Correction to: Representation of light pressure resultant force and moment as a tensor series

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Correction to: Celest Mech Dyn Astr (2017) 128:483–513 DOI 10.1007/s10569-017-9758-8

In the paper Nerovny et al. (2017), the commentaries about a convergence of series which represent the absolute value function and corresponding equations contain several mistakes (Sect. 2, from Eqs. (4) to (6)).

The series Eq. (3)

$$\begin{aligned} |\hat{\mathbf {n}}\cdot \hat{\mathbf {s}}|=|x| = \frac{2}{\pi } - \frac{4}{\pi }\sum \limits _{n=1}^{\infty }\frac{(-1)^n T_{2n}(x)}{-1+4n^2} \end{aligned}$$

of Chebyshev polynomials of the first kind for \(|\hat{\mathbf {n}}\cdot \hat{\mathbf {s}}|=|x|\le 1\) is absolutely convergent. If we define \(x=\cos y\), than \(T_{2n}=\cos 2ny\), \(|T_{2n}|\le 1\), and we get the ordinary Fourier series which is majorizable by the following convergent series:

$$\begin{aligned} \frac{2}{\pi } -\frac{4}{\pi }\sum \limits _{n=1}^{\infty }\frac{1}{-1+4n^2}. \end{aligned}$$

Additionally, for any x the original series is an alternating Leibniz series. Its partial sum differs from |x| less or equal than the absolute value of the first neglected term.

These are the steps to produce a power series of absolute value function from Eq. (3):

$$\begin{aligned} |\hat{\mathbf {n}}\cdot \hat{\mathbf {s}}|&= -\lim \limits _{N_{\max }\rightarrow \infty } \frac{4}{\pi }\sum \limits _{n=1}^{N_{\max }}\sum \limits _{k=0}^{n-1}\frac{(-1)^n(-1)^k n (2n-k-1)!}{(-1+4n^2)k!(2n-2k)!}4^{n-k}(\hat{\mathbf {n}}\cdot \hat{\mathbf {s}})^{2(n-k)}=\\&(\text {let}\ m = n-k)\\&=-\lim \limits _{N_{\max }\rightarrow \infty }\sum \limits _{m=1}^{N_{\max }} \frac{(-1)^m 4^{m+1}}{\pi (2m)!}\sum \limits _{n=m}^{N_{\max }}\frac{n(n+m-1)!}{(-1+4n^2)(n-m)!} (\hat{\mathbf {n}}\cdot \hat{\mathbf {s}})^{2m}. \end{aligned}$$

That’s why the Eqs. (4) and (5) from Nerovny et al. (2017) should be written as follows:

$$\begin{aligned} |\hat{\mathbf {n}}\cdot \hat{\mathbf {s}}|= & {} \lim \limits _{N_{\max }\rightarrow \infty }\sum \limits _{m=1}^{N_{\max }} B_m (\hat{\mathbf {n}}\cdot \hat{\mathbf {s}})^{2m}\approx \sum \limits _{m=1}^{N_{\max }} B_m (\hat{\mathbf {n}}\cdot \hat{\mathbf {s}})^{2m},\\ B_m\approx & {} -\frac{(-1)^m 4^{m+1}}{\pi (2m)!}\sum \limits _{n=m}^{N_{\max }}\frac{n(n+m-1)!}{(-1+4n^2)(n-m)!}, \end{aligned}$$

and in equations for \(N_{\max B}\), Eqs. (6) and (34), the \(\lfloor (N_{\max }-1)/2 \rfloor \) term should be replaced by \(N_{\max }\).

The results of calculations in Sects. 7 and 8 are not affected by this error because in the numerical calculations we used correct relations presented in this erratum.

Reference

  1. Nerovny, N., Zimin, V., Fedorchuk, S., Golubev, E.: Representation of light pressure resultant force and moment as a tensor series. Celest. Mech. Dyn. Astron. 128, 483–513 (2017). doi:10.1007/s10569-017-9758-8

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Correspondence to Nikolay Nerovny.

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The online version of the original article can be found under doi:10.1007/s10569-017-9758-8.

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Nerovny, N., Zimin, V., Fedorchuk, S. et al. Correction to: Representation of light pressure resultant force and moment as a tensor series. Celest Mech Dyn Astr 129, 555–556 (2017). https://doi.org/10.1007/s10569-017-9791-7

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