ISSN 1812-3368. ȼɟɫɬɧɢɤ ɆȽɌɍ ɢɦ. ɇ.ɗ. Ȼɚɭɦɚɧɚ. ɋɟɪ. «ȿɫɬɟɫɬɜɟɧɧɵɟ ɧɚɭɤɢ». 2012
50
=
arccos
r
p
+
arccos
0
r
p
;
(7)
k
0
q
0
sgn
M M
0
0
1/ 2
( )
k q
g t dt
M
M
³
= arccos
q
r
+ arccos
0
r
q
.
(8)
Ɋɢɫ. 1. Ɋɚɫɩɨɥɨɠɟɧɢɟ ɫɬɚɰɢɨɧɚɪɧɵɯ ɬɨɱɟɤ ɨɩɬɢɱɟɫɤɨɝɨ ɢɧɬɟɝɪɚɥɚ ɜ ɚɫɢɦɩɬɨ-
ɬɢɱɟɫɤɨɦ ɩɪɢɛɥɢɠɟɧɢɢ
ɍɪɚɜɧɟɧɢɟ (7) ɨɩɪɟɞɟɥɹɟɬ ɬɪɚɟɤɬɨɪɢɸ ɥɭɱɚ, ɩɪɢɲɟɞɲɟɝɨ ɜ ɬɨɱɤɭ
ɧɚɛɥɸɞɟɧɢɹ, ɨɬɪɚɡɢɜɲɢɫɶ ɨɬ ɩɪɟɩɹɬɫɬɜɢɹ. ɍɪɚɜɧɟɧɢɟ (8) ɞɥɹ ɫɬɚɰɢɨ-
ɧɚɪɧɨɣ ɬɨɱɤɢ
Q
R
= k
0
q
ɨɩɢɫɵɜɚɟɬ ɬɪɚɟɤɬɨɪɢɸ ɥɭɱɚ, ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ
ɢɞɭɳɟɝɨ ɢɡ ɬɨɱɤɢ ɢɫɬɨɱɧɢɤɚ ɜ ɜɵɛɪɚɧɧɭɸ ɬɨɱɤɭ ɧɚɛɥɸɞɟɧɢɹ. Ɂɞɟɫɶ
(
r
0
,
M
0
), (
r
,
M
) —
ɩɨɥɹɪɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɟɱɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɢ ɬɨɱɤɢ
ɧɚɛɥɸɞɟɧɢɹ.
Ʌɟɜɚɹ ɱɚɫɬɶ ɪɚɜɟɧɫɬɜɚ ɫɨɞɟɪɠɢɬ ɜɵɪɚɠɟɧɢɟ, ɫɜɹɡɚɧɧɨɟ ɫ ɜɥɢɹɧɢ-
ɟɦ ɞɢɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɫɥɨɹ:
0
k q
g
=
2
0
( )
k q
–
g
(
t
).
Ɂɞɟɫɶ
g
(
t
)
=
Z
2
(
H
–
H
0
)
2
2
( )
t a
U
ª
º
¬
¼
,
ɝɞɟ
Z
—
ɱɚɫɬɨɬɚ;
H
,
H
0
—
ɞɢɷɥɟɤ-
ɬɪɢɱɟɫɤɚɹ ɩɪɨɧɢɰɚɟɦɨɫɬɶ ɫɥɨɹ ɢ ɫɪɟɞɵ;
a
–
ɪɚɞɢɭɫ ɰɢɥɢɧɞɪɚ;
r =
=
U
(
M
)
—
ɩɨɥɹɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨɜɟɪɯɧɨɫɬɢ ɞɢɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɫɥɨɹ;
k
0
—
ɜɨɥɧɨɜɨɟ ɱɢɫɥɨ ɜɧɟɲɧɟɣ ɫɪɟɞɵ.
ȿɫɥɢ ɫɥɨɣ ɨɬɫɭɬɫɬɜɭɟɬ, ɥɟɜɚɹ ɱɚɫɬɶ ɪɚɜɟɧɫɬɜɚ (8) ɨɱɟɜɢɞɧɨ ɩɪɟ-
ɜɪɚɳɚɟɬɫɹ ɜ
0
,
M M
ɬ. ɟ. ɭɪɚɜɧɟɧɢɟ (8) ɜɵɪɨɠɞɚɟɬɫɹ ɜ ɢɡɜɟɫɬɧɨɟ
ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɩɪɚɜɨɣ ɫɬɚɰɢɨɧɚɪɧɨɣ ɬɨɱɤɢ ɨɩɬɢɱɟɫɤɨɝɨ ɢɧɬɟɝɪɚɥɚ
ɤɥɚɫɫɢɱɟɫɤɨɣ ɡɚɞɚɱɢ ɪɚɫɫɟɹɧɢɹ ɩɨɥɹ ɬɨɱɟɱɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɧɚ ɢɞɟɚɥɶ-
ɧɨ ɩɪɨɜɨɞɹɳɟɦ ɤɪɭɝɨɜɨɦ ɰɢɥɢɧɞɪɟ [1]:
0
M M
= arccos
r
q
+ arccos
0
q
r
,
(9)