 5 \times 10^7$| samples per second during that overflight for a signal receiver bandwidth of 150 kHz (taken as 0.05 of the nominal observing frequency, in accord with the bandwidth of a dipole antenna). Assuming performance comparable to Planet Lab’s download rate of 280 Megabits s⁻¹ would provide five bits per sample, or a dynamic range of |$2^5$|⁠.

The required mean power to transmit a dual polarization base-band signal sampled at a rate |$2 \pi \nu _{\rm obs}$| to a ground-based telescope of diameter D at a range R is

$$\begin{eqnarray} P_{\rm tr} &=& {64 \pi \nu _{\rm obs} R^2 (S/N) k_B T_{\rm data} \over D^2 G_{\rm tr}} \\&\approx & {\text{60 mW} \over G_{\rm tr}} {S/N \over 10} {T_{\rm data} \over \text{30 K}} \left({R \over \text{2000 km}}\right)^2 \left({\text{12 m} \over D}\right)^2, \end{eqnarray}$$

(4)

where |$S/N$| is the receiver signal-to-noise ratio, |$T_{\rm data}$| the data receiver noise temperature, and |$G_{\rm tr}$| is the transmitter gain. The ALMA Band 1 receiver, operating at 35–50 GHz, somewhat above the Ka (26–40 GHz) band, has a specified noise temperature of 32 K, justifying scaling to 30 K, but even a system noise temperature of 100 K would imply a power requirement of only |$\sim 0.2\,$| W.

The required power is modest, even with a dipole transmitting antenna (⁠|$G_{\rm tr} \approx 1$|⁠). Even a 1U Cubesat intercepts about 10 W of sunlight, and a larger satellite more. This could provide |$\gt 1\,$| W of photoelectric power with a duty factor (q.v. equation 17) |$\approx 0.67$|⁠.

The data transmission rate may be reduced by orders of magnitude if the received signal is processed on-board, taking advantage of Moore’s Law and the revolutions in electronics since the era of Alouette, RAE-1, and RAE-2. Rather than transmitting the base-band signal, it would only be necessary to transmit the scientific information of interest. This may be the total received power as a function of time, the autocorrelation of the received signal that indicates the presence of a transient event (because of the delay between direct and ionospheric-reflected signal), or the location of a steady source (again because the delay depends on its zenith angle).

3 ORBITAL LIFETIME

The mass of wire (equation 2) is small compared to the typical mass |$\sim n\,$|kg of a nU Cubesat, but the projected area of the antennae, each of length |$\lambda /2$|⁠, is |$2 \times 2 r \lambda /2 \approx 400\,$| cm|$^2$| (equation 1), several times the projected area of a Cubesat. Equating the work done by atmospheric drag to the decrease in energy of the satellite (noting that half the work done by gravity goes to increasing its kinetic energy), the orbital altitude h decreases at a rate

$$\begin{eqnarray} {{\rm d}h \over {\rm d}t} &=& {4 C_d r L \rho _a \over M_{{\rm Cube}}} \sqrt{GM_{{\oplus}} R_{\rm orb}} \\&\approx & 0.022 {\text{1 kg} \over M_{{\rm Cube}}} {\rho _a \over 10^{-16}\, \text{g cm}^{-3}}\ \rm {cm\,s^{-1}}, \end{eqnarray}$$

(5)

where |$R_{\rm orb}$| is the orbital radius (from the centre of the Earth), |$M_{{\rm Cube}}$| is the mass of the Cubesat, |$\rho _a$| is the atmospheric density, |$M_{{\oplus}} \approx 6.0 \times 10^{27}\,$| g is the mass of the Earth, and the drag coefficient |$C_d$| is taken as unity.

The scale height of the atmosphere at altitudes of interest (800–1200 km) is about 20 km because of its elevated temperature. As a result, the characteristic orbital lifetime

$$\begin{eqnarray} t_{\rm orb} \equiv {\text{20 km} \over {\rm d}h/{\rm d}t} \approx 3\ {M_{{\rm Cube}} \over \text{1 kg}} {10^{-16} \rm {g\,cm}^{-3} \over \rho _a}\ \text{y}. \end{eqnarray}$$

(6)

At these altitudes |$\rho _a$| is sensitive to the Solar cycle and activity (Jacchia 1970; Roberts 1971), and also depends on time of day (but not much on season at equatorial latitudes). It is more useful to specify the air density than the geometrical altitude, and it must be recognized that the orbital decay time (equation 6) may decrease rapidly and unpredictably with Solar activity.

4 DEPLOYMENT

The antennae must be extended by centrifugal force by setting the telescope rotating. Because of its small size, not much angular momentum can be imparted to the Cubesat by forces applied to its surfaces, but even a small initial angular momentum can begin the process of extension by rotating the Cubesat. As the antennae extend, thrusters at their ends produce increasing torques.

Several problems must be addressed:

The thrusters must be simple, light, and cheap.
The thrusters must continue to act over an extended time, perhaps hours or days, as the antennae gradually extend, increasing their lever arms.
The thrusters must remain tangentially oriented. The tiny torsional stiffness of the thin wire antennae that connect them to the Cubesat is insufficient to align them. Nor could tubular (or partial tubular) antennae be both stiff enough and have walls thick enough for handling within the mass budget.

The first two problems are solved by using Knudsen cells (Garland, Nibler & Shoemaker 2009) as the thrusters. A low vapour pressure compound, such as naphthalene, would gradually escape through an aperture at one end of each cell, with its recoil providing the thrust.

The third problem is solved by connecting the ends of the antennae with fine electrically insulating fibre, such as aramid, as shown in Fig. 1, and fixing the Knudsen cells to the fibres. Aramid fibres are available as thin as 170 dtex (1 dtex is defined as a mass of 1 g/10 km) corresponding to a radius of about |$60\, \mu = 0.006\,$| cm (this is also expressed as a length per unit mass |$\text{Nm} = 60$|⁠, where 1 Nm is 1 m g⁻¹). These fibres have the negligible total mass of about 2.5 g. Each fibre has a tensile strength of tens of N, orders of magnitude greater than its tensile load at an angular rotation rate of 3.6 s⁻¹ (equation 9); it is only necessary that the rotation rate be much greater than the orbital angular frequency |$\omega _{\rm orb} \approx 10^{-3}\,$| s|$^{-1}$| in low Earth orbit to maintain the geometry.

As the antennae extend these fibres will also be made taut by centrifugal force. Thrusters tied to them would be aligned tangentially, so their recoil forces spin up the entire system, keeping it taut and stable. The moment of inertia of the four-armed (two |$\lambda /2$| dipole antennae) telescope shown in Fig. 1 is

$$\begin{eqnarray} I = {4 \pi \over 3} L^3 r^2 \rho _{\rm Al} \approx 8 \times 10^7\ \text{g-cm}^2, \end{eqnarray}$$

(7)

where equation (1) has been taken for the wire radius r.

Free molecular flow from Knudsen cells imparts an angular momentum

$$\begin{eqnarray} {\cal L} = L m_p \sqrt{2 k_B T \over \pi m_g}, \end{eqnarray}$$

(8)

where |$m_p$| is the mass of propellant gas exhausted, |$m_g$| its molecular weight, and T its temperature T, and hence the vapour pressure and evaporation time, are determined by the radiative properties of the outsides of the Knudsen cells. For naphthalene at 300 K the rotation rate

$$\begin{eqnarray} \omega = {{\cal L} \over I} \approx 3.6 {m_p \over \text{10 g}}\,\rm s^{-1}. \end{eqnarray}$$

(9)

For |$m_p = 10\,$| g the load on the wire at the Cubesat is |$\pi r^2 \omega \rho _{\rm Al} L^2/2 \approx 1.5\,$| N and the tensile stress |$\omega ^2 \rho _{\rm Al} L^2/2 \approx 1.1 \times 10^8\,$|dyne cm⁻², less than a tenth of the tensile strength of aluminum. The peripheral velocity |$\omega L \approx 90 \,$| m s⁻¹.

If the telescope plane is inclined at an angle i to its orbital plane its spin angular momentum and plane precess (as a result of the Earth’s gravitational torque) around its orbital angular momentum at a rate

$$\begin{eqnarray} \omega _{\rm pre} = - {\omega _{\rm orb}^2 \over \omega } \cos {i} \approx - 3.5 \times 10^{-7} \cos {i}\ \text{s}^{-1}, \end{eqnarray}$$

(10)

or about one radian per month for the assumed parameters. Spin precession slews the broad dipole antenna pattern on the sky at an angular rate |$\omega _{\rm pre} \sin {i}$| with angular amplitude i. Significantly faster or slower precession can be obtained by choice of |$m_p$| and hence of the rotation rate |$\omega$|⁠. The rotational plane of a telescope whose orbit is not equatorial will also precess because of Earth’s equatorial bulge, but (if its spin and orbit are aligned) at a much slower rate than given by equation (10).

5 SOURCES

There are no known extra-Solar System point sources of radiation in the 1–10 MHz range, but sources observed at higher frequencies may also be detectable at these lower frequencies. Promising candidates include radio galaxies and AGN. The mean emission of the Crab pulsar was detected at 26.5 MHz with a flux density |$\sim 1000\,$| Jy by Andrew et al. (1964) before its discovery as a pulsar!

5.1 Attenuation

Galactic absorption is significant (Cane 1979) at frequencies of a few MHz, and at lower frequencies sets an effective horizon within the Galactic disc. This need not preclude observation of old neutron stars that may radiate at these lower frequencies, even though they are not detected in pulsar searches at VHF and UHF frequencies. There are |$\sim 10^8$| neutron stars in the Galactic disc, so that the nearest is likely at a distance of |$\approx 20\,$| pc. The path from such a close source has an absorption optical depth of only |$\sim 0.2$| of that through the full thickness of the Galactic disc, so the radiation from such a nearby source would be much less attenuated and scattered than that from outside the disc.

The opacity at 3 MHz (Spitzer 1962)

$$\begin{eqnarray} \kappa _\text{3 MHz} = \left\lbrace \begin{array}{@{}l@{\quad }l@{}}1.05 \times 10^{-17} n_e^2\ \text{cm}^{-1} & T = 100\ \text{K} \\ 1.8\phantom{0} \times 10^{-21} n_e^2\ \text{cm}^{-1} & T = 10^4\ \text{K}, \end{array}\right. \end{eqnarray}$$

(11)

where the electron density |$n_e$| (in cm|$^{-3}$|⁠) is assumed to come from singly ionized species. In a weakly ionized (⁠|$n_e = 0.03\,$| cm|$^{-3}$|⁠) cool (100 K) cloud the absorption length is |$\sim 30\,$| pc and varies nearly |$\propto T^{3/2}$|⁠, while in a warm (⁠|$10^4$| K) ionized intercloud medium (⁠|$n_e = 0.01\,$| cm|$^{-3}$| in pressure equilibrium with a cool neutral cloud with |$n_\text{H} = 1\,$| cm|$^{-3}$|⁠) the absorption length is |$\sim 150\,$| kpc.

Cool clouds may be opaque, but cosmic ray electrons within them produce an internal source of radio radiation. Condensation into clouds likely increases their cosmic ray density and magnetic field, so they may be net emitters in comparison to the extra-Galactic background. The warm ionized intercloud medium is transparent, transmitting the extra-Galactic background.

5.2 Steady sources

Electron cosmic rays emit incoherent but non-thermal radio synchrotron radiation, and provide a well-understood background. The absorption of this radiation by interstellar ionized gas diagnoses the spatial and temperature distribution of that gas (Ellis & Hamilton 1966; Weber et al. 1971; Alexander et al. 1975; Cane 1979). The proposed system would extend these observations to lower frequencies where the interstellar plasma absorption, varying as the |$-2$| power of frequency, is greater.

Occultation by the Earth’s limb would permit angular resolution on small angular scales in one dimension, as in the early days of X-ray astronomy occultation by the Moon localized X-ray sources on the sky, and permitted their first identifications. This would enable mapping of the distribution of the Galactic non-thermal radio synchrotron radiation, and hence of the distribution of electron cosmic rays. The anisotropic response of the dipole antenna would permit low-resolution mapping.

It may also be possible to infer the absorption along paths to very bright discrete sources, such as Cygnus A, by observing the change of the sky-integrated signal as it enters or leaves Earth occultation. This radio galaxy has a synchrotron radiation spectrum with a flux density at 74 MHz of 1040 Jy and a spectral index between 74 and 327 MHz of −1.49 (Lazio et al. 2006). Even if there is a spectral turnover, a mean spectral index of −1 would imply a flux density of about 25 kJy. This steady source would be readily observable at 3 MHz. A 10 000 s integration would provide |$B t_{\rm int} \approx 1.5 \times 10^9$| and a detection threshold of about 1 kJy (equation 3). Its spectrum would be diagnostic of the interstellar medium along its line of sight.

An additional possible source might be interstellar magnetic reconnection, dissipating the energy of the interstellar field, amplified by magnetohydrodynamic turbulence. These processes are ubiquitous in conducting plasmas, from laboratory magnetic fusion reactors to the Solar photosphere and corona and wind, and stellar and quasar accretion discs. Electric fields resulting from changing magnetic fields accelerate electrons to relativistic energies, and plasma instabilities lead to turbulent resistivity. Electrons accelerated to Lorentz factors |${\cal O}(10^3)$| have synchrotron frequencies of about 3 MHz, making them possible sources of radiation in this band. It is unclear how rapidly such a source would vary; it might be steady on instrumental time-scales.

6 THE TOPSIDE IONOSPHERE

The spatial structure of the topside ionosphere (Banks, Schunk & Raitt 1976; Pignalberi et al. 2020; Prol et al. 2022) may be probed by observing the reflection of 3 MHz radiation from a strong steady natural source or an artificial beacon. Natural sources include the radio galaxies Cen A and Cyg A and the SNR Cas A. If the reflective layer is tilted by gravity waves (or otherwise), the source’s effective elevation varies, and can be inferred from the phase difference and interference between the reflected and direct signals from steady sources of known direction.

The low Galactic latitude Cas A (⁠|$b = -1.96^\circ$|⁠) may be observable only at frequencies |$\gtrapprox 10\,$| MHz because of interstellar absorption (Stanislavsky et al. 2023), that may also be significant for Cyg A (⁠|$b = 5.6^\circ$|⁠). At larger zenith angles (lower elevations) the ionosphere is reflective at higher frequencies, less absorbed by interstellar plasma, and grazing reflection at these frequencies can also be used to study the topside ionosphere.

In the simplest possible model the ionosphere is static (except for the effect of the diurnal variation of the Solar ionizing radiation) and its density contours are spherical (horizontal in the flat-ionosphere approximation). However, the elevation |$\Delta$| (and zenith angle |$\theta$| vary as the satellite moves in its orbit, so the path difference |$\delta \ell$| and phase difference |$2 \pi \delta \ell /\lambda$| vary with time, comparatively rapidly. For a steady source the received power will oscillate as the phase difference between these two paths changes as |$\Delta$| varies with the satellite motion. This is the same principle as that of the Sea Interferometer (Bolton & Slee 1953) of early radio astronomy, itself a long-wave realization of Lloyd’s mirror.

For an ideal static ionosphere the frequency of oscillation is determined by the satellite’s orbit and the direction to the source. Ionospheric oscillations (tilts of the reflecting surface, which is the critical density surface for sources at the zenith but is higher for sources at non-zero zenith angles) will manifest themselves as deviations of these oscillations from the predictions of the flat-ionosphere model. This would require only measurement of the received power, averaged over a fraction of the oscillation period (typically seconds), rather than processing or transmission of base-band data.

The elevation |$\Delta$| of a source at a known Right Ascension |$\alpha$| and Declination |$\delta$| as viewed from a point in an equatorial orbit (⁠|$\delta _{\rm orb} = 0$|⁠) with geocentric Right Ascension |$\alpha _{\rm orb} = \Omega t$|⁠, where |$\Omega$| is the orbital angular velocity, is given by

$$\begin{eqnarray} \cos {\Delta } = \cos {\delta }\cos {(\alpha -\Omega t)}. \end{eqnarray}$$

(12)

The amplitude of the sum of the direct and reflected fields, and hence of the received power, oscillates with a period |$P_{\rm osc}$|⁠. The rate of change of the delay between the two signals is

$$\begin{eqnarray} \begin{split} {\rm d}\delta t &= {2h \over c}{\rm d}\delta \sin {\Delta } = {2h \over c}\cos {\Delta }d\Delta \\ &= {2h \Omega \over c}\cos ^2{\delta } \cos {(\alpha -\Omega t)}\sin {(\alpha -\Omega t)}{\rm d}t, \end{split} \end{eqnarray}$$

(13)

$$\begin{eqnarray} {{\rm d} \delta t \over {\rm d}t} = {h \Omega \over c} \cos ^2{\delta } \sin {2(\alpha -\Omega t)}. \end{eqnarray}$$

(14)

A full cycle of this oscillation occurs after a time

$$\begin{eqnarray} {\rm d}\delta t = {2\pi \over \omega }, \end{eqnarray}$$

(15)

where |$\omega$| is the angular frequency of the radio radiation. This condition is met after an interval

$$\begin{eqnarray} P_{\rm osc} = {2\pi /\omega \over {\rm d}\delta t/{\rm d}t} = {cP_{\rm orb} \over \omega h}{1 \over \cos ^2{\delta }\sin {2(\alpha -\Omega t)}}, \end{eqnarray}$$

(16)

where |$P_{\rm orb}$| is the orbital period. For |$P_{\rm orb} = 100\,$| min, |$\omega = 2 \times 10^7$| s|$^{-1}$| (3 MHz) and |$h = 300$| km, the numerical factor is about 0.3 s. As many as |${\cal O}(10^4)$| cycles of this oscillation could be observed in the approximately half-orbit (3000 s) during which an astronomical source can be observed, so that ionospheric tilts |${\cal O}(10^{-4})$| radian might be detectable.

The approximately 5 per cent bandwidth of a half-wave dipole is wide enough that the oscillation phase varies by many radians across the band, so that the band would have to be divided into channels to be analysed separately. The oscillation frequency (or |$P_{\rm osc}$|⁠) of the received power is described by few tens of bits of data because the oscillation has a frequency of a few Hz. Transmitting these stored data, even for hundreds of frequency channels, would not require much bandwidth.

The delay also requires correction for propagation through layers in which the electron density is high enough that the signal group velocity is significantly less than c, and its path is bent by refraction. Because the electron gyrofrequency is 1–2 MHz, the effect of the geomagnetic field is significant and the ordinary and extraordinary modes must be considered separately within the ionosphere.

This analysis depends on the assumption that the ionospheric density contours are smooth, undisturbed by turbulence. The failure of these predictions (variation of the received intensity that is not periodic with the period of equation 16) would therefore be a novel probe of topside ionospheric turbulence that is not addressed by existing ionospheric diagnostics (Schreiter et al. 2023).

7 SHIELDING OF THE SUN

At low frequencies the Sun is an intense source of background in a dipole antenna. The Earth shields this background whenever the antenna is in its shadow. At equinoxes it is shielded by the solid Earth a fraction

$$\begin{eqnarray} f_{\rm shield} = {1 \over \pi } \sin ^{-1}{\left({R_{{\oplus}} \over R_{{\oplus}}+h}\right)} \approx 0.33 \end{eqnarray}$$

(17)

of the time, where we have taken |$h = 1000\,$| km.

The ionosphere increases |$f_{\rm shield}$| because it increases the effective (opaque) |$R_{{\oplus}}$|⁠. In addition, ionospheric refraction when the Sun is near the antenna’s horizon may further increase |$f_{\rm shield}$|⁠:

$$\begin{eqnarray} f_{\rm shield} = {1 \over 2} + {\phi - \psi \over \pi }, \end{eqnarray}$$

(18)

where

$$\begin{eqnarray} \phi = \cos ^{-1}{\sqrt{1 - {\nu _{p\text{-}\rm orb}^2 \over \nu ^2}}} \end{eqnarray}$$

(19)

is the maximum angle of ionospheric plasma refraction, |$\nu _{p\text{-}\rm orb}$| is the plasma frequency at the antenna’s altitude and |$\nu$| is the radiation frequency. The depression angle of the critical density horizon

$$\begin{eqnarray} \psi = \cos ^{-1}{\left({R_{{\oplus}} + h_{\rm crit} \over R_{{\oplus}} + h}\right)}, \end{eqnarray}$$

(20)

where |$h_{\rm crit}$| is the altitude at which the plasma frequency |$\nu _p(h_{\rm crit}) = \nu$|⁠; for observation to be possible |$h_{\rm crit} \lt h$|⁠.

Both |$\phi$| and |$\psi$| are small angles for a satellite in LEO. Averaging over the year, |$f_{\rm shield}$| is multiplied by a factor |$\approx 1-\varepsilon ^2/4 \approx 0.96$|⁠, where |$\varepsilon \approx 0.41\,$| rad is the obliquity of the Earth’s equator.

In contrast, deep space or GEO observatories like FIRST, SURO-LC, DARIS, Wind, STEREO, SunRISE, and OLFAR are illuminated by the Sun; their purpose is to observe it and its corona, but this background is a severe obstacle to observations of weak extra-Solar System sources. Refraction by interplanetary plasma that broadens the arrival directions of low frequency radiation by |$\Delta \theta \sim 10^\prime \sim 3\,$| mrad spreads its arrival time by |$\sim 1\, \text{AU} (\Delta \theta )^2/2c \sim 2.5\,$|ms, limiting the possible resolution of interferometry, even with long baselines.

8 DISCUSSION

Even without angular resolution, it would be possible to probe the Universe in this unexplored frequency band:

A dipole antenna would measure a sky average (weighted by its gain) temperature, yielding information about the interstellar medium not obtainable in any other manner. Models of the interstellar medium predict the antenna temperature of a dipole, and can be tested by its measurement.
A dipole antenna observing the reflection of an exospheric beacon would measure the variability of the topside ionosphere.

Angular resolution would provide additional information:

A single dipole telescope (or two orthogonal dipoles) whose rotation axis is not parallel to Earth’s would precess, even in equatorial orbit. This would sweep its dipole beam pattern across the sky, providing some angular resolution of steady emission like that of the interstellar medium.
Resolution could be obtained by aperture synthesis with a larger telescope comprising multiple dipoles. This could resolve interstellar cloud structure.
Aperture synthesis using multiple, widely spaced, dipoles in equatorial orbit could narrowly constrain the locations of any transients strong enough to be observed, but would be poorly matched to the broad angular scales of interstellar clouds. More closely spaced dipoles would be a better match to that angular structure. Knowledge of the dipoles’ locations (although not necessarily active station-keeping) to allow for unpredictable differences in atmospheric drag would be required; this could be provided by GPS.

An equatorial orbit would permit data downloads to a single ground station every orbit, minimizing data storage and download rate requirements. The orbit of such a satellite, locked to the Earth’s equatorial bulge, would not precess. If rotating in the Earth’s equatorial plane, its spin would also not precess.

Large, low frequency space-based antennae are plausible candidates for demonstration of in-space manufacturing and assembly. Such structures would need to be quite large (⁠|$\sim \,$|km) in order to provide even degree-scale resolution at these long wavelengths. Before such a costly and technically challenging demonstration it would be prudent to develop a ‘pathfinder’ Cubesat-scale telescope to explore the signal characteristics to be observed by a larger instrument.

An extended antenna wire poses a collision risk for other satellites. Its effective cross-section for a 1 m satellite would be |$\sim 50\,$| m|$^2$|⁠. This is much larger than the satellite’s |$\sim 1\,$| m|$^2$| cross-section for a small piece of space debris, but the proposal is only for one antenna while lethal debris are numerous. If the wire is thin enough collision might not be catastrophic to the satellite. Once the mission is over, the antennae could be detached from the Cubesat. No longer centrifugally extended, a slight (unavoidable) intrinsic curvature would crumple the thin wire into a compact tangle, with much reduced cross-section.

The mutual collison risk of multiple 3 MHz antennae, such as might be deployed to make an interferometer, would be minimized if all were in equatorial orbits, with rotation axes parallel to the Earth’s and each other’s. Then their mutual collision cross-section would be proportional only to the first power of their antenna length, rather than quadratic.

DATA AVAILABILITY

This theoretical work generated no original data.

ACKNOWLEDGEMENTS

We thank Joseph Lazio, David Palmer, and anonymous referees for calling our attention to earlier work on this subject.

Footnotes

This is a different meaning of ‘gain’ than that used by antenna engineers.

References

Alexander

J. K.

Brown

L. W.

Clark

T. A.

Stone

R. G.

Weber

R. R.

1969

ApJ

157

L163

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Article Contents

3 MHz space antenna

ABSTRACT

1 INTRODUCTION

2 THE ANTENNA

2.1 Parameters

2.2 Sensitivity

2.3 Data download

3 ORBITAL LIFETIME

4 DEPLOYMENT

5 SOURCES

5.1 Attenuation

5.2 Steady sources

6 THE TOPSIDE IONOSPHERE

7 SHIELDING OF THE SUN

8 DISCUSSION

DATA AVAILABILITY

ACKNOWLEDGEMENTS

Footnotes

References

APPENDIX A: TRANSIENT AND VARIABLE SOURCES

A1 Dispersion and broadening

A2 Detecting transients?

A3 Localization by ionospheric reflection

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