Hd1_Ceres_1801_Piazzi_17_obs_Theta=RA_plot.xmcd 6/21/2016 1

Transcription

1 BATCH LEAST SQUARES DIFFERENTIAL CORRECTION OF A HELIOCENTRIC ORBIT PART - TEST CASE SPECIFICATION WORKSHEET Roger L. Mansfield, June, 6 This worsheet defines a test case the HDC worsheet. HDC implements the equations of weighted batch least squares differential correction (DC) of a heliocentric orbit. This worsheet inputs observations of the orbit of a minor planet or asteroid from the file specified below, which replaces the generic file OBSERVATIONS.PRN. Here now is an outline of the steps we will follow in this worsheet:. Specify the observations and the weight matrix, W.. Specify the observer's coordinates and calculate the observer's ECI equatorial positions at the observation times. 3. Calculate the sun's ECI equatorial coordinates and then the sun's coordinates relative to the observer at the observation times. 4. Calculate the N-by- "observed" measurements vector, Y. 5. Specify the initial estimate of state, X o. 6. Write test case specification values to dis use by worsheet HDC. (To create other test cases HDC, simply duplicate this worsheet, rename as "HD", "HD3", etc., and modify as desired. HDC wors best with test cases specified in this way.) As a preliminary, we define some constants that we will need and set the Mathcad worsheet ORIGIN to so that subscripts start at unity rather than at zero. 8 ORIGIN π SecPerDeg 36. SecPerRad SecPerDeg 8 π TIPS ON READING MATHCAD WORKSHEETS. A Mathcad worsheet typically consists of text regions and math regions.. Text and math regions can be anywhere on a page. Text regions are just optional comments. Mathcad uses math regions to do its calculations, and according to the following rule: math regions are calculated in the order of left to right, then top to bottom. 3. In a math region, colon-equals ( := ) is lie an assignment statement in C. That is, the expression on the right is calculated and placed in the variable on the left, whereas equals ( = ) by itself is used to display on the right of the equals sign the value of the variable on the left. 4. Mathcad has functions just lie C does. Typically a function has its name and input variables inside of parentheses on the left of a colon-equals ( := ) and a sequence of vertical line segments with the assignment statements of the function on the right. Inside of a function, the assignment statements use left arrows <-- instead of colon-equals := to mae assignments. The last line of a function is its output argument, which may be a a scalar, a vector, or a matrix of variables. 5. My Mathcad worsheets generally are matted so that the flow is left to right, top to bottom of an 8.5" x " page when the worshet is printed. - But I lie to use a second 8.5" page side-by-side additional material, added later, that I might later delete, e.g., scratchpad or temporary calculations. - Sometimes I use this extra right-margin space to do, in parallel, calculations needed later on, or to simply add material without disturbing the main flow. - When my worsheet has the second, side-by-side page and I want to publish the worsheet, I print the worsheet as an Adobe.pdf file. I specify the "ledger" mat, which, thanfully, prints out both pages side-by-side. These tips themselves fit my second, side-by-side page convention: they are additionaly material that is not part of the main flow of the worsheet. SecPerRev SecPerDeg36. Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6

2 . Input the observations and sensor coordinates and specify the weight matrix, W. Specify time (JDT), right ascension (RA), and declination (DEC) observations through n. Note that RA and DEC are referrred to the true equator and equinox of 8 January. Obs READPRN ("8_Ceres_with_LMT+h_minus_obs_3,6.txt" ) Obs Retrieve observations matrix from text file. Then set observation n and measurement count N: n N rows( Obs) n We define and then invoe the procedural functions JED9, AzRA, and ElDec to extract and convert the JDT, AZRA, and ELDEC observation vectors, respectively, n observations. DayCount specifies the count of days from the beginning of the year, up through the last day of the previous month of any non-leap year. JED8 calculates the number of Julian ephemeris days corresponding to Year, Month, and Day. Note that JED8 is intended to be used with any Gregorian calendar date since 8 January., having JED = Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6

3 DayCount ( ) T JED8( YearMonthDay) JED Y 8 Year if Y Year JED JED 365 if mod( Y4 ) otherwise if otherwise mod( Y) = JED JED 366 if mod( Y4) = JED JED 365 otherwise JED JED 366 otherwise JED JED DayCount Day Month JED JED JED if mod( Y4 ) otherwise if Month if mod( Y) = JED JED if mod( Y4) = JED JED otherwise JED JED JED JED otherwise otherwise JDTCalc( ) i JDT JED8 Obs Obs Obs i i i i3 Obs i4 JDT JDT i i 4 Obs i5 44 Obs i6 864 JDT JDT JDTCalc( n) Epoch JDT Epoch Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6 3

4 Assume in this worsheet that RA is expressed in deg/min/sec rather than hours/min/sec: AzRA( ) i AZRA i AZRA Obs i7 Obs i8 6 Obs i9 36 RA AzRA( n) Right ascensions are in radians. ElDec( ) i ELDEC i ELDEC i ELDEC Obs i ELDEC i Obs i 6 if Obs i Obs i 36 DEC ElDec( n) Declinations are in radians. WEIGHT( ) W i j W if i = j ij W otherwise ij W WEIGHT( N). Input the observer's geographical coordinates* in MPC mat. Sensor READPRN ("Observers.txt" ) NSen rows( Sensor) Sensor NSen Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6 4

5 Compute the observer's ("sensor's") ECI equatorial positions at the observation times via function SENPOS. We now that Piazzi used a meridian circle to mae his observations of Ceres. Since the right ascension of a target celestial body, in hours, as measured by a meridian circle, is the local apparent sidereal time of the observer, we tae advantage of this fact when we compute the right ascensions of the observer in SENPOS. SENPOS( ) i θ RA i R if R i j NSen Sensor j = Obs i9 RC Sensor j3 RS Sensor j4 Sensor j λ RCcos( θ) RCsin( θ) RS Note that λ is computed but not used in SENPOS. This is because a meridian circle measures the right ascension of the celestial object directly, and this is also the local apparent sidereal time. So the arguments of the cos and sin functions in SENPOS become (θ) instead of (θ + λ). *MPC observatory codes and the observers in year 86: R SEN SENPOS( n) 83 = Bremen (Olbers) 58 = Goettingen (Harding) 8 = Lilienthal (Bessel) = Palermo (Piazzi) Set values to zero to initialize light-time correction. DELTA( ) i Δ. i Δ Δ DELTA( N) Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6 5

6 3. Calculate sun's ECI positions at the observation times. Here we will use C, UPM and HGEO as defined in previous worsheets, convenience, even though UPM is not really needed the very-nearly-circular orbit of the Earth around the Sun. We will need function C to calculate the first four c-functions function UPM. C( x) N while x. x x 4 N N c 3 c c c 3 x c c x while N x x 4 x 7 x x 56 x 6 x x 3 x 56 x 9 x 3 x 8 N N c c c 3 c 3 4 c c c c c c c c c c c c c 3 T We will need the unim, two-body path propagator function, UPM, which will be invoed by function HGEO μ. K μ Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6 6

7 UPM( Kq e i Ω ω Δt ) α K ( e) q p q( e) Δt s q Δs s while Δs. c C αs f qs K es 3 c Δt 4 Df q K es c 3 DDf K esc m if Df m otherwise 5f Δs Df m ( 4Df ) f DDf s s Δs P cos( Ω) cos( ω) sin( Ω) cos() i sin( ω) P sin( Ω) cos( ω) cos( Ω) cos() i sin( ω) P sin( i) sin( ω) 3 Q ( cos( Ω) sin( ω) sin( Ω) cos() i cos( ω) ) Q ( sin( Ω) sin( ω) cos( Ω) cos() i cos( ω) ) Q sin( i) cos( ω) 3 c Cαs rcosv q K s c 3 rsinv K psc rcosvp rsinvq Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6 7

8 We need function HGEO to calculate the ECI equatorial J. coordinates of the Sun. (HGEO actually calculates the heliocentric ecliptic coordinates of the Earth-Moon barycenter, and then corrects them to the geocenter. See [] a reference.) HGEO( JD) JD o JD JD o T c a..5t c e t c q a( e) μ.34 K μ 3 n Ka ω i Ω. L T c SecPerDeg 46.94T c SecPerDeg SecPerRev SecPerDeg T JD mod( L ω π) n Δt JD T r EM UPM( Kq e i Ω ω Δt ) mod T c 36. L M r EM r EM.3cos L M.3sin L M r EM3 T c Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6 8

9 But we will here substitute the improved HGEO model. (As noted in Reference, Reference 4 provides an improved model. HGEO was taen from Reference 4.) HGEO( JD) JD o JD JD o T c a.6.6t c e t c q a( e) μ.34 K μ 3 n Ka ω i Ω. L T c t c T c T JD mod( L ω π) n Δt JD T r EM UPM( Kq e i Ω ω Δt ) mod T c 36. L M r EM r EM.3cos L M.3sin L M r EM3 Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6 9

10 In order to go from the mean equator and equinox of J. to the true equator and equinox of 8 January., we will need precession and nutation matrices, which will be applied to the Sun's position vector in function SUNPOS. PRECESS is the precession model of Capitaine, et al. []. PRECESS( rjd) T ψ A ( JD ) T.7969T.445T 3 SecPerRad ε o SecPerRad ω A ε o SecPerRad.5754T.563T.7753T 3 SecPerRad.55643T.3849T.97T 3 χ A SecPerRad S sinε o C cosε o S sinψ A C cosψ A S 3 sinω A C 3 cosω A S 4 sinχ A C 4 cosχ A P C 4 C S S 4 C 3 P C 4 S C S 4 C 3 C C S S 4 S 3 P C 3 4 S S S 4 C 3 C S C S 4 S 3 P S 4 C S C 4 C 3 P S 4 S C C 4 C 3 C C S C 4 S 3 P S 3 4 S S C 4 C 3 C S C C 4 S 3 P S 3 S 3 P S 3 3 C C S C 3 P S 33 3 C S C 3 C Pr Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6

11 We define our nutation matrix function, NUTATE, as follows. For a reference, see [3], p. 3. NUTATE( rjd) d JD ε d Δψ.48sin.4sin Δψ Δψ d Δε.6cos.cos Δε N Nr Δε. Δψcos( ε) Δψsin( ε) Δψcos( ε). Δε Δψsin( ε) Δε d d We need the obliquity of the ecliptic, ε, at J., in order to transm the ECI ecliptic J. coordinates of the Sun to ECI equatorial J. coordinates ε ECEQ( r) MOr MO cos( ε) sin( ε) sin( ε) cos( ε) (Transms from ecliptic to equatorial.) Define geocentric observer positions R plot points in HDC worsheet. These are 3 additional equally-space ephemeris points, 3 days apart, starting at JDT epoch. EQEC( r) MO r SUNPOS( ) R (Transms from equatorial to ecliptic.) i R i MOHGEOJDT i R i NUTATEPRECESSR i JDTi JDT i SUNPOS i 3 t JDT 3( i ) i R R i MOHGEOt i R i NUTATEPRECESSR i ti t i Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6

12 To obtain the ECI equatorial J. coordinates of the Sun, we compute - times the HCI equatorial J. position vector of Earth (this explains the minus sign of R in SUNPOS). R SUNPOS R SUN SUNPOS( n) R SUN R (Contemporary values would agree well with the Astronomical Almanac, Section C.) Calculate the Sun's position vectors relative to the observer at the observation times. Write to dis the plot points R: R SEN R R SUN (The constant is the number of Earth radii in one A.U.) WRITEPRN ("RVALS.prn" ) R 4. Calculate the N-by- observed measurements vector, Y. YVALUES( n) j i n i j Y cos DEC j i RA i Y DEC i Y Y YVALUES( n) (Note that function YVALUES converts n observations to N = n measurements.) Y (Clic on the column vector Y and scroll up or down to see all entries.) Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6

13 5. Specify the initial estimate of state, X o. We will start with the HCI ecliptic values of position and velocity as obtained from the HGM-Heliocentric worsheets HH and HHC. Note also that because HDC expects position and velocity to be HCI equatorial true of date (TOD), we will need to convert them from ecliptic to equatorial now, and bac to ecliptic in worsheet HDC, bee we can transm to conic elements there. r v ε JDT MO cos( ε) sin( ε) sin( ε) cos( ε) ECEQ( r) MOr EQEC( r) MO r (Transms from ecliptic to equatorial.) (Transms from equatorial to ecliptic.) r ECEQ( r) v ECEQ( v) X o stac( rv ) X o (Note that the epoch of this state vector is "Epoch", as defined above, and that the units are A.U. position and A.U. per day velocity.) Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6 3

14 6. Write test case specification values to dis use by worsheet HDC. WRITEPRN ("NOBS.dat" ) n Number of observations. WRITEPRN ("TVALS.prn" ) JDT Observation times. WRITEPRN ("EPOCH.dat" ) Epoch Epoch of state vector solution. WRITEPRN ("WEIGHTS.prn" ) W Measurement weights matrix. WRITEPRN ("RVALS.prn" ) R Values of R. WRITEPRN ("YVALS.prn" ) Y Values of Y. WRITEPRN ("STATE.prn" ) X o WRITEPRN ("RMS.prn" ) ( ) WRITEPRN ("DELTA.prn" ) Δ State vector (to be corrected by HDC). RMS history state corrections by HDC (one entry each iteration). Initialize light-time corrections. (The array is given dimension N rather than n to mae it easy to extract the values from function FXA in worsheet HDC, by augmenting the N-by- array that contains FX and A.) REFERENCES [] Seidelmann, P. K. (Editor), Explanatory Supplement to the Astronomical Almanac, University Science Boos (99), p. 36. This model was superseded by the model in [4]. [] Astronomical Almanac the Year 5, p. B5. Precession model of Capitaine, et al. [3] H.M. Nautical Almanac Office, Royal Greenwich Observatory and Nautical Almanac Office, U.S. Naval Observatory, Planetary and Lunar Coordinates the Years 984-, London and Washington, January 983. See "Auxiliary Data," pp More accurate nutation models are available, but this one is deemed sufficient the present application. [4] Urban, Sean, and P. K. Seidelmann, (Editors), Explanatory Supplement to the Astronomical Almanac, University Science Boos, 3rd Edition (3), p Hd_Ceres_8_Piazzi_7_obs_Theta=RA_plot.xmcd 6//6 4