6 Faraday rotation

The simple model described above gives fits to the data with a typical residual of $0.5~\rm {TU}$. An elementary calculation shows that with such a residual, and a reasonable model of the Earth's magnetic field, we should be able to calculate the ionospheric rotation measure with a typical accuracy of about 0.2 ${\rm rad/m^2}$ at 327 MHz. Such a capability would permit correction of the ionospherically-induced rotation of the plane of polarization to better than 10$^\circ$- more than sufficient to permit meaningful polarimetry. However, tests of this capability were difficult to schedule because of observing pressure at the VLA and the need to reinstall a GPS receiver to obtain simultaneous GPS data. In particular, a valid test requires observation of a strongly polarized source over an extended period during which the Faraday rotation of the plane of polarization changes fairly quickly. The validity of the method is most convincingly demonstrated if the rate of change of the observed position angle dramatically changes during the observation period - such as would normally occur during sunrise or sunset.

To test the validity of our method, we obtained eight test observations between October 1996 and June 1999 of the strong and highly polarized pulsar PSR 1932+109. Of these eight trials, five showed insufficient change in the observed position angle of the pulsar over the duration of the observation to permit a useful test of the method. The three successful tests are summarized in Table 2.

 

 

Table 2: Observing Log of VLA Faraday rotation observations

Date	Time Range	VLA Config.	RMS Fit
	MST		Deg.
04 Apr. 1997	04:00-10:30	B	4 $.\!\!^\circ$9
27 Aug. 1998	16:00-21:30	B	8 $.\!\!^\circ$3
24 Jun. 1999	02:00-07:30	A	9 $.\!\!^\circ$4

The results of our first useful test are shown in Fig. 9. Observations began at 4 AM local time, well before sunrise, and continued until 10:30 AM. This time span permitted observations for a significant period before sunrise and through the majority of the sunrise period in during which the ionosphere's TEC greatly increases. The observations were taken in the gated pulsar mode so as to ensure maximum SNR, although this pulsar's emission is sufficiently strong that the experiment can be run in a normal ungated mode.

In Fig. 9 for the first 1.5 hours, essentially no change is seen in either the predicted or observed plane of polarization, reflecting the fact that these data were taken before dawn. At 5:30 AM local time, both observed and predicted position angle suddenly begin to increase, with this trend continuing steadily until the end of the experiment. In Fig. 9 we plot single five-minute integrations of the observed and modeled position angles. The data were taken in B-configuration (maximum arm length $\sim$6 km), which at times can be difficult to calibrate if the ionosphere is in a disturbed state. However, no disturbances were noted, and the calibration (using nearby point-like objects) of the phase, amplitude, and polarization of the antennas proceeded smoothly. Images of the pulsar in Stokes' I, Q, and U were made every 5 min to permit detailed tracking of the ionospheric rotation measure. The linear polarization of the pulsar is $> 80\%$, and it does not rotate appreciably during the each pulse, making measurement of the pulsar's polarized emission very straightforward.

Figure 9: The change of observed and predicted plane of polarization of the pulsar PSR 1932+109 through dawn on 4 April 1997. The standard deviation between the observation and the model for a single 5-min integration is 4 $.\!\!^\circ$7. The standard deviation of the mean of the 59 integrations, i.e. $4\hbox{$.\!\!^\circ$ }7/\sqrt{58}$, is 0 $.\!\!^\circ$6

The data from the GPS receiver located at the VLA site were processed through two special AIPS programs, LDGPS, and APGPS. The former program loads the GPS data into a GP table which is attached to the AIPS database containing the visibility data. This table contains the time, satellite PRN number, satellite azimuth and elevation, and the observed TECTAU and TECPHS. The latter program does the model fit, and calculates the ionospheric rotation measure (RM) for every source in the AIPS database, the results of which are then written into the AIPS CL table.

It was felt that more data were needed and two more successful tests were made using essentially the same setup. The second test was made on 27 August 1998 (see Fig. 10). This test ran through sunset, however, quite abnormal ionospheric conditions occurred. The TEC began to decrease as normally happens at sunset but then, about an hour after sunset, it increased to above its daytime level and a very strong north-south gradient set in. This unusual behavior was mirrored satisfactorily in both the predicted and observed Faraday rotation except for a large spike in the model prediction at 03:55 IAT. This was caused by the fact that the North-South gradient was extremely high, the highest that we have ever observed, while at the same time the GPS satellites were mostly east and west of the VLA site, making the modeling of this gradient rather unstable.

Figure 10: Same as Fig. 9 but for 27 August 1998. In this case the standard deviation for single integrations is 8 $.\!\!^\circ$3 and the standard deviation of their mean is 1 $.\!\!^\circ$2

A third test was run through sunrise on 24 June 1999 (see Fig. 11). In this case the ionospheric measurements were completely normal with the TEC being low and constant until sunrise, and then rising rapidly. The Faraday rotation predicted by the model reflected this situation. However, the observed rotation was about 30$^\circ$ above the prediction at the beginning of the test, some four hours before sunrise. The observed rotation then decreased and came into agreement with the prediction about an hour before sunrise, and the predicted and observed rotations then rose together after sunrise. Also plotted in Fig. 11 are the calculated values using values of the TEC taken from archived data stored at the Crustal Dynamics Data Information Center at the Goddard Space Flight Center. We downloaded the JPL Global Ionospheric Model Maps, and used these to calculate the predicted Faraday rotation, using the AIPS program TECOR.

The two models are in excellent agreement during the pre-dawn hours, but diverge significantly about two hours after sunrise. We note that the pulsar was at very low elevations after this time (25$^\circ$ at IAT = 13, dropping to 9$^\circ$ at the end of the experiment), so the differences may be related to the differing geometries utilized by these models. Because the global data are heavily averaged, travelling ionospheric disturbances, including sharp ionization waves, will be smoothed out. This explains the smoothness of the JPL model predictions. On the other hand, our model, utilizing data averaged only on 5 min timescales, is sensitive to TIDs and other localized disturbances, and this shows in the fluctuations in our model predictions. We note that neither model is in agreement with the observed data prior to sunrise - we have no viable explanation for the discrepancy at the beginning of this test.

More low-frequency observations of highly polarized objects will be needed for a better understanding of the limitations of our, and of the global, models.

Figure 11: Same as Fig. 9 but for 24 June 1999. From 11:00 IAT until the end of the test, the standard deviation of the single integrations was 9 $.\!\!^\circ$4. Also shown are the predictions using data from JPL's global ionospheric maps