Jupiter’s radio spectrum from 74 MHz up to 8 GHz

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Jupiter’s radio spectrum from 74 MHz up to 8 GHz Imke de Pater,a,* B.J. Butler,b D.A. Green,c R. Strom,d R. Millan,e M.J. Klein,f M.K. Bird,g O. Funke,g J. Neidho¨fer,h R. Maddalena,i R.J. Sault,j M. Kesteven,j D.P. Smits,k and R. Hunsteadl a

Department of Astronomy, and Department of Earth and Planetary Science, 601 Campbell Hall, University of California, Berkeley, CA 94720, USA b National Radio Astronomy Observatory, Socorro, NM 87801, USA c Mullard Radio Astronomy Observatory, Cavendish Laboratory, Madingley Rd, Cambridge CB3 0HE, UK d Astron, P.O. Box 2, 7990 AA Dwingeloo, and Astronomical Institute “A. Pannekoek,” Univ. of Amsterdam, the Netherlands e Physics Department, University of California, Berkeley, CA 94720, USA f Jet Propulsion Laboratory, Pasadena, CA 91109, USA g Radioastron. Inst., Universita¨t Bonn, Auf dem Hu¨gel 71, 53121 Bonn, Germany h Max-Planck-Institut fu¨r Radioastronomie, Auf dem Hu¨gel 69, 53121 Bonn, Germany i National Radio Astronomy Observatory, Green Bank, WV 24944, USA j CSIRO, Epping, NSW 2121, Australia k Dept. of Mathematics, Applied Mathematics and Astronomy, PO Box 392, UNISA, 0003 South Africa l School of Physics, University of Sydney, NSW 2006, Australia Received 12 February 2002; revised 18 November 2002

Abstract We carried out a brief campaign in September 1998 to determine Jupiter’s radio spectrum at frequencies spanning a range from 74 MHz up to 8 GHz. Eleven different telescopes were used in this effort, each uniquely suited to observe at a particular frequency. We find that Jupiter’s spectrum is basically flat shortwards of 1–2 GHz, and drops off steeply at frequencies greater than 2 GHz. We compared the 1998 spectrum with a spectrum (330 MHz– 8 GHz) obtained in June 1994, and report a large difference in spectral shape, being most pronounced at the lowest frequencies. The difference seems to be linear with log(␯), with the largest deviations at the lowest frequencies (␯). We have compared our spectra with calculations of Jupiter’s synchrotron radiation using several published models. The spectral shape is determined by the energy-dependent spatial distribution of the electrons in Jupiter’s magnetic field, which in turn is determined by the detailed diffusion process across L-shells and in pitch angle, as well as energy-dependent particle losses. The spectral shape observed in September 1998 can be matched well if the electron energy spectrum at L ⫽ 6 is modeled by a double power law E⫺a 共1 ⫹ 共E/E0 兲兲⫺b , with a ⫽ 0.4, b ⫽ 3, E0 ⫽ 100 MeV, and a lifetime against local losses ␶0 ⫽ 6 ⫻ 107 s. In June 1994 the observations can be matched equally well with two different sets of parameters: (1) a ⫽ 0.6, b ⫽ 3, E0 ⫽ 100 MeV, ␶0 ⫽ 6 ⫻ 107 s, or (2) a ⫽ 0.4, b ⫽ 3, E0 ⫽ 100 MeV, ␶0 ⫽ 8.6 ⫻ 106 s. We attribute the large variation in spectral shape between 1994 and 1998 to pitch angle scattering, coulomb scattering and/or energy degradation by dust in Jupiter’s inner radiation belts. © 2003 Elsevier Science (USA). All rights reserved. Keywords: Jupiter; Magnetosphere; Radio observations

1. Introduction Jupiter’s synchrotron radiation has been observed regularly at frequencies between 300 and 8000 MHz, usually as * Corresponding author. Fax: ⫹1-510-642-3411. E-mail address: [email protected] (I. de Pater).

stand-alone observations either in imaging or single-dish mode. Since Jupiter’s flux density exhibits substantial variations over time (e.g., Gerard, 1970, 1976; Klein et al., 1972, 1989, 2001; Klein, 1976; de Pater and Klein, 1989; Miyoshi et al., 1999), data taken at different times cannot simply be combined to form a radio spectrum. In addition, already in the early seventies some observations hinted at

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I. de Pater et al. / Icarus 163 (2003) 434 – 448

time variability in the planet’s radio spectrum (Klein, 1976), a speculation which has since then been confirmed during the period that Comet Shoemaker–Levy 9 (SL9) crashed into Jupiter (e.g., de Pater et al., 1995), and later by Galopeau and Gerard (2001) who showed that changes in the planet’s radio spectrum are not uncommon. Hence it is impossible to form a spectrum by simply scaling data to a common epoch using, e.g., the 13-cm database compiled by M.J. Klein. Radio spectra contain information on the energy distribution of the emitting electrons, a much needed parameter for model studies of Jupiter’s synchrotron radiation. No spectra have been published spanning a frequency range as large as 74 MHz up to 8 GHz. We conducted a brief campaign over a 2-week period in September 1998 to obtain such a spectrum by using a large variety of radio telescopes, each operating at specific frequencies between 74 MHz and 8 GHz. We assume in this paper that Jupiter’s spectrum did not change over this time period, an assumption that is probably justified since the flux densities at individual frequencies did not change (in particular, see the results at 2295 MHz in Table 1). We compare this spectrum with one between 330 MHz and 8 GHz, obtained in June 1994 just before Comet SL9 crashed into Jupiter.

2. Observations and data reduction techniques We observed Jupiter between 13 and 26 September 1998 to determine the planet’s radio spectrum from 74 MHz up to 8 GHz. The planet was near opposition, at a right ascension of ⬃23hr 30min and declination between ⫺4° and ⫺5°. The declination of the Earth with respect to Jupiter, DE, was ⬃2.45°. We observed Jupiter with 11 different telescopes: the VLA (Very Large Array, USA), CLFST (Cambridge Low-Frequency Synthesis Telescope, United Kingdom), WSRT (Westerbork Synthesis Radio Telescope, Netherlands), MOST (Molonglo Observatory Synthesis Telescope, Canberra, Australia), Parkes 64-m (Australia), NRAO 140-ft (National Radio Observatory, Green Bank, USA), the DSN and GAVRT antennas at Goldstone (the 34-m and 70-m NASA Deep Space Network antennas and the 34-m Goldstone–Apple Valley Radio Telescope at Goldstone, California, USA), MPIfR 100-m (Max Planck Institute fu¨ r Radioastronomie, Effelsberg, Germany), and HartRAO (Hartebeesthoek Radio Astronomy Observatory, South Africa). VLA observations at 74/333 MHz were carried out on September 19 and 20, 1998, and at 327/1365 MHz on September 15 and 16, 1998. These observations are described in detail by de Pater and Butler (2003). The CLFST observations were carried out on six consecutive nights between September 16/17 and 21/22, for over 5 h per night, at a frequency of 151 MHz. A general description of the CLFST is given by McGilchrist et al. (1990). Since the antennas of the telescope cannot physically point below ⫹5° Dec, Jupiter had to be observed ⬃9°

435

from the center of the primary beam. Because the telescope’s primary beam (half power beam width, HPBW) was ⬃17°, this could indeed be done. We note that the experiment was not straightforward, not only because Jupiter was ⬃9° removed from the center of the field, but also because the CLFST is a nearly east–west array of antennas, so that the synthesized beam is very elongated in the north–south direction, and distorted, for sources near 0° Dec. However, since Jupiter moved from night to night with respect to the background sources, we could successfully remove the background field and determine Jupiter’s flux density on each day with an accuracy of ⬃10% after smoothing the images. The final Jupiter flux densities were tied to the background sources 4C-04.89 and 4C-04.88, which were readily detected in the field of the observations of Jupiter on each night. The WSRT observed between September 14 and 19 at frequencies of 377 and 610 MHz. A description of the WSRT is given by, e.g., Ho¨ gbom and Brouw (1974). We observed Jupiter for ⬃6 h each day, switching between frequencies, and imaged a relatively large field around Jupiter on each day (2.5° at 610 MHz and 4° at 377 MHz), using the WSRT software NEWSTAR. Like the CLFST, WSRT is an east–west array of telescopes, and hence the synthesized beam is very elongated in the north–south direction. The observations initially were not corrected for Jupiter’s motion on the sky, which facilitated identification, self-calibration on, and subtraction of background sources. After the background sources were subtracted, the phases of the UV data were corrected so that the phase center followed Jupiter’s motion on the sky (using the MIRIAD software, Sault et al., 1995). We then self-calibrated the data on Jupiter and determined the planet’s total flux density with JMFIT in AIPS. The MOST data were taken in a manner analogous to that during the impacts of Comet Shoemaker–Levy 9 in 1994, as decribed by Dulk et al. (1995) and de Pater et al. (1995). Also for MOST the synthesized beam is very elongated in the north–south direction. The total integration time was 6 h. The integrated flux densities of Jupiter and the background source 4C-04.88 (observed in the same field) were determined using KVIEW in the KARMA package (Gooch, 1995). The single dish NRAO 140-ft and Australian Parkes telescope data were obtained as described for the 1994 SLq observations by Wong et al. (1996), but the total observing time per day was much shorter than in 1994. We therefore combined the data taken on consecutive days (for Parkes on all 3 days) before we determined the total flux density. For details on the MPIfR 100-m telescope measurements we refer the reader to Bird et al. (1996), and for the HartRAO observations to Millan et al. (1998). The observations with NASA’s Deep Space Network (DSN) antennas at Goldstone, California, were made at 2295 MHz on nine nights from September 8 through 22. Jupiter was observed 6 to 10 h each night with either the DSN 70-m, the 34-m research and development antenna

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(DSS13), or the 34-m antenna operated by the GoldstoneApple Valley Radio Telescope (GAVRT) science education partnership.1 The GAVRT antenna is equipped with dual frequency feeds and receiving systems operating at 2295 and 8480 MHz. Three of the observing sessions were led by GAVRT-trained teachers and conducted by middle and high school students from their classrooms (Bollman et al., 2001). The results of their observations are reported in this paper. All of the Goldstone observations followed procedures established for the ongoing NASA/JPL Jupiter Patrol described in Klein et al. (1989). Jupiter’s radio emission consists of a thermal and nonthermal component. Since we are only interested in the nonthermal (synchrotron) component, we subtracted the thermal flux density from the data, assuming the planet’s emission can be approximated by a uniform disk with a disk-averaged brightness temperature TB. We used the Rayleigh–Jeans approximation to Planck’s radiation law to calculate the thermal flux density corresponding to TB at each wavelength. At 3.5 and 6.1 cm we adopted TB (3.5 cm) ⫽ 191.4 ⫾ 5.8 K and TB (6.1 cm) ⫽ 232.4 ⫾ 8.2 K, as determined from fits to VLA UV data by de Pater et al. (2001). Based upon data and model calculations presented in the latter paper, we adopted the following brightness temperatures at the longer wavelengths: 282 ⫾ 8 K at 11 cm, 285 ⫾ 8 K at 13 cm, 350 ⫾ 30 K at 20 cm, 420 ⫾ 40 K at 36 cm, ⬃500 K at 50 cm, and ⬃700 K at 90 cm. Jupiter’s synchrotron radiation exhibits a well-known quasisinusoidal variation with a period equal to the planet’s rotation period (⬃10 h), referred to as the “beaming curve.” For the single dish data, we derived an average flux density from the beaming curve using the algorithm described by Klein et al. (1989) for the 2295 MHz data, and that of de Pater et al. (1995) and Wong et al. (1996) for the data obtained with the MPIfR 100-m telescope and the NRAO 140-ft and Parkes telescopes. Since the constants An and Pn in their equations depend on the declination of Earth with respect to Jupiter, we used values for An and Pn as derived by M.J. Klein from 70-m data at 13 cm at DE ⬇ 2.45° (e.g., Klein et al., 1989). We assumed the constants to be similar at higher and lower frequencies. At 3.5 cm no beaming curve could be distinguished in the data, so here we took a straight average. The images constructed from interferometer data were all averaged over time, so here we obtained average values without fitting a beaming curve to the data.

1 The Goldstone–Apple Valley Radio Telescope (GAVRT) science education project is a partnership involving NASA, the Jet Propulsion Laboratory, and the Lewis Center for Educational Research (LCER) in Apple Valley, CA. Working with the Lewis Center over the Internet, GAVRT students conduct remotely controlled radio astronomy observations using a 34-m antenna at Goldstone.

2.1. Size correction factors When the spatial extent of Jupiter’s radio emision is a nonnegligible fraction of the beam width of a single dish radio telescope, the flux density has to be corrected. For an extended uniform disk, as applicable to Jupiter’s thermal emission, the correction factor CT was determined using the formalism of Ulich and Haas (1976). The values are typically ⱗ3% (Table 1). As detailed above and in Table 1, Jupiter’s thermal disk temperature is approximately 190 K at 3.5 cm, rising to ⬃500 K at 50 cm. We divided the expected thermal flux density by CT to obtain the “observed” thermal flux density, ST. The total observed nonthermal flux density, SNT, is then SNT ⫽ Sobs ⫺ ST, where Sobs is the flux density as observed at the telescope. We then need to correct SNT for the decrease in intensity due to the finite beam width of the telescope. Since the brightness distribution of Jupiter’s nonthermal radio emission is very similar at frequencies between 330 MHz and 5 GHz (de Pater, 1991), we determined the multiplication factor, CNT, for Jupiter’s nonthermal emission from observations of Jupiter at a wavelength of 90 cm, where the thermal contribution is negligible (⬃1%) compared to the total radiation. We convolved 90-cm images, taken with the VLA and WSRT in 1994 with different array configurations,2 down to a lower resolution and determined the peak flux density in the images. This peak flux density represents the total flux density as would be observed by a single dish with a HPBW similar to the gaussian beam used to convolve the radio images. In addition, we analyzed data from the past few years when Jupiter was observed on the same nights with the DSN 34-m and 70-m antennas at Goldstone. The purpose was to test the algorithm by comparing the 70-m data, which is more sensitive to the correction, with 34-m observations made on the same night. Observations from 26 nights were used to calculate values for the 70-m correction factors, C NT共70兲 ⫽ C NT共34兲 * 关S 34/S 70兴,

(1)

where S34 and S70 are the observed flux densities from the two antennas on the same night. The results were “binned” into five intervals of [HPBW/RJ] (1RJ ⫽ 1 Jovian radius) and the average values are plotted in Fig. 1, together with the VLA/WSRT derived values at larger [HPBW/RJ]. We superpose an exponential fit to the data, C NT ⫽ 1 ⫹ x 1exp共共 x 2 ⫺ 关HPBW/R J兴兲/x 3兲,

(2)

with x1 ⫽ 0.33, x2 ⫽ 7.91, and x3 ⫽ 4.22. The agreement with the fitted curve (solid line) is consistent with the error bars on the DSN data points. This graph has been used in the past by e.g., de Pater et al. (1995), Millan et al. (1998), Bird et al. (1996), Funke et al. (1997), and Klein et al. (2001).

2 The original data used to determine the size correction factor were published by de Pater et al. (1995) and Sukumar et al. (1998).

I. de Pater et al. / Icarus 163 (2003) 434 – 448

437

Table 1 Jupiter flux densities Frequency (MHz)

Wavelength (cm)

Telescope

Date (UT 1998)

74 74 74 151 151 151 151 151 151 151 327.5 327.5 333.0 333.0 330 377 377 377 609 609 609 609 609 843 1350 1370 1370 1370 1480 1480 1480 2295 2295 2295 2295 2295 2295 2295 2295 2295 2295 2295 2295 2295 2695 2695 2695 2695 4850 4850 4850 4850 8480 8480 8480 8480 8580 8580 8580 8580 8580 8580 8580 8580

405 405 405 199 199 199 199 199 199 199 91.6 91.6 90.1 90.1 90.9 79.6 79.6 79.6 49.3 49.3 49.3 49.3 49.3 35.6 22.2 21.9 21.9 21.9 20.3 20.3 20.3 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 13.1 11.1 11.1 11.1 11.1 6.19 6.19 6.19 6.19 3.54 3.54 3.54 3.54 3.50 3.50 3.50 3.50 3.50 3.50 3.50 3.50

VLA VLA VLA CLFST CLFST CLFST CLFST CLFST CLFST CLFST VLA VLA VLA VLA VLA WSRT WSRT WSRT WSRT WSRT WSRT WSRT WSRT MOST Parkes NRAO 140-foot NRAO 140-foot NRAO 140-foot NRAO 140-foot NRAO 140-foot NRAO 140-foot NASA DSN 34-m NASA DSN 34-m NASA DSN 70-m NASA DSN 70-m NASA DSN 34-m NASA DSN 34-m NASA GAVRT 34-m NASA GAVRT 34-m NASA GAVRT 34-m NASA DSN 70-m NASA DSN 34-m NASA DSN 34-m NASA DSN MPIR 100-m MPIR 100-m MPIR 100-m MPIR 100-m MPIR 100-m MPIR 100-m MPIR 100-m MPIR 100-m NASA GAVRT 34-m NASA GAVRT 34-m NASA GAVRT 34-m NASA GAVRT 34-m HartRAO HartRAO HartRAO HartRAO HartRAO HartRAO HartRAO HartRAO

19 Sep 20 Sep Average 16-17 Sep 17-18 Sep 18-19 Sep 19-20 Sep 20-21 Sep 21-22 Sep Average 15 Sep 16 Sep 19 Sep 20 Sep Average 18 Sep 19 Sep Average 14 Sep 15 Sep 18 Sep 19 Sep Average 18 Sep 26-28 Sep 23-24 Sep 25-26 Sep Average 23-24 Sep 25-26 Sep Average 9 Sep 10 Sep 13 Sep 14 Sep 15 Sep 16 Sep 18 Sep 19 Sep 20 Sep 20 Sep 22 Sep 23 Sep Average 15-16 Sep 18-19 Sep 19-20 Sep Average 15-16 Sep 18-19 Sep 19-20 Sep Average 18 Sep 19 Sep 20 Sep Average 13-14 Sep 15 Sep 16 Sep 17 Sep 18 Sep 19 Sep 20-21 Sep Average

a

Total flux densitya at 4.04 AU (Jy)

Nonthermal flux densityb at 4.04 AU (Jy)

4.97 ⫾ 0.3 4.72 ⫾ 0.3

4.96 ⫾ 0.3 4.71 ⫾ 0.3

4.5 ⫾ 0.5 6.4 ⫾ 0.6 5.3 ⫾ 0.5 5.9 ⫾ 0.6 5.5 ⫾ 0.5 5.3 ⫾ 0.5

4.5 ⫾ 0.5 6.4 ⫾ 0.6 5.3 ⫾ 0.5 5.9 ⫾ 0.6 5.5 ⫾ 0.5 5.3 ⫾ 0.5

5.25 ⫾ 0.06 5.37 ⫾ 0.06 5.22 ⫾ 0.06 5.12 ⫾ 0.06

5.15 ⫾ 0.06 5.27 ⫾ 0.06 5.12 ⫾ 0.06 5.02 ⫾ 0.06

5.3 ⫾ 0.3 4.7 ⫾ 0.3

5.2 ⫾ 0.3 4.6 ⫾ 0.3

5.8 ⫾ 0.6 5.3 ⫾ 0.4 4.4 ⫾ 0.5 5.0 ⫾ 0.6

5.5 ⫾ 0.6 5.1 ⫾ 0.4 4.2 ⫾ 0.5 4.8 ⫾ 0.6

5.5 ⫾ 0.2 5.60 ⫾ 0.15 5.47 ⫾ 0.16 5.50 ⫾ 0.16

5.1 ⫾ 0.2 4.80 ⫾ 0.15 (0.17) 4.65 ⫾ 0.16 (0.17) 4.68 ⫾ 0.16 (0.17)

5.46 ⫾ 0.16 5.11 ⫾ 0.16

4.49 ⫾ 0.16 (0.18) 4.55 ⫾ 0.16 (0.18)

5.906 ⫾ 0.050 5.891 ⫾ 0.050 5.976 ⫾ 0.050 5.876 ⫾ 0.048 5.955 ⫾ 0.053 5.992 ⫾ 0.060 5.926 ⫾ 0.053 5.890 ⫾ 0.057 5.886 ⫾ 0.055 5.903 ⫾ 0.051 5.917 ⫾ 0.051 5.900 ⫾ 0.055

4.016 ⫾ 0.060 (0.08) 4.001 ⫾ 0.060 (0.08) 4.086 ⫾ 0.060 (0.08) 3.986 ⫾ 0.059 (0.08) 4.065 ⫾ 0.062 (0.08) 4.102 ⫾ 0.064 (0.08) 4.036 ⫾ 0.061 (0.08) 4.000 ⫾ 0.063 (0.08) 3.996 ⫾ 0.062 (0.08) 4.013 ⫾ 0.060 (0.08) 4.027 ⫾ 0.061 (0.08) 4.010 ⫾ 0.062 (0.08)

6.02 ⫾ 0.14 6.06 ⫾ 0.18 6.00 ⫾ 0.15

4.01 ⫾ 0.14 (0.19) 4.06 ⫾ 0.18 (0.23) 3.99 ⫾ 0.15 (0.19)

9.25 ⫾ 0.11 9.22 ⫾ 0.17 9.13 ⫾ 0.10

3.73 ⫾ 0.11 (0.37) 3.69 ⫾ 0.17 (0.44) 3.56 ⫾ 0.10 (0.37)

19.30 ⫾ 0.07 19.37 ⫾ 0.05 19.16 ⫾ 0.08

2.00 ⫾ 0.08 (0.6) 2.08 ⫾ 0.06 (0.6) 1.89 ⫾ 0.09 (0.6)

19.56 ⫾ 0.19 19.77 ⫾ 0.10 19.71 ⫾ 0.14 19.74 ⫾ 0.21 19.64 ⫾ 0.18 19.71 ⫾ 0.15 19.74 ⫾ 0.13

2.15 ⫾ 0.19 (0.6) 2.38 ⫾ 0.10 (0.6) 2.32 ⫾ 0.14 (0.6) 2.35 ⫾ 0.21 (0.6) 2.25 ⫾ 0.18 (0.6) 2.32 ⫾ 0.15 (0.6) 2.35 ⫾ 0.13 (0.6)

Nonthermal flux densityc at 4.04 AU (Jy)

Correction factors CT, CNT

4.84 ⫾ 0.16

5.50 ⫾ 0.25

5.13 ⫾ 0.05 4.9 ⫾ 0.2

4.9 ⫾ 0.3 5.1 ⫾ 0.2 4.80 ⫾ 0.15 (0.17) 4.67 ⫾ 0.11 (0.13) 4.52 ⫾ 0.11 (0.14) 1.003, 1.003, 1.015, 1.015, 1.003, 1.003, 1.003, 1.003, 1.003, 1.015, 1.003, 1.003,

1.001 1.001 1.044 1.044 1.001 1.001 1.001 1.001 1.001 1.044 1.001 1.001

4.028 ⫾ 0.01 (0.05) 1.011, 1.161 1.011, 1.161 1.011, 1.161 4.02 ⫾ 0.09 (0.12) 1.034, 1.434 1.034, 1.433 1.034, 1.433 3.66 ⫾ 0.07 (0.34) 1.016 1.251 1.015 1.251 1.015 1.251 1.982 ⫾ 0.06 (0.6) 1.01, 1.01, 1.01, 1.01, 1.01, 1.01, 1.01,

1.10 1.10 1.10 1.10 1.10 1.10 1.10

2.30 ⫾ 0.06 (0.6)

Total flux density, scaled to the nominal distance of 4.04 AU. Total nonthermal flux density, scaled to the nominal distance of 4.04 AU, and corrected for telescope resolution effects. A thermal contribution was subtracted, equal to a disk brightness temperature of 191.4 ⫾ 5.8 K at 3.5 cm, 232.4 ⫾ 8.2 K at 6 cm, 282 ⫾ 8 K at 11 cm, 285 ⫾ 8 at 13 cm, 350 ⫾ 30 K at 20 cm, 420 ⫾ 40 K at 36 cm, ⬃500 K at 50 cm, and ⬃700 K at 90 cm (de Pater et al., 2001). The total uncertainty (including the uncertainty in the thermal contribution) is indicated in brackets. c Average nonthermal flux density at 4.04 AU. The total uncertainty (including the uncertainty in the thermal contribution) is indicated in brackets. b

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I. de Pater et al. / Icarus 163 (2003) 434 – 448

Fig. 1. Graph of the correction factor CNT to correct single-dish data for Jupiter’s size. The correction factor CNT is plotted as a function of the telescope HPBW expressed in Jovian radii, based upon VLA/WSRT (filled triangles) and NASA DSN data (open circles), as described in the text. The solid line is a best-fit exponential curve (Eq. 2) through the data.

2.2. Flux density calibration We used Cygnus A as our primary flux calibrator at 74 MHz, with an adopted flux density of 1.6127 kJy (Baars et al., 1977). At other frequencies we used 3C48, 3C123, or 3C161, with flux densities which have been tied to Cygnus A by, e.g., Baars et al. (1977) and Ott et al. (1994). However, 3C48 as well as many other 3C calibrators do vary over time. Perley (2001, personal communication) developed an internal VLA calibration scale at various epochs; we used the Perley 1999.2 scale for 3C48 to calibrate our data. The flux density values adopted for 3C48 at the various frequencies are listed in Table 2, and shown graphically in Fig. 2. To check the overall calibration scale, we also observed the 3C sources 3C123, 3C161, 3C446, and 3C454.3. At the VLA and Parkes we observed the source J2325-121, which was used as a phase calibrator for some of the VLA observations, and serves simultaneously as a general check on the calibration scale. The flux densities for these sources as derived from our observations are summarized in Table 2. Fig. 2 shows graphs of the radio spectra of the five 3C sources (including 3C48) and J2325-121. The lines are

best-fit spectra to the data points, using (as e.g., Baars et al., 1977): logS v关 Jy兴 ⫽ z 0 ⫹ z 1 log␯ 关MHz兴 ⫹ z 2 log2␯ 关MHz兴.

(3)

Values for z0, z1, and z2 for the various sources are given in Table 3. All derived spectra and observed data points look very decent, which gives us confidence in our Jupiter observations. We note that the only exceptions to this statement are the WSRT 610 MHz points for 3C123 and 3C161; 3C446 at this frequency looks normal. We could not find a reason why the flux density of these two sources at this frequency was so low, and discarded them in our fit of Eq. (3). One may notice the absence of 3C source calibrators at 151 and 843 MHz in Table 2. At these frequencies we relied on the flux density of two 4C sources in Jupiter’s field: 4C ⫺04.89 (PKS 2336-045) and 4C ⫺04.88 (PKS 2332-049). Since these sources were observed simultaneously with Jupiter in the same field with the VLA (74, 327-333 MHz), MOST (843 MHz), WSRT (377 MHz), and CLFST (151

I. de Pater et al. / Icarus 163 (2003) 434 – 448

439

Table 2 Calibrator flux densities Source

Frequency (MHz)

Wavelength (cm)

Telescope

Flux density (Jy)

3C48

74 327 333 377 609 1365 1370 1480 2295 2695 4850 4860 8440 8480 8580 74 327.5 333 609 1365 1370 1480 2295 2695 4850 4860 8440 8480 74 327.5 333 609 1350 1365 1370 1480 2295 2695 4850 4860 8440 8480 74 327.5 333 377 609 1365 1370 1480 2295 2695 4850 4860 8440 8480 74 327.5 333 1365 2695 4850 4860 8440 8480 74 327.5 333 1350 1365 4860

405 91.7 90.1 79.6 49.3 22.0 21.9 20.3 13.1 11.1 6.19 6.17 3.55 3.54 3.50 405 91.6 90.1 49.3 22.0 21.9 20.3 13.1 11.1 6.19 6.17 3.55 3.54 405 91.6 90.1 49.3 22.2 22.0 21.9 20.3 13.1 11.1 6.19 6.17 3.55 3.54 405 91.6 90.1 79.6 49.3 22.0 21.9 20.3 13.1 11.1 6.19 6.17 3.55 3.54 405 91.6 90.1 22.0 11.1 6.19 6.17 3.55 3.54 405 91.6 90.1 22.2 22.0 6.17

VLA VLA VLA WSRT WSRT VLA NRAO 140-foot NRAO 140-foot NASA DSN MPIR 100-m MPIR 100-m VLA VLA NASA DSN VLA VLA VLA VLA WSRT VLA NRAO 140-foot NRAO 140-foot NASA DSN MPIR 100-m MPIR 100-m VLA VLA NASA DSN VLA VLA VLA WSRT Parkes VLA NRAO 140-foot NRAO 140-foot NASA DSN MPIR 100-m MPIR 100-m VLA VLA NASA DSN VLA VLA VLA WSRT WSRT VLA NRAO 140-foot NRAO 140-foot NASA DSN MPIR 100-m MPIR 100-m VLA VLA NASA DSN VLA VLA VLA VLA MPIR 100-m MPIR 100-m VLA VLA NASA DSN VLA VLA VLA Parkes VLA VLA

67.5 ⫾ 1 43.26 42.81 39.83 29.46 16.38 16.34 15.08 10.60 ⫾ 0.15 9.305 5.443 5.432 3.162 3.16 ⫾ 0.03 3.110 334.1 ⫾ 1.8 163.2 ⫾ 2.0 145.7 ⫾ 0.3 75.1 ⫾ 5.2 49.01 ⫾ 0.2 48.31 ⫾ 0.08 45.58 ⫾ 0.17 31.77 27.53 ⫾ 0.24 16.24 ⫾ 0.15 17.59 ⫾ 0.4 8.09 ⫾ 0.2 9.40 98.0 ⫾ 0.8 51.2 ⫾ 0.2 49.88 ⫾ 0.15 23.3 ⫾ 1.6 20.01 20.06 ⫾ 0.15 19.03 ⫾ 1.2 17.86 ⫾ 1 12.53 ⫾ 0.12 11.04 ⫾ 0.07 6.72 ⫾ 0.04 6.31 ⫾ 0.1 3.56 ⫾ 0.11 3.74 ⫾ 0.04 27.8 ⫾ 0.7 14.31 ⫾ 0.04 13.57 ⫾ 0.03 12.1 ⫾ 0.9 9.75 ⫾ 0.7 6.90 ⫾ 0.02 6.33 ⫾ 0.02 6.10 ⫾ 0.02 4.88 ⫾ 0.03 4.62 ⫾ 0.04 4.59 ⫾ 0.12 4.73 ⫾ 0.01 4.75 ⫾ 0.01 4.67 ⫾ 0.05 11.7 ⫾ 1.0 14.80 ⫾ 0.05 14.49 ⫾ 0.04 13.62 ⫾ 0.03 13.25 ⫾ 0.12 13.01 ⫾ 0.34 12.98 ⫾ 0.02 11.93 ⫾ 0.05 11.95 ⫾ 0.10 28.3 ⫾ 0.3 9.13 ⫾ 0.03 8.78 ⫾ 0.02 1.95 ⫾ 0.06 2.21 ⫾ 0.003 0.41 ⫾ 0.01

3C123a

3C161

3C446

3C454.3

J2325-121

a b

We note that 3C123 at the VLA is partially resolved at the higher frequencies. Flux density, adopted as a standard, was derived from Eq. (3) and the parameters in Table 3.

Comments Flux Flux Flux Flux Flux Flux Flux

standard standard standard standard standard standard standard

Flux Flux Flux Flux

standard standard standard standard

Flux standard

Flux standard

Flux standard

Flux standardb

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I. de Pater et al. / Icarus 163 (2003) 434 – 448

marized in Table 1. In this table the first column shows the frequency, column 2 the wavelength, column 3 the telescope used, and column 4 the date of the observation. The total flux density, corrected to the standard geocentric distance of 4.04 AU, is given in column 5, and the total nonthermal (synchrotron) intensity is given in column 6 for each observing session. Column 7 reports the nonthermal flux density averaged over the various observing sessions, at each frequency. The uncertainties in the latter two columns also contain an uncertainty (given in brackets) induced by Jupiter’s (subtracted) thermal contribution. The last column contains the correction factors CT and CNT used in those single dish observations where Jupiter’s spatial extent relative to the telescope beam did require such a correction. Jupiter’s radio spectrum as constructed from all these data is shown in Fig. 4a. It appears that Jupiter’s radio spectrum is very flat at frequencies ⱗ 1 GHz, all the way down to 74 MHz. At higher frequencies the flux density drops off quite rapidly. Fig. 2. Graphs of the radio spectra of 3C48, 3C123, 3C161, 3C446, 3C454.3, and J2325-121 (based upon Table 2). The lines are best-fit curves to the data points, using Eq. (3) and Table 3.

MHz), they are the best calibrators one could wish for. The flux density scale for these sources was derived from the VLA and WSRT data, together with previously published values (Table 4). Spectra for these sources are shown in Fig. 3. Our observed flux density values at 74, 330, and 377 MHz blend in remarkably well, and extend these sources’ radio spectra to lower frequencies. We fitted the data according to Eq. (3), and used the results to determine the flux density at 151 and 843 MHz, which were then used to calibrate Jupiter’s flux density at these frequencies (at 843 MHz only 4C ⫺04.88 was in the field of view). 3. Results The flux densities for Jupiter’s synchrotron radiation as determined during our September 1998 campaign are sum-

4. Discussion 4.1. Model fits The shape of Jupiter’s radio spectrum is determined by the intrinsic spectrum of the synchrotron radiating electrons, the spatial distribution of the electrons and Jupiter’s magnetic field, Iv ⬀

冕

Emax

j共E, ␣ , L兲 B ⬜F

Emin

冉冊

v dE, vc

where Iv is the synchrotron volume emissivity at frequency v. The differential electron flux, j(E, ␣, L) is a function of energy E, pitch angle ␣, and location L in Jupiter’s magnetic field, where L refers to McIlwain’s L-shell parameter. For synchrotron radiating electrons the energy E ⫽ ␥rmec2 (me the electron’s rest mass, ␥r the relativistic correction factor and c the speed of light). B⬜ is the magnetic field strength

Table 3 Spectral parametersa of calibrator sources Source

Freq intervalb (MHz)

z0

z1

z2

3C48 3C123 3C161 3C446 3C454.3 J2325-121 4C ⫺04.88c 4C ⫺04.89c

74–8580 74–8480 74–8480 74–2695 327–8480 74–8440 74–5000 74–5000

1.162 ⫾ 0.001 3.161 ⫾ 0.011 2.023 ⫾ 0.026 2.167 ⫾ 0.068 1.147 ⫾ 0.026 1.996 ⫾ 0.043 1.828 ⫾ 0.085 1.337 ⫾ 0.069

0.855 ⫾ 0.001 ⫺0.12 ⫾ 0.007 0.304 ⫾ 0.018 ⫺0.312 ⫾ 0.048 0.042 ⫾ 0.016 0.043 ⫾ 0.031 ⫺0.311 ⫾ 0.064 0.080 ⫾ 0.055

⫺0.266 ⫾ 0.001 ⫺0.111 ⫾ 0.001 ⫺0.171 ⫾ 0.003 ⫺0.038 ⫾ 0.008 ⫺0.014 ⫾ 0.003 ⫺0.182 ⫾ 0.006 ⫺0.104 ⫾ 0.012 ⫺0.144 ⫾ 0.011

Spectral parameters according to Eq. (3): log Sv[Jy] ⫽ z0 ⫹ z1 log2v [MHz] ⫹ z2 log2v [MHz]. Equation (3) was fitted over the frequency range indicated. c Parameters are derived by fitting the flux densities in Table 4 (the 151 and 843 MHz points were omitted in the fit). a

b

(4)

I. de Pater et al. / Icarus 163 (2003) 434 – 448

441

Table 4 Flux densitiesa of 4C ⫺04.88 and 4C ⫺04.89 Source

Other name

Frequency (MHz)

Wavelength (cm)

Flux density (Jy)

Referenceb

4C ⫺04.88

PKS 2332-049

4C ⫺04.89

PKS 2336-045

74 151 178 330 365 377 408 843 1.400 2.700 4.850 5.000 74 80 151 160 178 330 365 377 408 1400 1410 2650 4850

405 199 169 91 82.2 79.6 73.5 35.6 21.4 11.1 6.2 6.0 405 375 199 188 169 91 82.2 79.6 73.5 21.4 21.3 11.3 6.19

7.49 ⫾ 0.15 4.5 ⫾ 0.2 3.9 ⫾ 0.6 2.43 ⫾ 0.014 2.147 ⫾ 0.043 1.85 ⫾ 0.13 1.83 ⫾ 0.06 1.05 ⫾ 0.03 0.594 ⫾ 0.021 0.31 ⫾ 0.02 0.192 ⫾ 0.014 0.23 ⫾ 0.02 9.65 ⫾ 0.14 8.0 ⫾ 1.1 6.7 ⫾ 0.3 7.1 ⫾ 1.1 5.4 ⫾ 0.7 4.22 ⫾ 0.013 4.155 ⫾ 0.1 3.9 ⫾ 0.3 3.87 ⫾ 0.12 1.461 ⫾ 0.052 1.3 ⫾ 0.15 0.70 ⫾ 0.08 0.493 ⫾ 0.028

This paper This paperc Gower et al. (1967): 4C This paper Douglas et al. (1996): Texas This paper Large et al. (1981): MRC This paperc Condon et al. (1998): NVSS Wall et al. (1976): PKS Griffith et al. (1995): PMN Wall et al. (1976): PKS This paper Slee (1995): Culg (Poss. variable) This paperc Slee (1995): Culg Gower et al. (1967): 4C This paper Douglas et al. (1996): Texas This paper Large et al. (1981): MRC Condon et al. (1998): NVSS Shimmins et al. (1966): PKS Shimmins et al. (1966): PKS Griffith et al. (1995): PMN

a

We used the flux density values as published in the original reference. 4C ⫽ 4C catalogue; Texas ⫽ Texas catalogue; MRC ⫽ Molonglo Reference Catalogue; NVSS ⫽ NRAO VLA Sky Survey; PKS ⫽ Parkes Survey; PMN ⫽ Parkes–MIT–NRAO survey; Culg ⫽ Culgoora survey. c This flux density, adopted as a standard for the CLFST and MOST data, was derived from Eq. (3) and the parameters in Table 3. The error in brackets corresponds to the error in the data. b

perpendicular to the line-of-sight. The integration limits Emin and Emax should span the entire range of electron energies. We typically used Emin ⫽ 0.5 MeV and Emax ⲏ 80 MeV. The integration is

冉冊 冕

v v F ⫽ vc vc

⬁

K 5/3共t兲dt,

(5)

v vc

j共␣e 兲 ⫽ j共␣e ⫽ ␲/2兲

冋冘 X Q 冉sin ␣ ⫺ BB 冊 册 , e

2

n

n

n

sn

e

l

(7)

where K5/3 is the modified Bessel function of order 5/3 and the critical frequency is vc ⫽ 共3/4␲兲 共qB/me 兲␥2r (q ⫽ electron charge). The radial dependence of j(E, ␣, L) is determined via diffusion theory, where the differential electron flux is related to the phase space density f(␮, L), j共E, ␣ e ⫽ ␲ / 2, L兲 ⫽ f共 ␮ , L兲 p 2,

a generic lifetime ␶ ⫽ ␶0. We calculate the diffusion for electrons with ␣e ⫽ ␲/2, and relate the latitude distribution j(␣e) to j(␣e ⫽ ␲/2) via (Roberts, 1965)

(6)

with ␮ the first adiabatic invariant, p the particle’s momentum, and ␣e the pitch angle of the electrons in the magnetic “equatorial plane” (locus of minima in the magnetic field strength, Be, along the field lines). The time evolution of f(␮, L), and hence the electron flux j(E, ␣, L), is calculated using radial diffusion theory (Birmingham et al., 1974; de Pater and Goertz, 1990), with the radial diffusion coefficient DLL ⫽ D0L3, and a generic loss term for particles given by

where Bl is the magnetic field strength at the absorption altitude (edge of the loss cone) and Xn a normalizing factor, so that Qn represents, for each term, the fractional contribution to the equatorial omnidirectional flux (see, e.g., de Pater et al., 1997). The quantity sn ⱖ 0, and each term n is set to zero for particles inside their loss cone. A value sn Ⰷ 0.5 simulates a “pancake” particle distribution, i.e., a distribution where the electrons are concentrated near the magnetic equator. When sn ⫽ 0 the particle distribution is isotropic in pitch angle, except that the loss cone is empty. One typically uses two terms in Eq. (7) (e.g., Roberts, 1976), although both terms undergo a pronounced change at Amalthea’s orbit (de Pater et al., 1997). The energy dependence of Jupiter’s electron flux is usually modeled by (e.g., Van Allen, 1976)

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I. de Pater et al. / Icarus 163 (2003) 434 – 448

Fig. 3. Graphs of the radio spectra for 4C ⫺04.88 and 4C ⫺04.89, using the data in Table 4. The values measured in this paper are indicated by filled symbols. Note that the values at 151 and 843 MHz were derived from the polynomial fit (the solid line) to the data.

冉

j共E, ␣ e ⫽ ␲ / 2, L ⫽ 6兲 ⫽ C 1E ⫺a 1 ⫹

冊

E E0

⫺b

,

(8)

with C1, a, b, and E0 constants to be fitted to the data. The observed flux density, Sv, at frequency ␯ is obtained by integrating Eq. (4) along the line of sight and over the solid angle ⍀B subtended by the source region. In our models we specify the electron distribution at a particular location in Jupiter’s radiation belts, for example in the magnetic equatorial plane at L ⫽ 6 and jovian longitude ␭III ⫽ 140°. From there the distribution is evolved inwards using radial diffusion theory, and as a function of longitude by assuming the electrons to drift around Jupiter along contours of constant magnetic field strength in a multipolar magnetic field configuration, such as the O6 model of Connerney (1992, 1993). The intrinsic electron spectrum softens considerably while electrons diffuse inwards due to synchrotron radiation losses (see, e.g., calculations by de Pater and Goertz, 1990; and Fig. 5 discussed below). The electron spectrum is further modified by, e.g., absorption by satellites, pitch angle scattering, coulomb scattering, and energy degradation by dust (e.g., MogroCampero, 1976; de Pater, 1981; de Pater and Goertz, 1990; Santos-Costa and Bourdarie, 2001). The modifications to the electron spectrum are tightly coupled to the diffusion process, even though the diffusion process itself may not depend on energy (and indeed usually is taken to be independent of energy). There are many parameters to tweak (e.g., de Pater and Goertz, 1990): the radial diffusion parameters D0 and ␶0 (assuming a fixed L-dependence), the intrinsic energy distribution of the electrons at a particular L-shell, satellite absorption/scattering effects, energy degradation by dust, lifetime of electrons against pitch angle

scattering, and coulomb losses. Modelers usually try to match the integral particle flux J(E ⬎ 20 MeV, ␣e ⫽ ␲/2, L) as observed by the Pioneer spacecraft (Van Allen, 1976) and tweak the various parameters to try to match all observed synchrotron radiation characteristics (i.e., spatial distribution of the radio emission, beaming effect, wavelength dependence, and polarization characteristics). It is clear that Jupiter’s radio spectrum is thus determined by the energydependent spatial distribution of the electrons within the magnetic field, which in turn is determined by a combination of the energy-dependent loss mechanisms, and the processes that lead to diffusion across L-shells and in pitch angle. We compare three different models to our data: (1) De Pater (1981) and de Pater et al. (1997) developed a model as described above, where the L-dependence of Jupiter’s electron distribution was derived from an analytical model for radial diffusion based upon the Pioneer data (Goertz et al., 1979), augmented by losses due to synchrotron radiation and absorption/scattering by satellites and Jupiter’s ring. The authors adopted an energy spectrum for the particle flux using a ⫽ 0, b ⫽ 1.6 and E0 ⫽ 100 MeV in Eq. (8), D0 ⫽ 3.5 ⫻ 10⫺9 s⫺1, and ␶0 ⫽ 4 ⫻ 106 s. After integration of the radio emission along the lines of sight a full 2-D image of the radiation is produced. At the time, these models gave a reasonable fit both to the radio spectrum between 0.6 and 5 GHz and to the VLA radio images at 1.4 GHz. The spectrum of the electron flux seemed somewhat harder, though, than measured by the Pioneer spacecraft (Van Allen, 1976). As noted by de Pater and Goertz (1990), however, there is enough uncertainty in the spacecraft data from different instruments that this may not be a big concern. The resulting longitude-averaged radio spectrum is superposed on Fig. 4a as a dashed line. We note that the integration limits in Eq. (4) in this model were taken as Emin ⫽ 0.5 MeV and Emax ⫽ 80 MeV. (2) De Pater and Goertz (1990) developed a numerical code for radial diffusion of energetic electrons, which they used to simulate Jupiter’s radio emission. They adopted an energy spectrum for the electrons based upon spacecraft data, with a ⫽ 1.2, b ⫽ 3.5, and E0 ⫽ 100 MeV at L ⫽ 6 (Eq. (8)). Good fits to the integral particle flux J(E ⬎ 20 MeV, ␣e ⫽ ␲/2,L) were obtained with D0 ⫽ 3.5 ⫻ 10⫺9 s⫺1 and ␶0 ⫽ 8.6 ⫻ 106 s. The synchrotron emission was then calculated at every point in Jupiter’s magnetic equator (assumed to be dipolar in the calculation), and the emissions were integrated along the line-of-sight to produce a onedimensional (equatorial) scan of the radio emission. From fits to both the radio and spacecraft data the authors determined an optimal set of parameters for the diffusion coefficient and loss terms due to generic losses, satellite absorption, pitch angle and coulomb scattering, and energy degradation by dust. We extended their energy degradation out to the satellite Thebe and assumed the product of particle radius ␶d and ring optical depth in the gossamer rings ␶R to be rd␶R ⬇ 5 ⫻ 10⫺5 ␮m. In the main ring we used the

I. de Pater et al. / Icarus 163 (2003) 434 – 448

443

Fig. 4. Jupiter’s radio spectrum as observed in September 1998. (a) Models (1), (2), and (3) (see text), normalized to the 1.4 GHz data point, are superposed, as described in the text. The spectral parameter E0 ⫽ 100 (Eq. (8)) in all models, and the diffusion parameter D0 ⫽ 3.5 ⫻ 10⫺9 s⫺1. The parameters a, b, ␶0, and Emax (Eq. 8) are indicated for each model. (b) Superposed are three model calculations (calculations as in model (2)), where the dot– dashed line is the same model as in (a) (a ⫽ 1.2, b ⫽ 3.5, E0 ⫽ 100, D0 ⫽ 3.5 ⫻ 10⫺9 s⫺1, and ␶0 ⫽ 8.6 ⫻ 106 s). The integration limit is Emax ⫽ 500 MeV, but there is no difference for Emax ⫽ 80 MeV or 500 MeV. The dashed and solid lines are calculations where the parameters a and ␶0 have been changed (Emax ⫽ 80 MeV): In both dashed and solid lines a ⫽ 0.4; for the dashed line ␶0 ⫽ 8.6 ⫻ 106 s, and for the solid line ␶0 ⫽ 6 ⫻ 107 s. (c) The integral particle flux, J(E ⬎ 20 MeV, ␣e ⫽ ␲/2, L), as a function of jovian distance (or effectively L-shell) for the same calculations as shown in (b). The y-axis scale is in electrons cm⫺2 s⫺1, but one can scale the model curves up and down by choosing different values for C1 in eq. (8). The filled circles represent the observed integral particle flux as measured by the Pioneer 11 spacecraft (Van Allen, 1976).

same value as de Pater and Goertz, rd␶R ⬇ 5 ⫻ 10⫺4 ␮m. The radio spectrum based upon this model is superposed on Fig. 4a by the dot– dashed line. The integration limits in Eq. (4) were taken between 0.5 and 500 MeV, but with a spectrum as steep as this one the precise value for Emax is not important (spectra are essentially the same for Emax ⫽ 80 or 500 MeV). (3) De Pater and Brecht (2001) replaced the analytic expression for the radial dependence of electrons in de Pater

et al. (1997) model by the numerical version of de Pater and Goertz (1990). They thus calculated the entire 2D radio brightness, in contrast to de Pater and Goertz (1990), who only calculated the radiation from Jupiter’s main radiation ring. The electron spectrum in de Pater and Brecht’s model was approximated by a ⫽ 0.4, b ⫽ 3, and E0 ⫽ 100 MeV, a set of parameters which together with the diffusion coefficient (D0 ⫽ 3.5 ⫻ 10⫺9 s⫺1) and a typical particle lifetime against local losses of ␶0 ⫽ 6 ⫻ 107 s match both the

444

I. de Pater et al. / Icarus 163 (2003) 434 – 448

Fig. 5. Graphs of the differential electron flux resulting from model (3). The flux scale can be changed by changing C1 in Eq. (8). The x-axis is in jovian radii at a jovian longitude ␭III ⬇ 140°. (a) Electron flux j(E, ␣e ⫽ ␲/2, L) as a function of distance for different energies, as indicated. (b) Electron flux j(E, ␣e, L) as a function of the particle’s equatorial pitch angle, ␣e, at three approximate locations in the magnetosphere, as indicated. (c) Electron flux j(E, ␣e ⫽ ␲/2, L) as a function of energy at different (approximate) L-shells.

spacecraft and radio data reasonably well. As in model (2), we included energy degradation by Jupiter’s ring in this model; however, we used rd␶R ⬇ few ⫻ 10⫺5 ␮m in the main ring, and a factor of 10 smaller in the halo and gossamer (Amalthea ⫹ Thebe) rings to better match the spatial distribution of Jupiter’s radio emissions (location of the main radiation peaks). These numbers actually agree better with optical and infrared measurements of the Jovian ring than the values used in model (2) (Showalter et al., 1987; de Pater et al., 1999). Changing rd␶R, however, does not noticeably affect the resulting radio spectrum. Energydependent satellite absorption effects, pitch angle scattering,

and coulomb losses were all incorporated as in models (1) and (2). The longitude-averaged result from this model is shown by the solid line in Fig. 4a. The integration limits Emin ⫽ 0.5 MeV and Emax ⫽ 160 MeV. The value of Emax does influence the high-frequency part of Jupiter’s radio spectrum: Choosing Emax ⫽ 80 rather than 160 MeV would lower the 10-GHz spectral point, relative to the 1.4 GHz point, by 0.5 Jy. We conclude that the shape of Jupiter’s radio spectrum depends on the intrinsic energy distribution of the electrons at a particular L-shell, combined with the detailed diffusion process across L-shells. This diffusion process, including

I. de Pater et al. / Icarus 163 (2003) 434 – 448

any energy-dependent loss mechanisms (such as scattering, absorption, and radiation), leads to the energy-dependent spatial distribution of the electrons within Jupiter’s magnetosphere that determines the radio spectrum. Depending on the electron energy spectrum, the choice of the integration limit Emax in Eq. (4) can be important at high frequencies (we always kept Emin at 0.5 MeV). Finally, one should calculate the full 2D radio brightness distribution (models 1 and 3), rather than only the equatorial emissions (model 2); the off-equatorial emissions contribute a significant fraction of the total emission, and, as expected, do influence the radio spectrum. To better illustrate this effect, we superpose three model (2) calculations on Fig. 4b: the dot– dashed line is the same as in Fig. 4a, i.e., a ⫽ 1.2, b ⫽ 3.5, E0 ⫽ 100 MeV, D0 ⫽ 3.5 ⫻ 10⫺9 s⫺1, ␶0 ⫽ 8.6 ⫻ 106 s, and Emax ⫽ 500 MeV. The dashed line is a calculation with a ⫽ 0.4, b ⫽ 3, E0 ⫽ 100 MeV, D0 ⫽ 3.5 ⫻ 10⫺9 s⫺1, ␶0 ⫽ 8.6 ⫻ 106 s, and Emax ⫽ 80 MeV; the solid line is a calculation with a ⫽ 0.4, b ⫽ 3, E0 ⫽ 100 MeV, D0 ⫽ 3.5 ⫻ 10⫺9 s⫺1, ␶0 ⫽ 6 ⫻ 107 s, and Emax ⫽ 80 MeV. If the integration limits for the latter two curves were extended up to 500 MeV, the 10 GHz model points would be raised by ⬃1 Jy (relative to the 1.4 GHz point), and the 0.1 GHz model point would be lowered by ⬃0.5 Jy. In Fig. 4c we show the integral particle flux, J(E ⬎ 20 MeV, ␣e ⫽ ␲/2, L), as a function of distance for the same calculations, in “arbitrary” units. Each of these J curves can be scaled up and down by adjusting the parameter C1 in Eq. (8). The filled circles represent the data points as measured by Pioneer 11 (Van Allen, 1976). If a ⫽ 0.4, a reasonable fit to J(E ⬎ 20 MeV, ␣e ⫽ ␲/2, L) can only be obtained if ␶0 ⬇ 6 ⫻ 107 s, as in model (3). The solid lines in Figs. 4b and 4c show model (2) with ␶0 ⫽ 6 ⫻ 107 s. We note that this model does not fit the observed radio spectrum, in contrast to the full 2D model (3). In Fig. 5 we show different curves for the differential electron flux from Model 3 that best fits all the data (synchrotron radiation and integral particle flux from Pioneer 11). Fig. 5a shows j(E, ␣e ⫽ ␲/2, L) as a function of distance for different energies. Fig. 5b shows j(E, ␣e, L) as a function of the particle’s equatorial pitch angle, ␣e, at three locations in the magnetosphere, and Fig. 5c shows j(E, ␣e ⫽ ␲/2, L) as a function of energy at different L-shells. Several processes affect the energy distribution, the dominant of which are as follows: particles gain energy while diffusing inwards, and they lose energy and/or are removed from the radiation belts due to synchrotron radiation losses (most severe at high energies), pitch angle scattering (most severe at low energies), and absorption by moons, the ring, and Jupiter’s atmosphere. The turnover at low energies at L ⬃ 1.6 is caused primarily by pitch angle scattering (see, e.g., graphs and equations by de Pater and Goertz, 1990). We note that a hardening of the intrinsic electron energy spectrum at L ⫽ 6, an increase in the diffusion coefficient or an increase in the particle lifetime against local losses or pitch angle scattering harden the radio spectrum. The absorption

445

effects by Amalthea and Thebe appear to soften the spectrum slightly. 4.2. Comparison of spectra obtained in 1994 and 1998 It is interesting to compare spectra taken at different times to evaluate suggestions that Jupiter’s radio spectrum may change over time. If it does change, the question arises whether or not it is correlated with time variations in Jupiter’s total flux density. Klein et al. (1989, 2001) have measured Jupiter’s 13-cm flux density since the early seventies, and found clear long-term time variations in the flux density (time scales of years). Bolton et al. (1989) noticed that the long-term time variations were highly correlated with the solar wind ram pressure, and de Pater and Goertz (1994) were able to model the variations in Jupiter’s flux density by varying the electron flux at L ⫽ 20 –50 in proportion to 2 公Nswvsw , where Nsw and ␯sw are the number density and velocity of solar wind particles, respectively. Dunn et al. (2003) show that the correlation of Jupiter’s flux density with solar wind parameters breaks down after ⬃1990, which suggests that processes inside Jupiter’s magnetosphere may be more important than previously assumed. Galopeau and Gerard (2001) were the first to conclusively show that Jupiter’s radio spectrum varies over time, in addition to the one-time event of the Comet Shoemaker– Levy 9 impacts. Galopeau and Gerard measured Jupiter’s emission over a 5-yr time span at frequencies of 1.41, 1.67, and 3.30 GHz, and found that the spectral index ␣ varied between ⫺0.7 and ⫹0.2, where S␯ ⬀ ␯␣. To extend this study in frequency, we compare our 1998 spectrum in Fig. 6a with that taken in June 1994, before Comet Shoemaker– Levy 9 crashed into Jupiter (de Pater et al., 1995; Millan et al., 1998). Fig. 6b shows a difference spectrum. Note that we considered the original error bars of the nonthermal flux densities in this latter graph (Table 1); i.e., we ignored the uncertainties introduced by subtracting the thermal contribution, since this does not affect the relative change in flux density (for both data sets we used the thermal disk temperatures as specified in Table 4). The plots clearly show a considerable change between the two years. The spectrum below 2 GHz is much flatter and lower in absolute intensity in 1998 compared to 1994. At 8 GHz the flux density did not change much, and at 5 GHz the flux density was higher in 1998 than in 1994. Although the correction term CNT at 5 GHz was quite large in 1998 (CNT ⫽ 1.43, Table 1), it cannot explain the difference in observed flux densities. Since Jupiter’s apparent size was much larger in 1998 (near opposition) than in 1994, the correction factor must be larger. If the correction factor had been the same, the flux densities at 5 GHz would have been equal. We overplotted on Fig. 6b a linear fit to all the data, ⌬S ⫽ x ⫹ y log ␯ Jy,

(9)

with x ⫽ ⫺3.5 ⫾ 0.3 MHz, y ⫽ 1.0 ⫾ 0.1, and ␯ in MHz. Our results show that Jupiter’s spectrum changed signifi-

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Fig. 6. Jupiter’s radio spectrum as measured in September 1998 (in red) and June 1994 (in blue; from de Pater et al., 1995, and Millan et al., 1998). Superposed on the 1998 data (red lines) is model (3), for two different Emax (integration limit in Eq. (4)). Superposed on the 1994 data (blue dashed line) is the same model, but with the generic lifetime against local losses ␶0 ⫽ 8.6 ⫻ 106 s, instead of ␶0 ⫽ 6 ⫻ 107 s. The solid line is a calculation with a ⫽ 0.6 instead of 0.4 in Eq. (8), keeping ␶0 ⫽ 6 ⫻ 107 s. In all calculations E0 ⫽ 100, and D0 ⫽ 3.5 ⫻ 10⫺9 s⫺1. (b) The difference in Jupiter’s radio spectrum between 1998 and 1994 (1998 minus 1994). Superposed is a best fit curve according to Eq. (9).

cantly between 1994 and 1998, with the largest change at the lowest frequencies. In fact the difference spectrum appears to be linear in log ␯. We modeled the 1994 and 1998 spectra in Fig. 6a; the various models are superposed on the graph. The red line superposed on the 1998 spectrum is model (3), discussed above. The difference between the dashed and solid lines is caused by a difference in the cutoff energy Emax in Eq. (4) (80 versus 160 MeV). A good match to the spectral shape of the June 1994 data can be obtained either by decreasing the generic lifetime against local losses from ␶0 ⫽ 6 ⫻ 107 s to ␶0 ⫽ 8.6 ⫻ 106 s while keeping a ⫽ 0.4 (blue dashed line, superposed on the 1994 spectrum), or by changing the intrinsic energy spectrum of the particles, such that a ⫽ 0.6 rather than 0.4, while keeping ␶0 ⫽ 6 ⫻ 107 s (blue solid line on Fig. 6a). In all models we kept b ⫽ 3, E0 ⫽ 100 MeV, and D0 ⫽ 3.5 ⫻ 10⫺9 s⫺1. For the 1994 spectra we kept Emax ⫽ 80 MeV. The model spectra were all scaled to the 20 cm data points by varying C1 in Eq. (8). It is interesting to consider the consequences of a softening in the electron spectrum (a ⫽ 0.4 3 0.6) and/or a decrease in the particle lifetime against local losses (␶0 ⫽ 6 ⫻ 107 3 8.6 ⫻ 106 s), which both result in a decrease in the radio intensity, opposite to what we observed. We note that Galopeau and Gerard (2001) typically did observe a softening in the radio spectrum when the total radio intensity decreased at 3.3 GHz, consistent with either a softening of the electron spectrum or decrease in the particle lifetime against local losses. They did not find a correlation between spectral index and radio intensity at lower (1.4 GHz) fre-

quencies, while we find below a few GHz that the spectrum hardens when the radio intensity decreases. So essentially it appears as if C1 in Eq. (8) decreases when the spectrum hardens. Since, in our observations, the radio intensity at the low frequencies decreased the most between 1994 and 1998, we suspect that pitch angle scattering, coulomb scattering, and/or energy degradation by dust play an important role in Jupiter’s inner radiation belts. We note that an overall increase in pitch angle scattering may prevent particles from coming into the inner radiation belts (de Pater and Goertz, 1990), so the processes that lead to these relatively large changes in radio intensity and spectral shape must operate only, or primarily, in the synchrotron radiating region itself. Such a notion is not new: Bolton (1991) arrived at a similar conclusion in an attempt to explain short-term time variability in Jupiter’s synchrotron radiation.

5. Conclusion In this paper we reported observations of Jupiter and several calibrator sources at frequencies spanning a range from 74 MHz up to 8 GHz. We used eleven different telescopes spread around the world, each uniquely suited to observe at a particular frequency. We thus obtained the first spectrum of Jupiter spanning such a broad range in frequency. We compared our spectrum with one obtained in June 1994, spanning frequencies from 330 MHz up to 8 GHz. The spectral shape changed drastically between the

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two dates: in 1998 the intensity shortwards of 5 GHz had decreased significantly compared with June 1994, particularly towards the lower frequencies. Based upon a comparison of our spectra with calculations of Jupiter’s synchrotron radiation, we suggest that pitch angle scattering, coulomb scattering, and/or energy degradation by dust play a much more important role in Jupiter’s inner radiation belts than hitherto assumed. Although observations at frequencies below a few hundred MHz become increasingly more difficult to carry out, certainly on a routine basis, it would be worthwhile to regularly monitor Jupiter at these frequencies to investigate its time variability. Such a program would yield a wealth of information regarding the relative importance of the variety of physical processes that operate in Jupiter’s inner radiation belts. Two new upcoming facilities would be ideal for this experiment: the ATA (⬃300 MHz–10 GHz), the Allen Telescope Array, which is being built at the Hat Creek Radio Observatory by the SETI institute and the Radio Astronomy Laboratory at the University of California in Berkeley, and LOFAR (10 MHz–250 MHz), the Low Frequency Array, which is being developed by radio astronomy groups in the Netherlands (ASTRON) and the USA (Naval Research Laboratory and MIT Haystack Observatory).

Acknowledgments This research was supported by NASA Grant NAG56890 to the University of California, Berkeley. Some of the data were obtained with the VLA (Very Large Array) of the National Radio Astronomy Observatory (NRAO), which is operated by Associated Universities, Inc., under a cooperative agreement with the National Science Foundation. The JPL contribution to this paper was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The WSRT is operated by ASTRON with financial support from the Netherlands Organization for Scientific Research (NWO). The MOST is operated by Sydney University with support from the Australian Research Council.

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