Chebyshev Polynomial

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

Chebyshev polynomials of the first kind, denoted $T_n(x)$ , are defined on $x \in [-1, 1]$ by the recurrence (Abramowitz and Stegun 1964, Ch. 22):

T_0(x) = 1, \qquad T_1(x) = x, \qquad T_{n+1}(x) = 2x\,T_n(x) - T_{n-1}(x), \quad n \ge 1

Equivalently, $T_n(x) = \cos(n \arccos x)$ . They are orthogonal with respect to the weight $w(x) = 1/\sqrt{1-x^2}$ :

\int_{-1}^{1} \frac{T_m(x) T_n(x)}{\sqrt{1-x^2}} \, dx = \begin{cases} 0, & m \neq n \\ \pi, & m = n = 0 \\ \pi/2, & m = n \neq 0 \end{cases}

The second kind $U_n(x)$ satisfies $U_n(\cos\theta) = \sin((n+1)\theta)/\sin\theta$ , but the first kind dominates astrodynamics applications.

A key property: among all monic polynomials of degree $n$ on $[-1, 1]$ , the scaled Chebyshev polynomial $2^{1-n}T_n(x)$ has the smallest maximum absolute value (the minimax property). This makes Chebyshev approximation nearly as good as the optimal minimax polynomial, and vastly simpler to compute.

Evaluation

The derivative $T_n'(x)$ is obtained via the recurrence for the second kind: $T_n'(x) = n U_{n-1}(x)$ . For numerical evaluation the Clenshaw recurrence is standard—it evaluates $\sum a_n T_n(x)$ in $O(N)$ with excellent numerical stability, avoiding explicit computation of each $T_n$ .

JPL Ephemeris Storage

The practical importance of Chebyshev polynomials in astrodynamics is tied to the JPL planetary and lunar ephemerides (DE/LE series). After numerically integrating the equations of motion for the solar system (variable-step Adams-type integrator), JPL fits the resulting positions and velocities with Chebyshev polynomials over contiguous time spans and stores only the coefficients (Standish 1990; Vallado 2022, Sec. 5.4).

For DE-245 and DE-405, the span lengths are:

4 days for the Moon
8 days for Mercury and the Earth-Moon libration
16 days for Venus, Earth, and the Sun
32 days for all other planets

The user who wants a planet's position at time $t$ simply locates the correct coefficient block, maps $t$ to $\tau \in [-1,1]$ , and evaluates the Chebyshev sum. This representation is compact (hundreds of coefficients per body per span vs. thousands of raw tabulated positions), differentiable (velocity and acceleration via $T_n'$ and $T_n''$ recurrences), and highly accurate (sub-meter for the Moon, sub-200 m for the Sun).

Orbit Approximation and Boundary Constraints

In trajectory design, Chebyshev polynomials are used to represent the position and velocity components of a reference orbit (e.g., a parking orbit about Earth or a distant retrograde orbit about the Moon) as smooth, differentiable functions of time. This allows the transfer-arc boundary conditions to be expressed as algebraic constraints on a finite set of Chebyshev coefficients—a much smaller optimization problem than pointwise constraint enforcement (Gomez et al. 2001, Vol. III, Sec. 4.3).

The same approach is used to fit quasi-periodic invariant tori: since a torus is a smooth surface parameterized by angles, expanding its embedding functions in Chebyshev series converges exponentially with the number of coefficients, enabling efficient storage and evaluation of high-dimensional dynamical structures.

First vs. Second Kind

Property	$T_n$ (First kind)	$U_n$ (Second kind)
Definition	$T_n(\cos\theta) = \cos(n\theta)$	$U_n(\cos\theta) = \frac{\sin((n+1)\theta)}{\sin\theta}$
Weight function	$1/\sqrt{1-x^2}$	$\sqrt{1-x^2}$
Derivative relation	$T_n' = n U_{n-1}$	—
Primary usage	Approximation, ephemeris	Numerical analysis (Gauss-Chebyshev quadrature)

References

Abramowitz and Stegun, 1964, Handbook of Mathematical Functions, Ch. 22 (Chebyshev polynomials: recurrence, orthogonality, minimax property)
Vallado, 2022, Fundamentals of Astrodynamics and Applications, Sec. 5.4 (JPL ephemeris Chebyshev representation; span lengths for DE-245/DE-405; accuracy for Moon and Sun)
Standish, 1990, The Observational Basis for JPL's DE 200, the Planetary Ephemerides of the Astronomical Almanac, Astron. Astrophys. 233:252–271 (Chebyshev coefficient fitting procedure for JPL ephemerides)
Gomez et al., 2001, Dynamics and Mission Design near Libration Points, Vol. III, Sec. 4.3 (Chebyshev representation of JPL ephemerides for CR3BP simulations)
Press et al., 1992, Numerical Recipes in C, Sec. 5.8 (Clenshaw recurrence for Chebyshev evaluation)