The residual gravity acceleration effect in the Poincaré dodecahedral space

Free Access

Table 1:

Statistical characteristics of coefficients a_i of the dominant (ith order) term in the radial and orthogonal components of the residual acceleration $\ddot{r}$ in perfectly regular well-proportioned spaces, for approximately isotropic displacements $\vec{r}$ ^a.
space	$\mbox{term}^{b}$	$\parallel/\perp^{c}$	$\left<a_i\right>$	$\sigma_{\left<a_i\right>}^{d}$	$\sigma_i^{e}$	$\gamma_i^{f}$
3-torus	(r/L_a)³	$\parallel$	0.00	0.00	6.11	0.58
3-torus	(r/L_a)³	$\perp$	6.39	0.00	6.82	-0.42
octahedral	$(r/R_{{\rm C}})^3$	$\parallel$	0.01	0.01	3.18	-0.58
octahedral	$(r/R_{{\rm C}})^3$	$\perp$	3.33	0.00	1.26	-0.42
${\rm tr. cube}$	$(r/R_{{\rm C}})^3$	$\parallel$	-0.05	0.05	14.24	0.58
${\rm tr. cube}$	$(r/R_{{\rm C}})^3$	$\perp$	14.94	0.02	5.63	-0.42
${\rm dodec/num}^{g}$	$(r/R_{{\rm C}})^5$	$\parallel$	-0.30	0.53	288.29	0.75
${\rm dodec/alg}^{h}$	$(r/R_{{\rm C}})^5$	$\parallel$	0.00	0.01	288.26	0.74
${\rm dodec/num}^{g}$	$(r/R_{{\rm C}})^5$	$\perp$	286.33	0.22	121.65	-0.37

^a Coefficients a_i as defined in Eqs. (14), (19), and (20); these are approximately constant with respect to r; the constant factor of Gm/L_a² for the T³ model or $Gm/R_{{\rm C}}^2$ for the other models has been ignored here; all values shown are dimensionless.
^b Dominant ith power of displacement, as derived in this paper.
^c Radial $\parallel$ or orthogonal $\perp$ component.
^d Standard error in the mean $\sigma_{\left<a_i\right>} = \sigma_i/\sqrt{N-1}$ for $N \gg 1$ test particles.
^e Sample standard deviation.
^f Sample skewness $\gamma_i = \left<[(a_i- \left<a_i\right>)/\sigma_i]^3 \right>$ .
^g From 70-bit significand numerical calculations using Eq. (9).
^h Using the algebraic expression in Eq. (21).

Source LaTeX | All tables | In the text

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.