Manuscript En01-B@2 · En01-B-v2.tex · proposition · f226d1eb

Proposition: Unsmeared continuous-sphere rate

Beyond quadratic localization: an explicit control › Exact homogeneous-sphere calculation · lines 1571–1611 · proof included or linked

En01-B@2#prop:sphere

For a continuous homogeneous sphere,

Γ_{0} (d) = K_{G} H (d / R),

where

H (u) = {u^{2} /2 - 3 u^{3} /16 + u^{5} /160, 6/5 - 1/ u, 0 \leq u \leq 2, u \geq 2.

In particular $A_{0} = G M^{2} / (2ℏ R^{3})$ .

Proof. The intrinsic density autocorrelation is

w_{0} (r) = (\frac{3 M}{4 π R ^{3}})^{2} \frac{π}{12} (4 R + r) (2 R - r)^{2} 1_{{r \leq 2 R}} .

It is the overlap volume of two spheres times the squared density. Its Newtonian pairing at displacement $d$ is

J (d) = 4 π \int_{0}^{2 R} \frac{r ^{2} w _{0} ( r )}{max ( r , d )} d r .

Integrating the polynomial separately below and above $r = d$ gives

J (d) = \frac{M ^{2}}{R} (\frac{6}{5} - \frac{u ^{2}}{2} + \frac{3 u ^{3}}{16} - \frac{u ^{5}}{160})

for $u \leq 2$ , and $J (d) = M^{2} / d$ otherwise. The rate is $G [J (0) - J (d)] /ℏ$ . Its quadratic coefficient follows at $d = 0$ .

Displayed formulas in this passage

unlabeled · line 1574
$Γ_{0} (d) = K_{G} H (d / R),$
unlabeled · line 1578
$H (u) = {u^{2} /2 - 3 u^{3} /16 + u^{5} /160, 6/5 - 1/ u, 0 \leq u \leq 2, u \geq 2.$
unlabeled · line 1590
$w_{0} (r) = (\frac{3 M}{4 π R ^{3}})^{2} \frac{π}{12} (4 R + r) (2 R - r)^{2} 1_{{r \leq 2 R}} .$
unlabeled · line 1598
$J (d) = 4 π \int_{0}^{2 R} \frac{r ^{2} w _{0} ( r )}{max ( r , d )} d r .$
unlabeled · line 1603
$J (d) = \frac{M ^{2}}{R} (\frac{6}{5} - \frac{u ^{2}}{2} + \frac{3 u ^{3}}{16} - \frac{u ^{5}}{160})$