两个新的Hansen系数的递推公式

Two New Recursion Formulae of Hansen Coefficients

推导给出了两个新的Hansen系数X_(k)^(-(n+1),m)的递推关系:{2k√1-e^(2)X^(-n,m)_(k)=(m-n)eX^(-(n+1),m-1)_(k)+2mX^(-(n+1),m)_(k)+(m+n)eX^(-(n+1),m+1)_(k),(R5){(m+1)(m-n-1)(n-m)e^(2)X^(-(n+1),m-2)_(k)_(m+1)(m+n-1)(2m-n-1)e^(2)X^(-(n+1),m)_(k)_(m-1)(m-n+1)(n+2m+1)e^(2)X^(-(n+1),m)_(k)+(m+1)(m-1)[4m+2me^(2)-4k(1-e^(2))^(3/2)]X^(-(n+1),m)_(k)+2k(m-1)(n+2m+1)e√1-e^(2)X^(-n,m+1)_(k)+2+(m+1)(2m-n-1)e√1-e^(2)X^(-n,m-1)_(k)_(m-1)(n+m+1)(n+m)e^(2)X^(-(n+1),m+2)_(k)=0,(R6)其中,n、m和k是Hansen系数X_(k)^(-(n+1),m)的3个指标,e为轨道偏心率.递推公式(R5)可以执行普通Hansen系数的向后递推,需要一行初值,公式简单.递推公式(R6)可以执行偏心率函数的向前递推,需要两行初值,比Vakhidov给出的递推公式明显简单.算例说明,这两种递推是有效的.

Two new recursion formulae of Hansen coefficents X_(k)^(−(n+1),m) are deduced:{2k√1-e^(2)X^(-n,m)_(k)=(m-n)eX^(-(n+1),m-1)_(k)+2mX^(-(n+1),m)_(k)+(m+n)eX^(-(n+1),m+1)_(k),(R5){(m+1)(m-n-1)(n-m)e^(2)X^(-(n+1),m-2)_(k)_(m+1)(m+n-1)(2m-n-1)e^(2)X^(-(n+1),m)_(k)_(m-1)(m-n+1)(n+2m+1)e^(2)X^(-(n+1),m)_(k)+(m+1)(m-1)[4m+2me^(2)-4k(1-e^(2))^(3/2)]X^(-(n+1),m)_(k)+2k(m-1)(n+2m+1)e√1-e^(2)X^(-n,m+1)_(k)+2+(m+1)(2m-n-1)e√1-e^(2)X^(-n,m-1)_(k)_(m-1)(n+m+1)(n+m)e^(2)X^(-(n+1),m+2)_(k)=0,(R6)where n,m and k are three indexes of Hansen coefficients X_(k)^(−(n+1),m),and e is the orbital eccentricity.Recursion formula(R5)can be used to perform the backward recursion of ordinary Hansen coefficents with one line of initial values,and it is simple.Recursion formula(R6)can be used to perform the forward recursion of eccentricity functions with two lines of initial values,and it is obviously simpler than Vakhidov’s recursion formula.Numerical example shows that these two new recursion formulae of Hansen coefficents are effective.