«Received ; accepted Observat´rio Astronˆmico-Departamento de Geociˆncias Universidade Estadual de o o e Ponta Grossa, Paran´, Brazil a Institute ...»
MEASURING THE SOLAR RADIUS FROM SPACE DURING
THE 2003 and 2006 MERCURY TRANSITS
J. R. Kuhn2
R. I. Bush3
I. F. Scholl2
Received ; accepted
Observat´rio Astronˆmico-Departamento de Geociˆncias Universidade Estadual de
o o e
Ponta Grossa, Paran´, Brazil
Institute for Astronomy, University of Hawaii, 2680 Woodlawn Dr. 96822, HI, USA Stanford University, Stanford, CA, 94305, USA –2– ABSTRACT The Michelson Doppler Imager (MDI) aboard the Solar and Heliospheric Observatory (SOHO) observed the transits of Mercury on May 07, 2003 and November 8, 2006. Contact times between Mercury and the solar limb have been used since the 17th century to derive the Sun’s size but this is the ﬁrst time that high quality imagery from space, above the Earth’s atmosphere, has been available.
Unlike other measurements (e.g. Kuhn et al. 2004) this technique is largely independent of optical distortion. The true solar radius is still a matter of debate in the literature as measured diﬀerences of several tenths of an arcsecond (i.e., about 500 km) are apparent. This is due mainly to systematic errors from diﬀerent instruments and observers since the claimed uncertainties for a single instrument are typically an order of magnitude smaller. From the MDI transit data we ﬁnd the solar radius to be 960”.12 ± 0”.09 (696, 342 ± 65km). This value is consistent between the transits and consistent between diﬀerent MDI focus settings after accounting for systematic eﬀects.
Subject headings: Sun: photosphere, Sun: fundamental parameters, Astrometry.
1. Introduction Observations of the interval of time that the planet Mercury takes to transit in front of the Sun provides, in principle, one of the most accurate methods to measure the solar diameter and potentially its long-term variation. Ground observations are limited by the spatial resolution with which one can determine the instant Mercury crosses the limb.
Atmospheric seeing and the intensity gradient near the limb (sometimes called the “black drop eﬀect”, see Schneider, Pasachoﬀ & Golub (2004) and Pasachoﬀ et al. (2005)) contribute as error sources for the precise timing required to derive an accurate radius. This is the ﬁrst time accurate Mercury’s transit contact times is measured by an instrument in space and improves at least 10 times the accuracy of classical observations (see Bessel (1832) and Gambart (1832)).
About 2,400 observations of those contacts from 30 transits of Mercury, distributed during the last 250 years, were published by Morrison & Ward (1975), and analyzed by Parkinson, Morrison & Stephenson (1980). Those measurements, collected mainly to determine the variations of rotation of the Earth and the relativistic movement of Mercury’s perihelion provided a time-series of the Sun’s diameter. Analyzing this data set, Parkinson, Morrison & Stephenson (1980) marginally found a decrease in the solar semi-diameter of 0”.14 ± 0”.08 from 1723 to 1973, consistent with the analysis of Shapiro (1980) which claimed a decrease of 0”.15 per century. Those variations are consistent with our (Bush, Emilio, & Kuhn 2010) null result and upper limit to secular variations obtained from MDI imagery of 0”.12 per century. Sveshnikov (2002), analyzing 4500 archival contact-timings between 1631 and 1973, found that the secular decrease did not exceed 0”.06 ± 0”.03.
Modern values found in the literature for the solar radius range from 958”.54 ± 0”.12 (S´nchez et al. 1995) to 960”.62 ± 0”.02 (Wittmann 2003). Figure 1 shows published a measurements of solar radius over the last 30 years (for a review see: Kuhn et al.
–4– (2004); Emilio & Leister (2005); Thuillier, Soﬁa, Haberreiter (2005)). Evidently these uncertainties reﬂect the statistical errors from averaging many measurements by single instruments and not the systematic errors between measurement techniques. For example, our previous determination of the Sun’s radius with MDI was based on an optical model of all instrumental distortion sources (Kuhn et al. 2004). The method described here has only a second order dependence on optical distortion (although it is sensitive to other systematics) and may be more accurate for this reason.
2. Data Analysis
The data consist of 1024x1024 pixel MDI-SOHO images from ﬁxed wavelength ﬁltergrams of Mercury crossing in front of the Sun. Images were obtained with a one-minute cadence in both transits, cycling between four instrument focus settings (focus blocks) in 2003 and two in 2006. For each image obtained at a given focus block, we subtracted a previous image of the Sun without Mercury using the same focus setting. This minimized the eﬀects of the limb darkening gradient and allowed us to ﬁnd the center of Mercury (Figure 2) more accurately. In a small portion of each image containing Mercury we ﬁtted a negative Gaussian. We adopt the center of the gaussian as the center of Mercury. Images that were too close to the solar limb were not ﬁtted because the black drop eﬀect inﬂates our center-determination errors. The position of the center of the Sun and the limb were calculated as described in Emilio et al. (2000). A polynomial was ﬁt to the x-y pixel coordinates of Mercury’s center during the image time-series. This transit trajectory was extrapolated to ﬁnd the precise geometric intersection with the limb by also iteratively accounting for the time variable apparent change in the solar radius (−0.221 ± 0.004 mas/minute in 2003 and 0.152 ± 0.003 mas/minute in 2006) between images.
The same procedure was performed independently for each focus block time-series. The –5– contact times were found from a least-squares ﬁt of Mercury’s center position trajectory xF B (t) and y F B (t) to the intersection with the limb near ﬁrst/second and third/fourth contact. Here F B are the focus blocks (3, 4, 5, 6). As a consequence we obtained eight diﬀerent contact times for 2003 and four for 2006, one for each focus block and two for each geometric contact. Figure 3 shows zoom images of the Sun containing Mercury close to the contact times for both 2003 and 2006 transits as well as the Mercury trajectory and the limb ﬁt.
FB 3 and 6 are far from an ideal focus, as Figure 4 upper and Figure 6 show, so that the shape of the limb darkening function is diﬀerent enough that it is diﬃcult to compare corresponding transit points with the near-focus images. Those focus block were not utilized in further analysis. We note also that FB 6 data contained a ghost image of Mercury which added new systematic errors to the data.
The total transit time was thus obtained with an accuracy of 4 seconds in 2003 and 1 second in 2006 from a comparison of FB 4 and FB 5 data. The correction to the radius is
found from the relation (Shapiro 1980):
where ω is the speed of Mercury relative to the Sun, T is the total length of the transit, R⊙ is the apparent value of the solar radius at 1 A.U. from the ephemeris for each transit instant (where we adopt 959”.645 (696, 000 ± 40km) for the nominal solar radius) and ∆T O−C is the diﬀerence between the observed and ephemeris duration of the transit.
Table 1 shows the ephemeris values used in this analysis. We’ve used the NASA ephemeris calculations provided by NASA for the SOHO Mercury transit observations. Ephemeris see http://sohowww.nascom.nasa.gov/soc/mercury2003/ –6– uncertainties for the absolute contact times are not greater than 0.18 seconds based on the SOHO absolute position error. Using the above equation we found the solar radius values to be 960”.03 ± 0”.08 (696, 277 ± 58km) and 960”.07 ± 0”.05 (696, 306 ± 36km) for the 2003 and 2006 transits respectively. Uncertainties were determined from the scatter in the two focus block settings from each epoch. We note that the observed absolute transit contact times are oﬀset 8 seconds in 2003 and 5 seconds in 2006, both inside our 2σ error.
Our accurate determination of the limb transit points, compared with the ephemeris predictions, implies that the orientation of the solar image is slightly rotated from the solar north pole lying along the y axis. We ﬁnd that the true solar north orientation is rotated 7′ ± 1′ counterclockwise in 2003 and 3′.3 ± 0′.3 arcmin in 2006. Table 2 summarizes our results from this analysis.
The solar radius in theoretical models is deﬁned as the photospheric region where optical depth is equal to the unity. In practice, helioseismic inversions determine this point using f mode analysis, but most experiments which measure the solar radius optically use the inﬂection point of the Limb Darkening Function (LDF) as the deﬁnition of the solar radius. Tripathy & Antia (1999) argue that the diﬀerences between the two deﬁnitions are between 200 to 300 Km (0”.276-0”.414). This explains why the two helioseismologic measurements in Figure 1 show a smaller value. The convolution of the LDF and the Earth’s atmospheric turbulence and transmission cause the inﬂection point to shift. The eﬀect (corrected by the Fried parameter to inﬁnity) is at the order of 0”.123 for r0 = 5cm and 1”.21 r0 = 1cm (Djafer, Thuilier & Soﬁa 2008) and depends on local atmospheric –7– turbulence, the aperture of the instrument and the wavelength resulting in an observed solar radius smaller than the true one (Chollet & Sinceac 1999). Many of the published values were not corrected for the Fried parameter which may explain most of the low values seen in Fig. 1.
We analyzed some systematics regarding the numerical calculation of the inﬂection point. The numerical method to calculate the LDF derivative used is the 5-point rule of the Lagrange polynomial. The systematic diﬀerence between the 3-point rule that is used by default in the IDL (Interactive Data Language) derivative routine, is 0.01 pixels for FB 5 up to 0.04 pixels for FB 6. In this work we deﬁned the inﬂection point as the maximum of the LDF derivative squared. From those points we ﬁt a Gaussian plus quadratic function and adopt the maximum of this function as the inﬂection point. Adopting the maximum of the Gaussian part of the ﬁt instead diﬀers 0.001 pixels for FB 3,4,5 and ﬁve times more for FB 6. The quadratic function part is important since the LDF is not a symmetric function. Figure 5 compares the gaussian plus quadratic ﬁt to ﬁnd the function maximum with only ﬁtting a parabola using 3 points near the maximum. They diﬀer 0.01, -0.06,
0.04 and -.05 pixels for FB 3, 4, 5 and 6 respectively. The statistical error sampling for all the available data after accounting for changing the distance is 0.0002 pixels for FB 5 and
0.0003 pixels for FB 6. Figure 6 shows a close up of Mercury for each focus block where we can visually see a spurious ghost image of FB 6 since this FB is far from the focus plane.
That can explain why the numerical methods gave more diﬀerence for this FB. By ﬁtting a parabola over the solar image center positions as a function of time, we also took into account jitter movements of the SOHO spacecraft. Solar limb coordinates and the Mercury trajectory were adjusted to the corrected solar center coordinate frame. This procedure made a correction of order of 0.01 pixels in our deﬁnition of the solar limb. Smoothing the LDF derivative changes the inﬂection point up to.35 pixels that is the major correction we applied in our previous published value in Kuhn et al. (2004). We describe all corrections –8– applied to this value at the end of this section.
MDI made solar observations in 5 diﬀerent positions of a line center operating wavelength of 676.78 nm. Conforming Fig 1. of Bush, Emilio, & Kuhn (2010) at line center, the apparent solar radius is 0.14 arcsec or approximately 100 km larger than the Sun observed in nearby continuum. We used a single ﬁltergram nearby the continuum in this work. Comparison with the composite linear composition of ﬁltergrams representing the continuum diﬀers from 0.001 arcmin. Neckel (1995) suggest the wavelength correction due to the (continuum) increases with wavelength. Modern models conﬁrm this dependence with a smaller correction. Djafer, Thuilier & Soﬁa (2008) computed the contribution of wavelength using Hestroﬀer & Magnan (1998) model. According to this model and others cited by Djafer, Thuilier & Soﬁa (2008) the solar radius also increases with wavelength (see Fig. 1 of Djafer, Thuilier & Soﬁa (2008)). The correction at 550 nm is on the order of 0”.02 and was not applied since the dependence of wavelength for the PSF contribution goes to the opposite direction and the correction is inside our error bars.
3.3. Point Spread-Function (PSF)
The MDI PSF is a complicated function since diﬀerent points of the image are in diﬀerent focus positions (see Kuhn et al. (2004) for description of MDI optical distortions).
In our earlier work the non-circular low-order optical aberrations are eﬀectively eliminated by averaging the radial limb position around the limb, over all angular bins. But in the case of the Mercury transit, only two positions at the limb apply so that it is important to recognize that the LDF (and the inﬂection point) will systematically vary around the limb.
–9– The overall contribution to the PSF of the solar image increases with increasing telescope aperture, and decreases with increasing observing wavelength as noted in the last section.
Djafer, Thuilier & Soﬁa (2008) computed the contribution of the MDI instrument point spread function (PSF) at its center operating wavelength of 676.78 nm assuming a perfectly focused telescope dominated by diﬀraction. The authors made use of the solar limb model of Hestroﬀer & Magnan (1998) to calculate the inﬂection point. They found that the MDI instrumental LDF inﬂection point was displaced 0”.422 inward at ‘ideal focus’ from the undistorted LDF, but the authors used a MDI aperture of 15 cm instead of actual aperture diameter of 12.5 cm. In addition, we do not ﬁnd the pixelization eﬀect cited by the authors.