3. Brightness and the magnetic field
In this section we concentrate on understanding the brightness of the magnetic equatorial radiation and the character of the magnetic field.
3.1. Brightness and in the dipole equator
As noted by Dulk et al. (1997) and in Paper I, at each point around Jupiter's magnetic equator, the magnetic declination (the angle between the plane perpendicular to the field and the plane perpendicular to Jupiter's rotational axis) is
where are the azimuthal and colatitudinal components of the field. The synchrotron radiation from relativistic electrons with pitch angles near 90o is concentrated in the plane perpendicular to the field, while the Earth is located in the plane perpendicular to the rotational axis when , and is 2.9o south of it when .
We have calculated the longitude profile of in three models of the magnetic field, O6, H4 and VIP4. In Paper I (Fig. 2) we compared these profiles at the locus in the magnetic equator where G, and showed that the H4 model, while not perfect, is more consistent with the observations than are the other two; therefore we use it to illustrate the relationship between and brightness.
The middle panel of Fig. 2 shows calculated from the H4 model. When longitudes where is positive are on the east limb, the beam of magnetic equatorial radiation is centered northward of the Jovigraphic equator, and when they are on the west limb the beam of radiation is centered southward of the Jovigraphic equator.
Most striking is near , the location of the large excursion of to its maximum value of +15o: little intensity is received at the Earth from that . The Earth would have to be 15o above or below Jupiter's equatorial plane to be directly in the beam of radiation.
When , as in the right panel, the Earth is equally far from the beam at east and west limb passage of all longitudes, thus the east and west limb brightness profiles are nearly equal. In the middle panel of Fig. 2, corresponds to the Earth being along the central line, the zero of . Then it is the absolute value of that determines the brightness observed, the same on east and west. The general correlation of brightness and is very good. The peak brightness occurs at , one of the two zero crossings of , a time when the Earth is directly in the beam. However, there is no similar peak at when again the Earth is in the beam. This difference is attributable to the slopes at the two zero crossings, the slope being inversely proportional to the depth along the line of sight of the radiation beamed toward Earth, and hence inversely proportional to the optical depth and the brightness. As shown in Paper I, an improved magnetic field model would have a steeper slope at than is in the H4 model, and much steeper than in the O6 or VIP4 models.
Now consider the left panel, where . When the large excursion at is on the east (west) limb, the Earth is 2.9o farther from (closer to) the beamed radiation. Thus the eastern brightness minimum is deeper than is the western brightness minimum. Noting that is negative everywhere from to 60o, and in that range the beam of magnetic equatorial radiation is southward directed on east limb passage, we can understand why the east limb brightness is the larger.
In the middle panel of Fig. 2 the short horizontal lines mark the zero crossings of in the H4 model, and there are vertical lines at , partly solid and partly dotted. The line at -2.9o (+2.9o) relates to east (west) limb brightness, and the distance of this line from the curve should be proportional to the brightness at east (west) limb passage. In general there is a very good proportionality between the two, for both limbs.
The distance between the curve and the vertical lines is the angle between the viewing direction and the direction of maximum intensity. For sources on the east and west limbs, the respective angles and are given by the relations:
The solid portions of the vertical lines denote longitudes where the east (west) limb brightness should be the larger, and this is borne out. The stars mark the longitudes where , i.e. where the Earth is directly in the beam, and there should be a maximum of intensity. Indeed, the principal maxima of brightness are in the region near 190o, that of the east limb is near the predicted value of 210o and that of the west limb is near the predicted value of 170o. On the other hand, for the reason discussed above, the predicted maxima near 60o do not exist.
The comparison in Fig. 2 is a qualitative one because the brightness profiles come from the dipolar magnetic equator, while the curve relates to the warped magnetic equator. Nevertheless the figure shows graphically that the east and west limb brightness variations with longitude and their changes with are well, but not perfectly explained by in the H4 model.
3.2. Brightness and in the true magnetic equator
In our series of two-dimensional images at varying CMLs, we have measured the maximum of intensity on both the east and west limbs. We now describe this maximum brightness and its changes with .
In Fig. 3, the symbols show the brightness temperature measured at the maxima on the east limb (squares) and west limb (diamonds) for three sets of observations. In the top panel, , the symbols for east and west limbs are well separated and the variation with is different at the two limbs. In the center panel , the data for the east and west limbs are moderately separated and they vary with similarly. And in the bottom panel, , the data for the east and west limbs are little separated and vary almost identically with .
The dashed and dot-dashed curves on Fig. 3 are calculations of and from Eqn. 2. They are consistent with the observations in their general form and in the fashion that they approach each other as approaches zero.
The physical relationship between brightness temperature and is unknown, and so the same is true for and . Therefore for the curves of Fig. 3 we have adjusted the amplitude of in degrees (right scale) to match approximately the amplitude of the E limb brightness variation, and smoothed them to the resolution of the observations in . We then used the same scaling for .
In Fig. 3 it is evident that the curves for and do not exactly reproduce the observations, particularly in the range of to 50o. As in Paper I we attribute most of the differences to uncertainties in the H4 model. We have made a similar comparison using the O6 and VIP4 models, and in general the match with the observations is poorer. In particular, the bottom panel of Fig. 4 shows the same brightness data as in Fig. 3, but a comparison is made between the H4 and VIP4 models. At most longitudes, particularly from 60o to 110o, the H4 model provides the better fit.
3.3. Residual asymmetry at
As the assymetry depends on , we expected that the May 1997 observations at the VLA when was as only -0.04o would show no asymmetry. However, the bottom panels of Fig. 3 and 4 show that a small asymmetry remains in the VLA observations, with the brightness at east limb passage being slightly but definitely larger than at west limb passage, particularly near . Similarly we expect no asymmetry from November 1997 observations at the ATCA when . However there is a similarly small asymmetry in the ATCA data, in the opposite sense to the VLA, at a level just above the noise.
There are several possible origins of this remaining asymmetry. 1) It is not real, but within the uncertainties of the observations. This seems not to be the case, at least for the VLA observations. We have estimated the level of the systematic errors by dividing the data into two independent sets. We find the asymmetry in both data sets at about the same level. 2) The asymmetry is real and is the result of the remaining, very small of -0.04o and +0.03o of the two sets of observations. This seems improbable, and there is no substantiation of the idea in existing magnetic field models. 3) The asymmetry is real and is due to a dawn-dusk effect on Jupiter. The VLA observations of May 1997 were made near eastern quadrature, and therefore biased by about 11o toward the dusk side of Jupiter, with the east limb being in 11o of sunlight. The ATCA observations of November 1997 were made near western quadrature, so the west limb was in 11o of sunlight. If this is the explanation, then an effect such as the solar radiation or a dawn-dusk electric field must enhance the synchrotron radiation from the limb that is in sunlight. We know of no mechanism to do this. 4) The asymmetry is apparent, with refraction in the Io torus being different for the east and west limbs. We estimate that when and crosses the east limb, the synchrotron radiation from the magnetic equator passes about 0.15 below the center of the Io torus, whose radius is about 1 and whose maximum density, about cm-3, varies with longitude. The density gradient in the torus then refracts the radiation southward. Conversely when and the region at crosses the west limb, the radiation passes about 0.15 above the center and at a different longitude in the torus, and the radiation is refracted northward. While the refraction at cm is very small, it may be enough to account for the small asymmetry observed.
3.4. Radius and latitude of the magnetic equator of 1.2 G
In the images at varying CMLs, we have measured the radius and latitude and brightness of the maximum of intensity on both the east and west limbs. For the reasons mentioned above and in Paper I, we consider that the radius and latitude of these maxima lie on the warped magnetic equator, and that they follow a locus of constant B of approximately 1.2 G.
Fig. 4 shows the radius, latitude and brightness temperature of the east and west limb intensity maxima as observed at the VLA in May 1997 when was near zero. We have similar observations from the other data sets at . In those observations (not shown here) the variations with of the radius and latitude are almost identical to the variations in Fig. 4. This is because the locus of maximum intensity is fixed by the field, and is completely independent of .
We see in Fig. 4 that, within the errors, the radius and latitude observed on the two limbs are the same. The variation of radius with is not well matched by calculations from either the H4 or VIP4 models. The difference between observations and models is very similar to that found in the 3-D reconstructions of Paper I, i.e. the increase in radius occurs about 30o earlier than in the models, and the latitude near 0o is higher than in the models.
While the form of the variation of radius with from the imaging here is the same as from the 3-D reconstruction, the values here are about 0.1 smaller. The reason for the difference is that images are biased (Dulk et al. 1997). For high resolution images, the bias is inwards because the radiation belt is, approximately, a torus and so there is a contribution to the brightness from the parts of the torus at smaller projected radii, in front of and behind the plane of the sky. There is not a similar bias in the results from 3-D reconstructions (Sault et al. 1997; Dulk et al. 1999).
A notable feature in the latitude data is the rapid shift from about +7o at to -3o at . A similar reduction of the observations with resolution in (here it is ) confirms and emphasizes the rapid shift. This rapid shift is an artifact resulting from two-dimensional imaging that the 3-D reconstruction avoids. The artifact results from the relatively sharp maximum of at : the emission seen when traverses a limb is actually from two regions, one north of the equator at somewhat less than 120o, and the other south of the equator at somewhat more that 120o. Thus there is a sudden switch when rotation dictates that the second region beams toward Earth more effectively than the first. This is evident in images on either side of 120o: the emission has a large north-south extent, with the maximum tending to the south with increasing . When we search for the maximum brightness in the images of limited angular and longitudinal resolution, it shifts rapidly from north to south. It is worth noting that the radius is nearly constant when the latitude shift occurs.
© European Southern Observatory (ESO) 1999
Online publication: June 6, 1999