System before tether was cut

 

*

Vorb – orbital velocity of the system

Vc – 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

Fci, Fci-1tighting 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 < wa

 

 

 

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


w ~ wa

 

 

w > wa

 


 

 

 



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

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

*   - angle of incidence