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Detection of Gravitational Redshift on the Solar Disk by Using Iodine-Cell Technique

With an aim to examine whether the predicted solar gravitational redshift can be observationally confirmed under the influence of the convective Doppler shift due to granular motions, we attempted measuring the absolute spectral line-shifts on a large number of points over the solar disk based on an extensive set of 5188-5212A region spectra taken through an iodine-cell with the Solar Domeless Telescope at Hida Observatory. The resulting heliocentric line shifts at the meridian line (where no rotational shift exists), which were derived by finding the best-fit parameterized model spectrum with the observed spectrum and corrected for the earth's motion, turned out to be weakly position-dependent as ~ +400 m/s near the disk center and increasing toward the limb up to ~ +600 m/s (both with a standard deviation of sigma ~ 100 m/s). Interestingly, this trend tends to disappear when the convectiveshift due to granular motions (~-300 m/s at the disk center and increasing toward the limb; simulated based on the two-component model along with the empirical center-to-limb variation) is subtracted, finally resulting in the averaged shift of 698 m/s (sigma = 113 m/s). Considering the ambiguities involved in the absolute wavelength calibration or in the correction due to convective Doppler shifts (at least several tens m/s, or more likely up to<~100 m/s), we may regard that this value is well consistent with the expected gravitational redshift of 633 m/s.


Introduction
Ever since Einstein (1911) predicted based on his general theory of relativity that the wavelengths of solar spectrum lines should be shifted redward by a proportion of 2 × 10 −6 (≈ 600 m s −1 in the velocity scale) as compared to the laboratory wavelengths, a number of solar physicists have tried to confirm this "solar gravitational redshift" observationally. Although very few people nowadays doubt the practical validity of this prediction from general relativity, given the remarkable success in detecting this effect to a high precision from groundbased experiments, such as the well-known Pound-Rebka experiment using the Mössbauer effect (Pound and Rebka, 1960) or the experiment using two hydrogen masers on the ground and on a spacecraft (Vessot et al., 1980), efforts of solar astronomers toward confirming this gravitational redshift in the spectrum of the Sun have been non-terminating and continuing up to now. Apart from the historical studies in the early time of first 50 years which are reviewed in Forbes (1961), several notable trials using modern techniques and/or observational data of higher quality have been done over the last 50 years, as briefly summarized in Table 1.
While the authors of these studies in Table 1 reported that they somehow detected the predicted redshift even to a quantitatively satisfactory degree, their results are not yet necessarily convincing, since they do not seem to have properly taken into consideration the effect of convective shifts due to granular motions and the closely related problem of line-shift determinations (i.e., definition of centroid wavelength) in case of asymmetric line profiles. Besides, most of these investigations are based on observations only on the disk center (or several disk points at best). Given that the well-known center-to-limb variation of the spectral line shift (i.e., the shift increasing redward toward the limb, often called "solar limb effect," first discovered by Halm (1907)) is regarded to be related with the photospheric velocity fields, it is highly desirable to study absolute line-shifts on a large number of disk points while considering the convective shifts (varying from point to point) at the same time. Namely, if the resulting residual shifts obtained from many points are confirmed to be nearly position-independent, it may suggest that shifts due to granular motions are successfully removed.
Recently, we carried out an extensive spectroscopic study of solar differential rotation based on the spectra collected at a large number (≈ 3000) of points over the solar disk while applying the iodine-cell technique for Doppler-shift measurements (Takeda and Ueno, 2011; hereinafter referred to as Paper I). Since these data are suitable for the purpose mentioned above, we decided to conduct absolute Doppler shift measurements by using these spectra, while paying special attention to their subset observed on the solar meridian where the rotational shift is irrelevant, in order to examine how and whether the solar gravitational redshift may be successfully confirmed by eliminating the shifts of velocity-field origin. This is the aim of this investigation.
The remainder of this paper is organized as follows. The observational data are explained in Section 2, and Section 3 describes our adopted method of analysis for absolute line-shift measurements. The resulting solutions and the simulation for the convective shifts to be subtracted are described in Sections 4 and 5, respectively, based on which the final results for the solar gravitational redshift are discussed in Section 6. The conclusion is summarized in Section 7. Finally, a supplementary section is presented as an appendix, where the limb effect empirically studied by Balthasar (1984) is reformulated for our application.

Observational Data
We use the same observational data as adopted in Paper I, which were obtained on 20 and 21 July 2010 by using the 60 cm Domeless Solar Telescope (DST) with the horizontal spectrograph of the Hida Observatory of Kyoto University (Nakai and Hattori, 1985) and the iodine cell placed in front of the spectrograph slit. One sequence is made up of 13 scanning observations with a step of ∆r = R/12 from the limb to the disk center at a fixed direction angle (θ), and this sequence was repeated at 48 θ-values (with an increment of 7. • 5), resulting in 13×48 points on the disk as a total. The whole set was similarly conducted on each of the two days but with different slit setting; i.e., E-W direction (20 July) and N-S direction (21 July). Since three spectra (at different positions along the direction of the slit) were extracted from one observation, 3744 (= 13×48×3×2) spectra in the 5188-5212Å region (with reference I 2 molecular lines imprinted) covering the entire solar disk were obtained. See Section 2 of Paper I for more details on the observational data. Prior to the analysis, a tentative wavelength calibration was done by using the representative solar lines, and an appropriate λ (wavelength 1 ) vs. x (pixel number) relation was assigned to each spectrum frame.
We examined the instrumental profile (P λ ) by fitting the observed lamp+I 2 spectrum (D obs λ ) with the modeled spectrum (D sim λ ≡ T λ ⊗ P λ ) where T λ is the transmission function of I 2 vapor and ⊗ means convolution. Regarding T λ , we adopt the spectrum (S/N ≈ 1000 with the resolving power of R ≈ 400000) recorded for the I 2 cell at Lick Observatory in the 5000-6300Å region by using the Fourier Transform Spectrometer at the McMath telescope of the National Solar Observatories at Kitt Peak (Marcy and Butler, 1992), which was kindly provided by Dr. G. W. Marcy (see also Takeda et al., 2002).
The instrumental profile was modeled by the sum of nine multiple Gaussian functions as P where v (in km s −1 ) is measured from the line center. Assuming a 0 = 1, we determined the 8 coefficients (a −4 , a −3 , a −2 , a −1 , a 1 , a 2 , a 3 , a 4 ) by requiring the best fit between the modeled and observed spectra in the 5202-5208Å region (Figure 1a). For a reference purpose, we also tried fitting with the Gaussian instrumental profile (∝ exp[−(v/v m ) 2 ]) by finding the best-match solution of the parameter v m . The shape of the resulting P (v) is depicted in Figure 1b, where we can see that the instrumental profile corresponding to our spectra (somewhat more rounded than the Gaussian) does not show any significant asymmetry that may affect absolute determinations of line shifts. The value of FWHM (≈ 5 km s −1 ) also suggests that the wavelength resolution is R ≈ 60000.

Spectrum Modeling
The iodine-cell technique has been extensively applied for precise Doppler shift determinations so far (e.g., spectroscopic search for exoplanets) since the pioneering study of Marcy and Butler (1992) who showed how to accomplish high precision Doppler analysis based on a stellar spectrum with many superimposed I 2 lines. However, its application has been limited essentially to "relative" measurements (i.e., with respect to a template spectrum whose absolute wavelength scale does not need to be precisely calibrated) so far. Since the determination of "absolute" shift is under question in this study, we have to modify Marcy and Butler's (1992) procedure (see Section 4 of their paper) appropriately. Namely, theoretically simulated spectrum with firmly established absolute wavelength scale must be used as the stellar template spectrum to be referenced.
Generally, in analogy with the profile modeling conventionally used in stellar physics, the theoretical solar intensity spectrum with imprinted I 2 lines at a disk point of direction cosine µ[≡ 1 − (x 2 + y 2 )/R 2 ] may be expressed as follows: (1) where k is an arbitrary normalization factor, I intr (λ; µ) is the intrinsic intensity at µ computed from a solar atmospheric model for given ξ (microturbulence) and A 1 , A 2 , A 3 , . . . (elemental abundances), M mt (λ; µ; ζ) is the broadening function corresponding to macroturbulence ζ, T I2 (λ) is the reference I 2 spectrum mentioned in Section 2, P (λ) is the instrumental profile (Figure 1b), and "⊗" denotes convolution (while "·" is usual multiplication). Note that such a line-broadening formulation based on the dichotomous micro-/macro-turbulence is nothing but a too simple modeling of actual velocity fields. Given that information for an adequate functional form of M mt is lacking, we move M mt (λ) to the outside of the square bracket in Equation (1) and further approximate M mt (λ; µ; ζ)⊗P (λ) by a single Gaussian macrobroadening function G(λ; v M ), which is parameterized by the macrobroadening parameter v M (e-folding width) as G(v) ∝ exp(−v 2 /v 2 M ). Thus, we adopt the following model for line-spectrum simulation.
) are the apparent wavelength shifts of solar lines and I 2 lines relative to the reference wavelength system (i.e., wavelengths given Table 2 for the former or in Table 3 for the latter), respectively.

Atomic Line Data for Spectrum Synthesis
As mentioned in Section 3.1, spectrum synthesis calculation for simulating the intrinsic specific intensity spectrum (I intr ) is important, which needs to be well reproduce the observed solar spectrum by adequately adjusting the elemental abundances and the macrobroadening parameter. Regarding the calculation of I intr (λ, µ), we use Kurucz's (1993) WIDTH9 program, which was modified by Y. Takeda to enable spectrum synthesis by including many lines, along with Kurucz's (1993) ATLAS9 solar photospheric model (LTE, plane-parallel model).
In addition, since the role of microturbulence is comparatively less significant in line broadening, ξ = 1 km s −1 is assumed as fixed. As to the atomic line data (wavelengths, oscillator strengths, etc.) necessary for this simulation, we adopt the data compiled by Kurucz and Bell (1995 Kurucz and Bell's database), we neglected (masked) this local feature in χ 2 evaluation. The finally adopted data for the important lines of appreciable strengths are summarized in Table 2, and how the observed disk-center spectrum (taken from the FTS solar spectrum atlas ;Neckel 1994Neckel , 1999 2 can be fitted with the theoretical calculated µ = 1 intensity spectrum by using these atomic data (with appropriately adjusted elemental abundances and the broadening parameter) is displayed in Figure 2, where the lines given in Table 2 are also identified.

Strengths and Formation Depths of Spectral Lines
It is necessary to have knowledge on the strength and related quantities (e.g., line-forming depth) of spectral lines existing in the relevant wavelength range (5188-5212Å), since the convective line shift (its correction is important as will be discussed in Section 5) appreciably depends on such individual line properties. For this purpose, by using the solutions of the elemental abundances derived as by-products of the synthetic spectrum fitting in Section 3.2 ( Figure  2), we inversely computed EW (equivalent width at the disk center) for all lines presented in Table 2, while assuming that each line is single and isolated (i.e., irrespective of whether they are actually blended or not). Besides, we computed two kinds of line-formation depth; one is the mean formation depth averaged over the profile ( log τ ) and the other is the formation depth at the line-center (log τ 0 ), which are defined as follows.
where R λ (≡ 1 − I λ /I cont ) is the line depth of the emergent intensity profile, τ 5000 (τ λ = 1) is the continuum optical depth at 5000Å corresponding to τ λ = 1 at wavelength λ within the profile, and the integration range is over the whole profile.
log τ 0 ≡ log τ 5000 (τ λ0 = 1) where τ λ0 is the optical depth at the line center. (All these quantities are computed for the µ = 1 ray at the disk center.) The resulting values of EW, log τ , and log τ 0 are given in Table 2.

Wavelength Scale of I 2 Reference Spectrum
Regarding the reference spectrum of I 2 molecular lines (T λ ), we adopt Marcy and Butler (1992)'s very high-resolution transmittance spectrum of Lick I 2 cell already mentioned Section 2. This standard I 2 spectrum in the relevant wavelength region, which we also used to fit the spectrum of our I 2 cell (Figure 1a), is given in Table 3 (electronic data). Though the original data are represented in terms of the vacuum wavelength scale (λ vac ), we converted it into air wavelength (λ air ; which is usually used in optical spectroscopy such as in Kurucz and Bell's atomic-line database we adopted) as Here, n air is the refractive index of air in the standard condition represented as a function of the wavenumber k cm −1 [= 10 8 /λ vac (Å)], for which we adopted the formula given in Cox (2000): n air = 1.+6432.8×10 −8 +2949810./(146.×10 8 −k 2 )+25540./(41.×10 8 −k 2 ). (7) While this Lick I 2 spectrum has so far been successfully used for very precisely detecting "relative" variations of radial velocity intended for exoplanet search or stellar seismology, it is not clear whether its "absolute" wavelength scale is sufficiently reliable. Since this point is of primary importance in this study, we examined its precision by comparing it with the theoretical I 2 spectrum by using "IodineSpec" program (Knöckel et al., 2004), which can model the wavelengths of the rovibronic structure of the I 2 B-X spectrum with very high precision (its uncertainty is on the order of only ≈ 30 MHz at ≈ 5200Å, which means a precision of ≈ 5 × 10 −8 or ≈ 15 m s −1 ; cf. their Figure 3), though the intensity distribution is not so precisely reproduced as wavelengths. The comparisons are displayed in Figures 3a (wide view) and 3b (magnified view of the 5196-5197Å region), where the vacuum wavelength scale (λ vac ) is used in the abscissa. We notice from Figure 3b that the Lick I 2 spectrum appears to be slightly shifted redwards relative to the theoretical one. This is quantitatively confirmed by a cross-correlation analysis (Figure 3c), showing that the average shift in this 5196-5197Å bin amounts to 0.0028Å or ≈ 160 m s −1 . We similarly examined all twenty-four 1Å-bins in the 5188-5212Å region and the histogram of the resulting shifts are depicted in Figure 3d, from which we conclude that the absolute wavelength scale of the Lick I 2 spectrum is systematically redder and should be corrected blueward by −167 m s −1 (simple average of 24 values; with a standard deviation of 42 m s −1 ). Besides, given that we transformed the vacuum wavelengths of reference I 2 spectrum into the air wavelengths by applying Equations (6) and (7), the air wavelengths of the atomic lines we adopted for simulating the solar spectrum have to be essentially on the same system. For the purpose of checking the consistency, we examined the λ vac /λ air ratios (or the refractive index in the air; n KB ) for all 719 lines (5188-5212Å region) included in Kurucz and Bell (1995) as where E upper and E lower are the excitation potentials of lower and upper levels in cm −1 , respectively, and λ air is the air wavelength inÅ (all these three kinds of data are presented in Kurucz and Bell's database). The comparisons of such evaluated n KB 's and the corresponding formula values (n formula ) resulting from Equation (7) are depicted in Figure 4. We can recognize from this figure that the consistency is satisfactorily good and the average of the relative difference [ (n KB − n formula )/n ] is only −4.47 × 10 −8 (with a standard deviation of σ = 6.46×10 −8 ) or −13 m s −1 (σ = 19 m s −1 ). 3 Thus, even though small, we formally apply a correction of −(−13) m s −1 to the air wavelengths of I 2 lines (cf. given 3rd column of Table 3) in order to maintain consistency with the wavelengths of atomic lines given in Table 2. (Regarding the sign of the correction, since λ formula air < λ KB air because of n KB < n formula , we have to shift λ formula air slightly redward to make it consistent with Kurucz and Bell's system.) Consequently, we conclude that the λ ref I2 values given in Table 2, which we adopt as the reference (air) wavelengths of I 2 lines, should be corrected blueward by δλ ref I2 = −154 m s −1 (net combination of −167 m s −1 and +13 m s −1 ). More precisely, since the wavelengths of I 2 lines directly affect λ ref , it is included in ∆λ abs as the term of positive sign (+λ ref I2 ; cf. Equation (3)), which means that ∆λ abs should be corrected by +δλ ref I2 = −154 m s −1 . In practice, the whole fitting analysis is conducted by literally using the (air) wavelengths of I 2 spectrum given Tables 3, and this correction is applied to the results at the last stage.

Results
Although our primary interest is only on the meridian points (x ≈ 0) unaffected by the rotational velocity, we carried out our absolute wavelength-shift analysis for all the available data points over the entire solar disk, except for those at the limb because the solutions there were found to be comparatively uncertain and unstable (i.e., the data at r 12 were not used; cf. Figure 2 in Paper I).
Our spectrum covering 5188-5212Å was divided into four segments of 5-6Å [regions (a), (b), (c), and (d), cf. Table 2], and ∆λ i abs was determined for each segment i (i = 1, 2, 3, 4; each corresponding to a, b, c, d, respectively) by applying the procedure described in Section 3.1. An example of how the observed solar+I 2 spectrum matches the modeled spectrum corresponding to the best-fit solutions of the parameters is demonstrated in Figure 5. Then, the (raw) radial velocity of the observed point on the solar surface (V raw r ) and its probable error (ǫ) were computed as and where we applied the correction of −154 m s −1 due to the absolute wavelength scale problem of the adopted I 2 spectrum (Section 3.3), c is the speed of light, and N = 4. Further, we added the heliocentric correction for the earth's motion (∆ hel ; cf. Section 4.1 in Paper I) and obtained V hel r in the heliocentric system as Figure 6 demonstrates, for the meridian case, how the results of V r obtained at different observational times become mutually consistent by applying the ∆ hel correction. The distribution of the probable error (ǫ) for each measurement is shown in Figure 7, which is directly comparable with Figure 9 of Paper I. As can be seen from Figure 7a, ǫ mostly ranges between ≈ 50 m s −1 and ≈ 100 m s −1 ; that is, by ≈ 2-3 times larger than the case of relative measurements in Paper I, reflecting the difficulty of absolute measurements. We do not see any clear dependence of ǫ upon the position (r) on the disk, except for a slight decrease near to the limb (Figure 7b). The results of our analysis [observational time, (r, θ) as well as (x, y) coordinates, V raw r , ∆ hel , and V hel r ] are presented in Table 4 (electronic data) for the meridian case of our concern, which will be discussed in Section 6.

Line Shift due to Convective Granular Motions
Although we have derived the absolute line-shift (V hel r ) reduced to the rest system of the Sun at each point on the disk, it alone is still insufficient for our main purpose of quantitatively confirming the gravitational redshift, since the systematic shift due to convective velocity fields has still to be subtracted.
A hot up-rising bright bubble (granular cell) originating from the deeper convective zone gradually decelerates its speed as it moves upward while dissipating its energy, and eventually stops and turns into cool downflows (dark intergranular lane). Since the contribution of the brighter former is larger than the darker latter in the integrated spectrum, the net effect generally makes a blue shift as seen from the observer. Since the extent of velocity contrast is depth-dependent (i.e., becoming less significant toward higher layers), such a blue shift tends to be more appreciable for weak deep-forming lines than strong high-forming lines, as seen in Figure 2 of Allende Prieto and García López (1998). For the same reason, the off-center region (forming deeper) of a line profile is more blue-shifted than the region of line-center core (forming higher), which results in an asymmetric profile with a "C"-shaped bisector (see, e.g., Dravins, 1982).
To make things further more complicated, not only the vertical velocity field but also the horizontal field has to be considered when studying the variation of the convective shifts over the disk. This results in a flatter decreasing rate of blue shift (from disk center toward the limb) compared to the case of the vertical field only; actually, even a peak of blue shift is occasionally observed not the disk center but at off-center around µ ≈ 0.8 (Schröter, 1959;Beckers and Nelson, 1978;Balthasar, 1984Balthasar, , 1985. Considering this complexity, we decided to adopt the following strategy: As far as the convective shift at the disk center (µ = 1) is concerned, we can evaluate it for any relevant line to a sufficient precision, thanks to the availability of a well-tuned two-component model. Meanwhile, since this kind of line shift at positions off the center (0 < µ < 1) is too difficult to simulate, we invoke the empirical work of Balthasar (1984), who extensively investigated the relative µ-dependence of the shifts for many spectral lines of different properties and classified them in terms of line-formation depth, which we reformulated in a more easy-to-use manner (cf. the Appendix). Accordingly, as successively described in the following sections, we (1) first simulate the absolute line-shifts at the disk center (µ = 1) for representative lines (Section 5.2), and (2) then combine them with the relative center-to-limb trends empirically elucidated by Balthasar (1984) (Section 5.3).

Absolute Shift at the Disk Center
For the purpose of simulating the disk-center intensity profile under convective velocity fields, we adopt the two-component model proposed by Borrero and Bellot Rubio (2002). Their model consists of two kinds of plane-parallel atmospheric models (with only vertical velocity fields) corresponding to hot/bright granular (g) and cool/dark intergranular (i) region. The details are presented in Appendix A in their paper, where T (temperature), P e (electron pressure), P g (gas pressure), ρ (density), and v (vertical velocity) are given as functions of depth (along with the depth-independent microturbulence and macroturbulence) for each of the two models. The structures of T (τ 5000 ) and v(τ 5000 ) for each model are depicted in Figure 8a.
By using the WIDTH9 program (as in Section 3) along with these models, we computed the emergent specific intensity profile I g (λ) and I i (λ) at the disk center, from which the final spectrum is eventually obtained as the combination of these two: where α is the filling factor assumed to be 0.24 according to Figure 3 of Borrero and Bellot Rubio (2002).
In order to check our computational procedure, we simulated the µ = 1 profiles of 22 Fe i lines given in Table 1 of Borrero and Bellot Rubio (2002), and confirmed that most 4 of the resulting line-shifts satisfactorily reproduced those of the observed line profiles of Neckel's (1994Neckel's ( , 1999) solar disk-center spectrum atlas. Figure 8b shows an example of such simulated I g , I i , and I comb (each being normalized with respect to the continuum level of its own) for the representative Fe i 5194.941 line.
Such computed I comb (λ) is further convolved with the instrumental profile (P ; cf. Section 2), resulting in an asymmetric broadened profile corresponding to our instrumental condition. Profile examples before and after I.P. convolution are displayed in Figure 8c again for the case of Fe i 5194.942.
The final step is to determine the line shift from such simulated (asymmetric) profile by applying the same procedure described in Section 3; i.e., fitting with the classically modeled symmetric profile while adjusting the line shift (δλ or δV ) along with the broadening width and the elemental abundance. This process is demonstrated in Figure 8d, where we can see that the match is satisfactory.
We applied the whole above-mentioned procedure to 18 lines of appreciable strengths, which we selected by the criterion of log τ < −1. Such determined line shifts (δV 1 ) at the disk center (µ = 1) for these lines are given in the last column of Table 2, where we can see that δV 1 is in the range of −310 m s −1 δV 1 −230 m s −1 . 5 4 More precisely, the predicted shifts of the 3 lines (Fe i 6574.2285, Fe i 6581.2100, and Fe i 6739.5220) out of 22 lines appear to be only slightly shifted bluewards compared to the observation; otherwise, the profile/wavelength match for the remaining lines was found to be very good. 5 It should be stressed that we adopted the fitting-based line-shift (δV fit ), while the line shift (δV min ) defined in terms of the wavelength of minimum intensity has been more commonly used so far. We also determined δV min by interpolating the lowest three profile points by a quadratic function (see, e.g., Allende Prieto and García López, 1998) for all these 18 lines in the present case, and found that the difference δV min − δV fit tends to be EW-depedent, which varies from −60 m s −1 (EW ≈ 70 mÅ) to +120 m s −1 (EW ≈ 230 mÅ) and can be approximated by the linear-regression relation δV min − δV fit (m s −1 ) = −129 + 1.02EW (mÅ). Since the typical extent of this difference is several tens of m s −1 , we may state that the results are not seriously influenced by the definition of line shift.

Center-to-Limb Variation of δV
We are now ready to derive the µ-dependent convective shift, δV µ , by combining the disk-center δV 1 computed in Section 5.1 and the center-to-limb variation formulated in the Appendix based on the results of Balthasar (1984).
For a line k, we first evaluate log τ k B (Balthasar's core-forming depth) from log τ k 0 (our line-center-forming depth) by solving Equation (23). Then, the coefficients b k and c k in the velocity unit can be derived from Equations (21), which are sufficient with Equation (19) to establish the relative µ-dependence of δV as Therefore, combining Equation (14) with δV k 1 which was already derived in Section 5.2, we can express the absolute convective shift for line k at any µ as Such obtained position/angle-dependent δV k µ results for the 18 lines are shown in Figure 11a (δV vs. µ) and 11b (δV vs. 1 − µ 2 ), where we can see that the tendencies are nearly similar to each other, except for two conspicuously strong Cr i lines (at 5206.038 and 5208.419Å) of low excitation (χ low ≈ 1 eV).
As the final step, we average these δV µ results over 18 lines in order to obtain the mean relation, while adopting as the weight 6 for line k. Accordingly, we finally obtain where · · · is the w k -weighted average over 18 lines. Such obtained δV µ , which is also depicted in Figures 11a and 11b by the thick red line, will be used in Section 6 for correcting V hel r for the convective shift.
6 This choice of weight (∝ 1/ τ ) is for the following reason. The precision of radial velocity determination is higher not only for stronger (deeper) but also for sharper (narrower) lines, which means that the radial velocity is mainly determined by sharp saturated lines at the flat part of the curve of growth. Meanwhile, "very" strong lines with appreciable damping wings do not necessarily contribute so significantly because of their less sharpness. We point out here that the extent of saturation (i.e., both of strength and sharpness), to which more weight should be given, is most well described by 1/ τ because it increases as a line gets stronger from the linear to the flat part of the curve of growth, while it then turns to decrease as a line further becomes very stronger and comes into the damping part of the curve of growth, which is the characteristic just what we want. (In contrast, 1/τ 0 is ever increasing monotonically with the growth of line strength.) Figure 11d describes well this situation.

Discussion
We now discuss the results of absolute line shifts obtained in Section 4, while taking into account the convective-shift correction estimated in Section 5. In the following description, x and y denote the Cartesian coordinates on the solar disk (cf. Figure 2a in vs. x), where the exhibited trends can be reasonably understood by the Doppler shift due to solar rotation (V hel r ∝ x; cf. Section 4.1 in Paper I). A rough inspection of these two figures suffices to realize that V hel r distributes around ≈ +500 m s −1 at x ≈ 0 (meridian line) or near to the disk center (where the data points are densely confined), which suggests that V hel r approaches the expected gravitational shift 7 of +633 m s −1 where the rotational Doppler shift is negligible.
In order to study this situation more in detail, we pay attention to the selected dataset corresponding to the observations on the meridian line (x ≈ 0), which are given in Table 4. The V hel r vs. y correlation for this subset is shown in Figure  10b, where we can confirm that V hel r values confine around ≈ +500 m s −1 (with a standard deviation of σ ≈ 100 m s −1 ) as mentioned above. In addition, we can recognize a tendency of slight increase of V hel r with |y|; i.e., ≈+400 m s −1 at the disk center and increasing toward the limb up to ≈+600 m s −1 . (Actually, such a change of V hel r from the limb to the center on the meridian line is already observed in Figure 6.) This trend is nothing but the "limb effect" already mentioned in (see, e.g., Figure 2 of Dravins, 1982). Interestingly, when we overplot the δV vs. y relation according to Equation (17) [describing δV as a function of µ (= 1 − (y/R) 2 ] derived in Section 5, its tendency is found to be similar to the behavior of V hel r (cf. the solid line in Figure 10b). Thus, V hel r is corrected for δV as and the resulting V hel,corr r is plotted against y in Figure 10d. We can see in this figure that the slight curvature observed in Figure 10b is more or less removed by this correction (especially in the northern hemisphere, though somewhat overcorrected in the southern hemisphere) and V hel,corr r is now nearly y-independent along the meridian line, which is just what we intended to achieve (i.e., to consistently detect the gravitational redshift at a number of different points by eliminating the position-dependent limb effect; cf. Section 1). The distribution histogram of V hel,corr r and the V hel,corr r (locally averaged value) vs. y relation are shown in Figures 10e and 10f, respectively. 8 The average of all the V hel,corr r (meridian) data turned out to be V hel,corr r = 698 m s −1 (with a standard deviation of σ = 113 m s −1 ). 7 This value of 633 m s −1 corresponds to an observer on the earth while the redshift is 636 m s −1 for the observer at infinity. 8 It is interesting to note that our V hel,corr r of 696 m s −1 at the disk center (cf. Figure 10f) is comparable to 684 m s −1 derived by Balthasar (1985) as the asymptotic value for the absolute disk-center line-shift at the strong-line limit of log τ B → −∞ (where the convective shift is expected to be insignificant).
Here, it may worthwhile to recall that several sources of errors are involved in our results: Each individual absolute V r measurement is accompanied by a probable error of ≈ 50-100 m s −1 (Figure 7). Also, the randomness of granular velocity fields (represented by "wiggly-line" character seen in high-resolution slit-spectrograms) produces an appreciable dispersion of V r (even though it must have been somewhat smoothed because spatial information over 42 ′′ has been averaged in our spectrum; cf. Section 2 in Paper I), which is estimated in the present case to be on the order of ≈ 100 m s −1 (Figures 10e and 10f). Yet, since these are random errors, they could (in principle) be reduced in the averaged V r by summing up as many data as possible; e.g., averaging of all 144 data on the meridian would result in a reduction by a factor of ≈ 10 (≈ √ 144). We thus estimate that the error component in V r due to such random sources would have been reduced to the ≈ 10-20 m s −1 level. Actually, however, systematic errors must be more important, which are hard to evaluate. For example, while we corrected the absolute wavelength-scale of Lick I 2 spectrum by 162 m s −1 , this value itself already has an uncertainty with a magnitude of several tens m s −1 (Figure 3d). Besides, although our convective-shift corrections turned out seemingly successful, we have no idea about how much systematic errors are involved in these, given that this correction differs from line to line with variations of ≈ 50-100 m s −1 (cf. Figure 9). Considering all what described above, we would modestly state that our final V hel,corr r value suffers inevitable ambiguities of no less than several tens m s −1 or maybe amounting up to 100 m s −1 .
Accordingly, while the resulting V hel,corr r of 698 m s −1 is slightly larger than the predicted gravitational redshift of 633 m s −1 by +65 m s −1 , this difference is not meaningful in view of the expected uncertainties of several tens m s −1 to 100 m s −1 as estimated above. We thus conclude that the solar gravitational redshift has been quantitatively confirmed in a reasonable manner by our absolute line-shift analysis using the iodine-cell technique, in combination with the simulated convective shift corrections.

Conclusion
Motivated by the situation that various past studies of solar gravitational redshift are not convincing because few of them seem to have properly considered the Doppler shifts due to inhomogeneous convective motions, we decided to examine whether the predicted solar gravitational redshift can be observationally confirmed by removing the appropriately simulated convective shift.
For this purpose, we attempted measuring the absolute spectral line-shifts on a large number of points over the solar disk based on an extensive set of 5188-5212Å region spectra taken through an iodine-cell with the Solar Domeless Telescope at Hida Observatory, where we paid attention only to the selected dataset observed on the solar meridian where the rotational Doppler shift is irrelevant.
In analogy with the analysis technique for precise "relative" shift measurements (from I 2 +object spectra) elaborated by Marcy and Butler (1992), we developed a method for measuring "absolute" shifts by using the theoretical solar intensity spectrum template, which is computed from a model atmosphere and parameterized by elemental abundances along with the broadening parameter; and the absolute shift can be established by finding the best-fit solution of the parameterized model spectrum with the observed spectrum.
The resulting heliocentric absolute shifts (V hel r ) on the meridian line turned out to be weakly position-dependent as ≈+400 m s −1 near the disk center and increasing toward the limb up to ≈+600 m s −1 (both with a standard deviation of ≈ 100 m s −1 ).
We then simulated the convective blue shifts for representative 18 lines by using Borrero and Bellot Rubio's (2002) two-component model for the disk center, combined them with the empirical relations of relative center-to-limb variation investigated by Balthasar (1984), and finally adopted the weighted-average over 18 lines. The resulting shift is δV ≈ −300 m s −1 at the disk center, gradually increases to ≈ −100 m s −1 at |y/R| ≈ 0.9, and shows a suddenly increase up to ≈ +200 m s −1 at the limb (|y/R| = 1).
Interestingly, the position-dependence of V hel r tends to be removed when the simulated convective blue-shift is subtracted, which finally makes the averaged shift for all the meridian data to be V hel,corr r = 698 m s −1 (σ = 113 m s −1 ). Considering the ambiguities involved in the absolute wavelength calibration or in the correction due to convective Doppler shifts (at least several tens m s −1 , or more likely up to 100 m s −1 ), we may state that this value is well consistent with the expected value of +633 m s −1 (due to the gravitational potential on the solar surface) as predicted from the general relativity theory. This means that we could confirm the solar gravitational redshift in a quantitatively satisfactory level within the precision of our analysis.
Acknowledgements Y. Takeda heartily thanks Dr. H. Knöckel for kindly providing the "IodineSpec" program, which turned out very helpful for checking the absolute wavelength scale of the reference I 2 spectrum.

Appendix Analytical Formulation of Balthasar's (1984) Empirical Limb-Effect
Balthasar (1984) carried out an empirical investigation on the center-to-limb variation of spectral-line shift on the solar disk and found that the trend of lineasymmetry as well as of line-shift depends on the formation heights of line cores (log τ B ). He tried to fit the observed shifts of many spectral lines by a quadratic polynomial in terms of 1 − µ (µ ≡ cos θ: direction cosine) as and determined the coefficients (a, b, c) for 6 different log τ B ranges (cf . Table IV therein), as shown in Figures 11a-11c. For practical convenience, we derived the analytical expression of a, b, and c with respect to log τ B by applying quadratic least-square fitting to these data as c (m s −1 ) = 1100. + 113.9(log τ B ) − 31.8(log τ B ) 2 .
One problem is that Balthasar (1984) did not clarify the definition of his "line-core formation depth" (expressed as "lgτ " therein but denoted here as log τ B ). We computed log τ and log τ 0 (the mean formation depth and the linecenter formation depth defined in Equations (4) and (5)) for all 143 lines given in Table I of Balthasar (1984) in the same way as we did in Section 3.3. These results are compared with his log τ B values in Figures 11d and 11e, where we can see that his log τ B is similar to our log τ 0 , though both are not in a linear relationship with each other. We found from least-squares analyses the following relations: In the practical application in Section 5, our log τ 0 is converted into log τ B by using Equation (23) for each line, from which the necessary coefficients (such as b or c) can be derived by applying Equation (21). Table 1. Solar gravitational redshift determinations over the last 50 years (expressed by R ≡ ∆ obs /∆ the ).
Authors: Blamont and Roddier (  Shown here are the atomic data of the lines of appreciable strengths (see also Figure 2), which were taken from Kurucz & Bell's (1995) compilation (with adjustments for 4 lines; cf. Section 3.2). Followed by first four self-explanatory columns, damping parameters are presented in columns 5-7: Gammar is the radiation damping width (s −1 ), log γ rad . Gammas is the Stark damping width (s −1 ) per electron density (cm −3 ) at 10 4 K, log(γe/Ne). Gammaw is the van der Waals damping width (s −1 ) per hydrogen density (cm −3 ) at 10 4 K, log(γw /N H ). In case where these damping parameters are not given in Kurucz and Bell (1995), they were computed as default values in Kurucz's WIDTH program (Leushin and Topil'skaya, 1987). Columns 8-10 give the strength-related quantities relevant to the disk-center spectrum (cf. Section 3.3): the equivalent width inversely computed from the best-fit abundance solution (cf. Figure 2) under the assumption of isolated single line, the mean depth of line formation, and the line-center formation depth. Presented in column 11 is the theoretically estimated convective blue-shift at µ = 1 (only for the representative 18 lines; cf. Section 5.2).  Figure 2. Demonstration of how the solar intensity spectrum simulated at µ = 1 for each wavelength region (a-d) can be fitted with the observed solar disk-center spectrum (Neckel, 1994(Neckel, , 1999    . Wavelength-dependence of the relative difference between n formula and n KB (two kinds of refractive indices) for all the spectral lines at 5188Å ≤ λ air ≤ 5212Å included in Kurucz and Bell's (1995) compilation, where n formula is the refractive index of air based on the standard formula given in Cox (2000) as given in Equation (7) while n KB is estimated from Kurucz and Bell's (1995) atomic data by Equation (8) intrinsic solar spectrum (computed) Figure 5. Graphical example [for the disk-center case at region (a)] of how the absolute wavelength shift can be determined by searching for the best-fit parameters such those accomplishing the best-fit (simulated) I λ,th [≡ G λ ⊗ (T λ · I λ,intr ); cf. Equation (2) (2002), which we adopted for the line-profile simulation. (b) Illustration of how the realistic asymmetric profile (green) is produced as a combination of granular (blue) and intergranular (red) profiles (being weighted according to the filling factor and the continuum intensity), for the case of  . Panels (a)-(c): Graphical representation of the depth-dependence of the coefficients (a, b, c) used for the polynomial fits of the limb effect, ∆λ = a + b(1 − µ) + c(1 − µ) 2 , which were empirically determined by Balthasar (1984;cf. Table IV therein). The quadratic dependence of these coefficients in terms of log τ B , which we derived from the least-squares fit are also depicted. Panels (d) and (e) show the relationships between the line-core-forming depth (log τ B ) adopted by Balthasar (1984) and the two kinds of line-formation depths ( log τ and log τ 0 ; i.e., mean depth of line formation and the line-center-forming depth) derived in this study. The least-squares fitted quadratic relations are also drawn.