The Correction of Atmospheric Dispersion in LAMOST*

Transcription

1 or ELSEVIER Chinese Astronomy and Astrophysics 30 (2006) CHINESE ASTRONOMY AND ASTROPHYSICS The Correction of Atmospheric Dispersion in LAMOST* LIU Gen-rong YUAN Xiang-yan National Astronomical Observatories,/Nanjing Institute of Astronomical Optics ~4 Technology, Chinese Academy of Sciences, Nanjing Abstract Atmospheric refraction and dispersion affect the observational quality and efficiency of the Large Sky Area Multi-object Fiber Spectroscopic Telescope (LAMOST). In this paper, by both ZEMAX simulation and numerical calculation, the atmospheric dispersion for LAMOST, which will be installed at the latitude 40.4 and has a 20 m focal length and F5 focal ratio, are calculated for the 0.75 h before and after the transit time. Various types of dispersion prism to be used for correcting the atmospheric dispersion in LAMOST were calculated and analyzed, from which one was finally adopted. The secondary spectrum of the dispersion prism, the residual dispersion during tracking of the telescope and the installation errors of the dispersion prism are investigated. Key words: telescope atmospheric effect 1. INTRODUCTION When a light ray from a celestial body goes through the atmosphere, atmospheric refraction will cause the ray to bend, changing its zenith distance and displacing the stellar image. And since the atmospheric refraction is dependent on the wavelength, atmospheric dispersion is resulted: making the point-like stellar image on the image surface to become a dispersed spectrum, spreading out the energy and blurring the originally distinct image. The image displacement caused by the atmospheric refraction can be partially corrected by automatic positioning of optical fibers, but the atmospheric dispersion can not be compensated in this way. When atmospheric dispersion exists, the proportion of the light rays entering the fiber decreases and therefore the observational efficiency is reduced. If the stellar image has a 2" Received ; revised version * A translation of Acta Astron. Sin. Vol. 46, No. 3, pp , ~06~S-see front matter 2006 Elsevier B. V All rights reserved. DOI: /j.chinastron

2 216 LIU Gen-rong et al. / Chinese Astronomy and Astrophysics 30 (2006") profile, and the diameter of the optical fiber is 3".3, (taking no account of the position error of the optical fiber,) then an energy loss will certainly result when the atmospheric dispersion exceeds 1".3. Hence the effect of atmospheric dispersion on the spectral observation and analysis is not negligible. Some studies have been made [1 3] relating to atmospheric refraction and dispersion for the Large Sky Area Multi-object Fiber Spectroscopic Telescope (LAMOST). In this paper, values of the atmospheric dispersion for the telescope that will be installed at Xing-long are simulated by the ZEMAX simulation software for optical designs and verified by numerical calculations. 2. THE ATMOSPHERIC DISPERSION IN LAMOST LAMOST has a focal length of 20m and a focal ratio of 5; its field of view is 5 when observing in the rage -10 < (~ < 60, and 3 when observing in the range 60 < (~ < 90. There are 4000 optical fibers on the focal surface, the diameter of each fiber is 3//.3. The time tracking of a celestial body covers the 0.75h before and after the transit time. The telescope will be installed at the Xing-long Station of Beijing Observatory, at latitude of 40.4 and a approximate altitude of 1000 m [4]. Stellar spectral analysis will be made on LAMOST in the wavelength range nm. In our dispersion analysis, the wavelength range is extended to nm, and we set A1 380nm, A2 1000nm. When the zenith distance is not too large, the following approximate formula for calculating the atmospheric refraction can be used: p ( - 1)tan. (1) In Eq.(1), p, z, z' and n denote the atmospheric refraction (in arcsecond), apparent zenith distance (in degree), true zenith distance (in degree) and atmospheric refl'activity, respectively. According to Eq.(1), atmospheric refraction is a function of the zenith distance. And the atmospheric dispersion is a function of wavelength. For A1 and A2, we have: Pl ~ (nxl - 1)tanz, P2 ~ (nx2-1) tanz, At) pl -p2 ~ (nxl - nx2) tanz An, tanz. (2) Eq.(2) is the atmospheric dispersion expressed in arcsecond, it is related to the difference of atmospheric refractivity at two different wavelengths. The atmospheric refractivity depends on the air temperature, humidity and atmospheric pressure. Standard reference gives, for 1000 m altitude, the average temperature is 8.5 C and the atmospheric pressure is mm Hg (0.8988bar). And the humidity is taken to be 0.5. Referring to Fig. 1, the zenith distance of an arbitrary star in the field of view, z, is related to the hour angle t by: cosz sin sin (~ + cos cos (~ cos t. (3) and (~ being the geographic latitude of the observer and the declination the star. In the course of +0.75h tracking, the hour angle varies in the range < t <

3 LIU Gen-rong et al. / Chinese Astronomy and Astrophysics 30 (2006") Z / 90"-8 \" \',\ / \ \ ~ /J Fig. 1 The spherical triangle showing the relation between the zenith distance and the hour angle For 11 values of the declination and 5 values of the hour angle and for the geographical latitude of 40.4, elevation of 100 m, and the local mean temperature, humidity and pressure, we calculated the atmospheric dispersion at the field centre of LAMEST (focal length 20 m, focal ratio F5). The results are given in Table 1. The table shows that the maximum dispersion (2".29) occurs at (~ -10, and the minimum dispersion (0".3), at (~ 40. For a given declination, the dispersion does not vary much when the hour angle varies cover a range of This means that although the hour angle at an off-centre point of the field of view differs from that at the center by up to 2.5, the variation of the dispersion from the edge to the center of the field will not be very large. Therefore, for the different fields of view in the same celestial region, just one dispersion prism will suffice for the dispersion correction. The following formula is the Elden model for calculating the atmospheric refractivity mentioned in Reference [5] p [ p ( t) 10 9] rt t A _A 2] x (4) Substituting the atmospheric refractivities calculated by Eq.(4) at two different wavelengths and Art into Eq.(2), the obtained results are basically as same as those in Table 1, so we can conclude that the atmospheric dispersion in LAMEST estimated by simulative calculations is correct. 3. CORRECTION OF ATMOSPHERIC DISPERSION IN LAMOST The focal surface of LAMEST is a spherical surface of 1.75m diameter, on which 4000 optical fibers are mounted. If a large dispersion corrector is to be placed in front of the focal surface, then we can image that the diameter of this dispersion corrector must be greater than 1.75 m. Because we must give consideration to both the large aperture and the large field of view; its design, material source, fabrication and testing will be very difficult, so this scheme is obviously unacceptable. Another scheme is the method of partitioning the

4 218 LIU Gen-rong et al. / Chinese Astronomy and Astrophysics 30 (2005") field of view, and this method in association with the focal reducer and image-field corrector was proposed by Su Ding-Qiang et al. [6] for large-field-of-view telescopes. Here we apply this method to the correction of atmospheric dispersion in LAMOST, i.e., we place a small dispersion corrector in front of each optical fiber to correct the atmospheric dispersion caused by the stellar image before it enters the fiber. Table 1 Atmospheric dispersion of LAMOST (in arcsecond) obtained by simulative calculations 5 t Ap 5 t Ap The Operating Principle of the Dispersion Prism As shown in Fig.2(a), the refracting prism will cause a dispersion in its principal section and a deflection of the optical axis. When two refracting prisms with the same vertex angle a are put together in opposite directions, if they are made of the same material, then the combination of the two refracting prisms will act as a plate glass, the light deflection is completely compensated, as shown in Fig.2(b). If the two refracting prisms are made of two different kinds of glass with the same refractivity at the central wavelength Ao and different dispersion coefficients, the light passing through this pair of prisms at the central wavelength Ao will have no deflection, but a dispersion will result because of the variation of refraction with the wavelength, i.e., the light at the longer wavelength and the light at the shorter wavelength will depart from the light at the central wavelength in opposite directions, as shown in Fig.2(c). And we can use the prism's dispersive property to correct the

5 LIU Gen-rong et al. / Chinese Astronomy and Astrophysics 30 (2006) atmospheric dispersion in LAMOST by making the two equal in magnitude and opposite in direction. n ~ n 2 ~_< ~. I --- ~Z (a) (b),(c) Fig. 2 Illustrative diagram of the dispersion prism In Fig.2(c), let the vertex angles of the two prisms be al and as, and the refractivities of the materials, nl and n2. If the incident angle is small, then as the light passes through separately the two prisms, the corresponding deflection angles at the wavelength A0 will be 61~o and 62Ao, respectively[r]: and the net deflection angle 6~o is: 61A0 (: 1 62~0 a2 (~IA0 -- 1), (5) (~0 -- 1), (6) ~A0 ~IA0 -- ~2A0. (7) It is known that 7gin o 7~2Ao, al a2, so 6~ o 0, i.e., at the wavelength Ao the incident light has a direct-vision effect and there is no deflection of the optical axis. At the wavelengths A1 and A2, we have: ala 1 -- ala 2 O~l(IglA1 -- IglA2), ~2A1 -- ~2A2 O~2(7~2A1 -- 7~2A2), (~1A1 -- ~2A1) -- (~1A2 -- (~2A2) O~1(Tg1A1 -- IglA2) -- O~2(Ig2A1 -- Ig2A2) (8) Eq.(8) is the width of angular dispersion of the dispersion prism. And, the dispersion coef- - r~ o - 1 ficient of the glass material is u nx o 1 Rewriting this as nx I - nx 2 and I~A1 -- I~A2 " // substituting into Eq.(8), we have: (~1A1 -- ~2A1) -- (~1A2 -- (~2A2) (: 1 ( IglA ) -a2( n2~0-1 ) y2al~0 -<a2~0 (9) Yl Y2 Yl Y2 From Eqs.(8) and (9), we can find that the angular dispersion caused by the dispersion prism is dominated by the vertex angle a and the difference of the dispersion coefficients of the two glass materials. It is proportional to the prism's vertex angle a and the difference

6 220 LIU Gen-rong et al. / Chinese Astronomy and Astrophysics 30 (2006) of u between the two glass materials. The dispersion increases with both increasing c~ and increasing difference of u. For a given dispersion value, we can take a smaller vertex angle c~ if we have a larger difference of u. So, to reduce the prism's vertex angle, we have to select as far as possible glass materials with the same nx 0 and a large difference in u. This rule is also applicable to prism systems composed of more than two prisms, and the method of calculating the dispersion is similar. 3.2 The Design and Comparison of Several Types of Dispersion Prism As shown in Fig.3, the dispersion prism in LAMOST is placed in a convergent path of light. As the light rays at different wavelengths pass through the prism, they are deflected by different angles, so the light beam will no longer converge to a point, rather, it will be spread out along the vertical direction. Meanwhile, like a plate glass in the convergent path of light, the dispersion prism will not only introduce various aberrations, but also enlarge the image spot. What we really want is a dispersion prism that has a defined dispersion in the image surface and a very small diffuse spot for the lights of different colors. For this purpose, we have analyzed and calculated the dispersion and diffuse spot size for various types of dispersion prism for a telescope of 20 m focal length and F5 focal ratio. According to the principle mentioned above, we selected the glass material QFI-BAK4 Is,9]. As shown in Fig.4, their refractivity curves cross each other at wavelength #m, where they have the same refractivity, We take this wavelength as the central wavelength for our calculations. Of course, other choices are possible. If the refractivity curves do not cross each other in the given wavelength range, then the light will have a deflection of the optical axis at A0, and this can be compensated by a surface tilt to give the effect of a direct-vision prism; but we will not discuss further on this issue. To make the effective aperture of the dispersion prism in front of each fiber not exceed 3 ram, the distance from the first surface of the dispersion prism to the focal surface is taken to be L 3 x F 15 ram. Fig.5 shows the configurations of several types of dispersion prism. Table 2 lists their configuration parameters and the results of spot diagram calculations of dispersion prisms that will each give rise to a dispersion of 2".2. The tilt angle of cemented surface given in the second column is positive clockwise, and negative counterclockwise. n ~1~ n2 i _.~ 2 ~=:27- ~" ;_,2:~:-- L-,! )v Q F I X(gm ) Fig. 3 Dispersion prism in a convergent path of light Fig. 4 Diagram of wavelength-refractivity curves

7 LIU Gen-rong et al. / Chinese Astronomy and Astrophysics 30 (2005") (a) (b) (c) (d) (e) (t) Fig. 5 Several types of dispersion prism table 2 Configuration parameters and calculated spot diameters for the different types of dispersion prism with the same 2".2 dispersion Serial number Parameters of configuration (ram) Spot diameter (arcsecond) R d n As A0 As Fig.5(a) Roo 3 QF /z~ 3 BAK4 {Tilted ) /Zoo L / 9 Fig.5 (b) Fig.5 (c) Fig.5 (d) Fig.5 (e) Fig.5 (f) R QF R BAK4 (Tilted ) R8 L/ 8 R~ 2.1 QF /Z~ 3.7 BAK4 (Tilted -45 ) /Z~ 2.1 QF1 (Tilted 45 ) Roo L / 7 R QF /Z~ 2.6 BAK4 (Tilted ) R~ 1.5 QF1 (Tilted ) /Z9.4 L ~ 9.4 R15 2 QF R BAK4 (Tilted ) R QF1 (Tilted 42.1 ) R6 L ~ 6 R QF R BAK4 (Tilted ) Rll QF1 (Tilted 25.4 ) /Z BAK4 (Tilted ) 1:t QF1 (Tilted 25.4 ) R5.2 L ~ 5.2 For the 6 types of dispersion prism shown in Fig.5, we make the following analysis and

8 222 LIU Gen-rong et al. / Chinese Astronomy and Astrophysics 30 (2006") comparisons: 1) The structure configuration and the tilt angle of cemented surface: The configurations (a) and (b) in Fig.5 are of 2-prism combination. The required dispersion is realized by relying on the tilt of the only one cemented surface in the middle, so the tilt angle is the largest, reaching about 60. Tile configurations (c),(d) and (e) are of 3-prism combination, the resulting dispersion is shared by two cemented surfaces, with tilt angles in the range 30 ~ 50. The configuration (f) is of 5-prism combination, the dispersion is realized by 4 cemented surfaces, with tilt angle of only For the same dispersion, the larger the number of components, the smaller the vertex angle a. 2) The structure configuration and the total optical length: In the configurations (a) and (c) of Fig.5, all the surfaces are plane, after combination they act as plate glasses. To insert a plate glass into a convergent path of light will make the image point displace backward, and the backward displacement is 2.2mm for the configuration (a) and 2.9mm for the configuration (c). Thus, while different dispersion prisms are used for different celestial areas, the focal surface of LAMOST should be correspondingly adjusted. For the configurations (b), (e) and (f), in tile case of no rotation, all tile surfaces are concentric spherical surfaces with the focal point as the center, so they will not cause a backward displacement of image point, and no focal surface adjustment is required. For the configuration (d), the cemented surfaces are plane, the front and rear surfaces are concentric spherical surfaces, and again no displacement of image surface will result. 3) The structure configuration and the size of image spot: The dispersion prism composed of plane surfaces will enlarge the image spot over the whole wavelength range. The larger the tilt angle of cemented surface, the greater the image spot. The dispersion prism composed of spherical surfaces will not cause enlargement of image spot at the wavelength k0. This is because the incident angle of the dispersion prism at the wavelength k0 is zero, the light will not be refracted on the other surfaces, but the enlargement of image spot exists at other wavelengths because of the differing refractivity. Also, increasing the number of pieces and reducing the incident angle on the cemented surfaces will make the image spot smaller, and the image quality can be much improved by making the tilt angles of cemented surfaces equal in magnitude and opposite in direction. A comprehensive comparison of the 6 types of dispersion prism indicates that the one composed of concentric spherical surfaces (configuration (b)) is superior to the dispersion prism with plane surfaces (configuration (a)), that the one containing two tilt angles (configuration (e)) is better than the one containing one tilt angle (configuration (b)), and that the one containing multi-pair tilt angles (configuration (f)) is better than the one containing one pair of tilt angles (configuration (e)). So the configuration (f) yields the best result, it gives the smallest image spot, the maximum diameter of image spot over the operating wavelength range is only 0".007, but its large number of components makes its manufacture difficult. From the angle of applications, the configuration (e) is good enough, with a spot diameter of 0".024 in the operating wavelength range. Its smaller number of components will mean a relatively small manufacturing error: this dispersion prism composed of three pieces of spherical surface prisms is the best choice. 4. SECONDARY SPECTRUM OF THE DISPERSION PRISM AND RESIDUALS DURING TRACKING Having selected the configuration of the dispersion prism, we have to make a further analysis

9 LIU Gen-rong et al. / Chinese Astronomy and Astrophysics 30 (2006) on: (1) the error of the dispersion prism itsel~the secondary spectrum; (2) the residual dispersion in the process of tracking. We now analyse these two questions one by one. 4.1 The Secondary Spectrum of the 2".2 Dispersion Prism From the calculated results in Table 1 we know that at the meridian of the (~ -10 celestial area, the atmospheric dispersion is 2".2. We put a 2".2 dispersion prism in the path of light, and make the dispersion of the prism to be opposite in direction with respect to the atmospheric dispersion, then, the image points in the colors A1 and A2 are coincident, but the image points of the other colors will appear as a dispersed spectrum in the vertical direction, and after the dispersion has been completely corrected at A1 and A2, the remaining dispersion is called the secondary spectrum. Table 3 lists the values for the secondary spectrum of the 2".2 dispersion prism, and the curve of the secondary spectrum is shown in Fig.6. As may be seen, the maximum of the secondary spectrum is ( ), about 0".2, almost one eleventh of the dispersion of the dispersion prism. Table 3 Secondary spectrum of the 2".2 dispersion prism A(#) Y)~(mm) l~m Fig. 6 Diagram of the secondary spectrum of the 2//.2 dispersion prism 4.2 The Residual Dispersion in the Process of Tracking During an astronomical observation, because of the diurnal motion of the celestial object, its azimuth, zenith distance and hence the atmospheric dispersion vary continuously. The direction of the atmospheric dispersion is always along the azimuth circle of the celestial object, with the blue end towards the zenith [2]. When the celestial body is located at the meridian, the atmospheric dispersion and the prism/s dispersion are equal in magnitude and opposite in direction. When the celestial body moves away from the meridian, its parallactic angle 0 changes (see Fig.l), the prism/s dispersion can no longer completely compensate the atmospheric dispersion and a part of residual dispersion remains. If the atmospheric

10 224 LIU Gen-rong et al. / Chinese Astronomy and Astrophysics 30 (2006) dispersion is denoted by the vector a, and the prism's dispersion is denoted by the vector b, then the vector of correction error is c a - b. As shown in Fig.7, to set the Y-axis in the direction of the prism's dispersion (the north-south direction), the northward is positive and the southward negative, and to set the X-axis in the west-east direction, the eastward is positive and the westward negative, then we have: as lalsin0, a s lalcos0, (lo) Am a~-bs lalsino, (11) Ay ay-by l alcoso-by. Where [ a [ Ap, 0 is the parallactic angle calculated by the formula sin 0 cos sin t Ap sin z ' is the atmospheric dispersion, as, ay, bs, by are the projections of the vectors a and b in the X, Y-direction, respectively, and Ax, Ay represent the residual dispersions during tracking. We Take 3 typical celestial areas (~ -10, 40, 90 ) and calculate the residual error during tracking after the dispersion correction. The results are given in Table 4. The results show that, when the atmospheric dispersion is corrected by a dispersion prism equivalent to the atmospheric dispersion, the atmospheric dispersion in the Y-direction is basically corrected, the maximum residual error in the ~ -10 celestial area is only 0."0513, but the component remaining in the X-direction is as much as 0".435: this is the main part of the residual dispersion. K.,- a) / b S z Atmo2h/cric dispersion / Fig. 7 Dispersion direction in tracking a..-x direction Fig. 8 Configuration of the 1//.6 dispersion prism 4.3 Design of the LAMOST Dispersion Prism and the Residual Error When LAMOST follows an object in a given sky area, the atmospheric dispersion in the Y-direction can be well corrected if the dispersion prism is properly selected. But if the dispersion prism has to be changed for different sky areas, it will be cumbersome. Fortunately, this is not necessary. To start with, we take just one type of dispersion prism for the whole observational area, with dispersion 1".6. It is mainly used for dispersion correction in areas where the atmospheric dispersion is greater than 1".6. Such areas are located at the two ends of the observational area, i.e., -10 < (~ < 10 and 70 < (~ < 90, and for the area 10 < (~ < 70 no correction is necessary. The proportion of the correctable area relative to the whole observational area is 34.7%.

11 LIU Gen-rong et al/ Chinese Astronomy and Astrophysics 30 (2006) Table 4 Residual dispersion error during tracking for 3 typical celestial areas Items of calculation Time angle Remnant error(arcsecond) ~5-10 ~5 40 ~5 90 Ap sin Ap cos Ax Ay For the design of the 1".6 dispersion prism we adopt the configuration 5(e). To reduce further its size, the distance from its first surface to the focal surface is temporarily taken to be 5 ram. Calculated with the focal ratio 5, the effective aperture of the dispersion prism defined by the on-axis light is 1, and considering the effect of the diameter of the optical fiber (i.e., the stellar image size) on the aperture, the actual effective aperture of its front surface is 1.3. The actual effective aperture of its rear surface is 0.5, so on the whole it has a conical shape. We take a new selection of composite glass, QFS-ZK3 with its greater difference in u that is more advantageous for the dispersion correction. The glass QFS-ZK3 has the same refractivity at wavelength A /,. Taking this wavelength as the central wavelength, the calculated result is given in Table 5, and the structure configuration is displayed in Fig.& If we adopt this scheme, then during tracking the dispersion prism will give a maximum residual correction error of 1" in the Y-direction and 0".435 in the X-direction, in addition to the prism% secondary spectrum of 0".145 (about one eleventh of 1".6). Then the residual correction error of the dispersion prism during tracking is 1".1 (RMS value). Table 5 Design of the 1".6 dispersion prism Configuration parameters Spot diameter (arcsecond) R d n A1 A0 A2 R5 1 QF R4(Tilted 60 ) 2 ZK3 R2(Tilted -60 ) 1.2 QF5 R0.8 L/ 0.8 One way to eliminate the error in the X-direction is to keep the direction of the main section of the dispersion prism to be always coincident with the direction of the atmospheric dispersion; but this is difficult to realize.

12 226 LIU Gen-rong et al. / Chinese Astronomy and Astrophysics 30 (2006") "[ There will be probably some difficulties in the fabrication of this 1//.6 dispersion prism; the method of its fabrication remains to be further investigated. 5. POSITION TOLERANCE FOR THE INSTALLATION OF THE DISPERSION PRISMS On the focal surface of LAMOST, 4000 optical fibers are mounted, and in front of each fiber a dispersion prism is installed [6]. Errors in their installation affect directly the effectiveness of the atmospheric dispersion correction. There are 6 position errors: 3 displacements and 3 tilts. We set the optical axis as the Z-axis, and the X-axis and Y-axis perpendicular to it. The displacements of the dispersion prism along the X- and Y-axis will enlarge the image spot, but will not change the dispersion; the displacement along the Z-axis will not only enlarge the image spot, but also change the dispersion; the tilts around the X- and Y-axis will cause a deflection of the optical axis, but will not give rise to any dispersion; and the tilt around the Z-axis will make the image tilted, make the dispersion correction in the Y-direction insufficient and cause an excessive component in the X-direction. The position tolerance for the installation is distributed as follows: 1) The displacements dx, dy, dz of the dispersion prism along the X-, Y-, Z-axis must each meet both of the requirements that the diameter of image spot < 0".03 and that the dispersion < 0//.2. 2) The tilts T~ and ~Fy around the X- and Y-axis cause separately a deflection of optical axis of < 0//.2. 3) For the tilt T~ around the Z-axis, the component caused by the image tilt is less than 0".2. Further analysis shows that although the tilts T~, Ty of the dispersion prism around the X- and Y-axis cause deflection of the optical axis, it can be compensated by automatic positioning of the fiber, hence as long as the enlargement of image spot does not exceed the allowed value, the tolerance on the tilts around the X- and Y-axis can be properly relaxed. Taking the 2".2 dispersion prism as an example, the result of the analysis on the positional tolerance for the installation is given in Table 6. Table 6 Positional tolerance for the installation As dx, dy have no contribution to the dispersion error, and T~, Ty can be compensated by automatic positioning of the fiber, so actually only dz and T~ dx dy dz Tx Ty Tz cause dispersion errors. The summation S (RMS) of 0.2ram 0.2 mm 0.3ram the dispersion errors caused by these two tolerance terms is: S ~ H. (12) 6. CONCLUSIONS AND DISCUSSION (1) Values of atmospheric dispersion for LAMOST calculated by respectively the ZEMAX simulation software for optical designs and the formula of atmospheric dispersion are basically equal to each other and lie in the range 0// ~ 2//.3. When the value of atmospheric dispersion reaches its maximum and the diameter of stellar image exceeds 1 H, the optical energy loss and energy reduction will result, hence it is necessary to make corrections for the atmospheric dispersion.

13 LIU Gen-rong et al. / Chinese Astronomy and Astrophysics 30 (2006") (2) For correcting the atmospheric dispersion, a dispersion prism with the structure shown in Fig.5(e) is suggested. This configuration gives rise to a very small image spot and no displacement of the image surface. (3) When the dispersion of the selected dispersion prism and the atmospheric dispersion are equal in magnitude and opposite in direction, the atmospheric dispersion can be well corrected, apart from a component remaining in the X-direction, which amounts to 0//.435 at most. The error in the Y-direction can be further reduced by increasing the types of dispersion prism, but to eliminate the error in the X-direction, we have to adopt a special mechanism on the dispersion prism, and this is difficult to realize. (4) Based on the calculation and analysis on the atmospheric dispersion values for LAMOST, a scheme of correcting the atmospheric dispersion is proposed in this paper, i.e., we use just one type of 1".6 dispersion prisms for the whole observational area; the remaining correction error during tracking will be 1".1 in the worst case and less than this value in most cases. Adopting this scheme, all the dispersion prisms are of a single type, less operations are required, and if we disregard the positioning error of the optical fibers and consider only the imaging quality of the optical system, the resultant image spot will have a diameter still less than 3".3 [4]. (5) According to the results presented in Table 6, the maximum dispersion error caused by the position error of the 2".2 dispersion prism is 0".213. Under the condition of the same position error, the dispersion error of the 1".6 dispersion prism will be less than this value. And the position tolerance of the dispersion prism given in Table 6 is generally realizable. ACKNOWLEDGEMENT We thank Academician Su Ding-qiang, Prof. Wang Ya-nan & Prof. Cui Xiang-qun for their help and instruction at the expense of their precious time, and for their concern and valuable comments on the manuscript of this paper. References 1 Su Ding-qiang, Wang Ya-nan, Acta Astrophy. Sinica, 1997, 17, Zhang Hao-tong, Technical Report of LAMOST, LAMOST-TR-BAO-L8-006, Zhang Ai-qun, Hu Jing-yao, Acta Astrophy. Sinica, 1997, 17, Shou-guan Wang, Ding-qiang Su, Yao-quan Chu, et al., Appl. Opt., 1996, 35, Zhang Xue-jun, Jiang YVen-han, Optical-electronic Engineering, 2002, 29, 1 6 Su Ding-Qian, Cao Chang-Xin, Liang Ming, SPIE, 1986, 628, Doodrovski A. Y., Theory of Optical Instruments, Beijing: Science Press, Su Ding-Qian, A&A, 1986, 156, Wang Ya-Nan, Su Ding-Qian, A&A, 1990, 232, 589