System before tether was cut

Vorb – orbital velocity of the system

V_c – capsule velocity.

where w - angular velocity

Tether length

Capsule is a material particle

q - degree of vertical deviation

Model of the tether

Tether is divided into N parts

Each part is exerted by next forces:

Fg – gravity force Fs – air drag force

Fc_i, Fc_i-1 – tighting force (from parts i+1 and i-1)

For each part we must solve system of equations

Where . For axis X

, where - air density.

The behavior of tether systems strongly depends on speed w and angle q at the moment of cutting the tether.

w < w_a

w_a – acceptable velocity, which provide the system to descend faster without tether curling

w ~ w_a

w > w_a

q,°	w,radian/s	T, s	H, km	Vatm, m/s	,°
- 0.55	-0.087	2500,00	294.54	7736.153	-0.0044
- 4.97	-0.071	2500,00	102.88	7870.356	-0.5009
-27.75	-0.068	2134.77	100.07	7841.792	-0.6538
-28.42	-0.067	2076.38	100.02	7853.078	-0.4502
-29.09	-0.066	2025.65	100.1	7889.012	-0.7203
-31.63	-0.062	1870.01	100.07	7898.699	-1.2857
-35.68	-0.054	1695.18	100.05	7852.437	-1.0876
-38.71	-0.047	1595.7	100.07	7883.782	-1.6178
-39.18	-0.046	1582.36	100.12	7884.907	-1.5416
-39.63	-0.045	1569.55	100.18	7851.654	-1.1541
-40.07	-0.044	1557.34	100.19	7862.066	-1.6279
-40.5	-0.042	1545.6	100.01	7880.388	-1.7694
-40.92	-0.041	1534.55	100.18	7904.365	-1.6847
-41.33	-0.04	1524.5	100.01	7850.917	-0.8665
-44.75	-0.028	1444.53	100.12	7846.288	-2.3742
-45.55	-0.025	1427.92	100.17	7811.891	-1.8289
-45.79	-0.024	1423.17	100.17	7879.122	-1.5613
-46.44	-0.02	1409.99	100.15	7872.515	-1.6017
-46.64	-0.019	1406.2	100.02	7869.419	-1.6239
-48.12	-0.002	1377.96	100.15	7880.931	-1.5227

q - degree of vertical deviation

w - angular velocity

T - time

H - Height

V_atm - capsule velocity at H=100 km (or T=2500 s)

- angle of incidence