# JED - December 2016 - 88

```E W

101

the Earth. To be consistent with our
spherical triangles, we will use capital
North
letters at the Angles and lower case for
Pole
the sides. But, of course, since this is
Threat
a plane triangle, the sides are physical
Location
lengths - rather than geocentric angles.
Latitude
E is at the satellite, F is at the threat,
of
Threat
and
G is at the center of the Earth.
Sub-vehicle
Point
The side e is the radius of the Earth
A
(6,371 km). The side f is the semi-major
Latitude
axis (the radius of the Earth plus the
of Sub-vehicle
b
satellite altitude = 10,560 km), angle G is
point
c
the side a from figure 2 above (21.57°),
and side g is the propagation distance
C
Center
a
between the satellite and the threat.
of Earth
B
The law of cosines for plane triangles
is: g2 = e2 + f2 - 2ef cos(G)
Fig 2: A spherical triangle is formed between the North pole, the sub-vehicle point and the
Plugging in values and solving for g:
threat location
g = sqrt[63712 + 105602 - 2 (6371)
(10560)(0.930)] = 5193 km
Cos a = cos b cos c + sin b sin c cos A
This
is
the
propagation
range
from the satellite to the threat.
Now that we know side a, side b and angle A, we can find
Figthe
2: A
sphericallaw
triangle
is formed between the North pole,
To find the elevation angle to the threat (angle E) from the
angle B from
(spherical)
of sines:
the sub-vehicle point and the threat location
plane triangle law of sines:

Longitude

The sine of angle B = sin b sin A / sin a

The Journal of Electronic Defense | December 2016

88

For example, let's choose a low Earth orbit with a period of 3
SinE/e = sinG/g,
hours. From table 2 in the June 2016 EW101 column, this satellite
so sinE = e SinG/g
will have a semi-major axis of 10,560 km. Let's specify that it has
Plugging in the values: E = arcsin[(6371)(0.368)/5193] = arca circular orbit. Then it will be at a constant altitude of 4,189 km.
sin(.451) = 26.8°
Our SVP is at 30° North latitude and 100° East longitude.
The target we are considering is on the surface of the Earth at
WHAT'S NEXT
45° North latitude and 120° East longitude
Next month, we will continue our discussion of space EW
The dimensions of the spherical triangle parts in the forwith the calculation of the range to the horizon and then the
mula are:
determination of the amount of time that the satellite can see
Angle A = 20°
the threat in a direct overhead pass. For your comments and sugSide b = 90° - 45° = 45°
gestions, Dave Adamy can be reached at dave@lynxpub.com. a
Side c = 90° - 30° = 60°
Plugging these values into the spherical triangle for sides:
Elevation
Propagation
Angle
Cos a = cos(45°) cos(60°) + sin(45°)
Satellite
E
Range
sin(60°) cos(20°)
= ( 0.707 x 0.5) + (0.707 x 0.866 x 0.940)
= 0.930
g
H
Threat
So side a = arccos (0.930) = 21.57°
Location
Now, from the spherical law of sines:
F
Sin (angle B) = sin(45°) sin(20°) /
f
sin(21.57°)
RE
e
= (0.707 x 0.342 ) / 0.368 = 0.657
So angle B = arcsin(0.657) = 41.08°
Side a
RE
This is the azimuth angle to the
From
target.
Figure 2

CALCULATING RANGE AND ELEVATION
TO THE THREAT LOCATION
Now consider figure 3. This is a plane
triangle in the plane that includes the
satellite, the threat and the center of

Center
of Earth

G

Fig 3: The elevation from nadir and range to a threat from a satellite can be determined from the
plane triangle defined by the satellite, the threat and the center of the Earth.

Fig 3: The elevation from nadir and range to a threat from a satellite
can be determined from the plane triangle defined by the satellite,
the threat and the center of the Earth.

```

# JED - December 2016

The View From Here
Conferences Calendar
Courses Calendar
From the President
The Monitor
Washington Report
World Report
Interview: Col Jeffrey N. Aldridge, USAF
Interview: US Rep. Joe Pitts
The Future of Airborne Electronic Attack
Regaining the Advantage – Cognitive Electronic Warfare
Technology Survey: Flightline and Portable EW Simulators
AOC EW Singapore Preview
New Products
Report from the 2016 EW GCC Conference
Crows in the Community
The Heat is On
EW 101
Report from the 2016 AOC CEMA Conference
2016 AOC Award Winners
AOC News
JED Quick Look
JED - December 2016 - cover1
JED - December 2016 - cover2
JED - December 2016 - 3
JED - December 2016 - 4
JED - December 2016 - 5
JED - December 2016 - The View From Here
JED - December 2016 - 7
JED - December 2016 - Conferences Calendar
JED - December 2016 - 9
JED - December 2016 - Courses Calendar
JED - December 2016 - 11
JED - December 2016 - From the President
JED - December 2016 - 13
JED - December 2016 - 14
JED - December 2016 - The Monitor
JED - December 2016 - insert1
JED - December 2016 - insert2
JED - December 2016 - 16
JED - December 2016 - 17
JED - December 2016 - 18
JED - December 2016 - 19
JED - December 2016 - 20
JED - December 2016 - 21
JED - December 2016 - 22
JED - December 2016 - 23
JED - December 2016 - 24
JED - December 2016 - 25
JED - December 2016 - Washington Report
JED - December 2016 - 27
JED - December 2016 - World Report
JED - December 2016 - 29
JED - December 2016 - Interview: Col Jeffrey N. Aldridge, USAF
JED - December 2016 - 31
JED - December 2016 - 32
JED - December 2016 - 33
JED - December 2016 - Interview: US Rep. Joe Pitts
JED - December 2016 - 35
JED - December 2016 - 36
JED - December 2016 - 37
JED - December 2016 - The Future of Airborne Electronic Attack
JED - December 2016 - 39
JED - December 2016 - 40
JED - December 2016 - 41
JED - December 2016 - 42
JED - December 2016 - 43
JED - December 2016 - 44
JED - December 2016 - 45
JED - December 2016 - 46
JED - December 2016 - 47
JED - December 2016 - 48
JED - December 2016 - 49
JED - December 2016 - 50
JED - December 2016 - 51
JED - December 2016 - 52
JED - December 2016 - 53
JED - December 2016 - 54
JED - December 2016 - 55
JED - December 2016 - Regaining the Advantage – Cognitive Electronic Warfare
JED - December 2016 - 57
JED - December 2016 - 58
JED - December 2016 - 59
JED - December 2016 - 60
JED - December 2016 - 61
JED - December 2016 - 62
JED - December 2016 - Technology Survey: Flightline and Portable EW Simulators
JED - December 2016 - 64
JED - December 2016 - 65
JED - December 2016 - 66
JED - December 2016 - 67
JED - December 2016 - 68
JED - December 2016 - 69
JED - December 2016 - 70
JED - December 2016 - AOC EW Singapore Preview
JED - December 2016 - 72
JED - December 2016 - 73
JED - December 2016 - New Products
JED - December 2016 - 75
JED - December 2016 - Report from the 2016 EW GCC Conference
JED - December 2016 - 77
JED - December 2016 - Crows in the Community
JED - December 2016 - 79
JED - December 2016 - The Heat is On
JED - December 2016 - 81
JED - December 2016 - 82
JED - December 2016 - 83
JED - December 2016 - 84
JED - December 2016 - 85
JED - December 2016 - EW 101
JED - December 2016 - 87
JED - December 2016 - 88
JED - December 2016 - 89
JED - December 2016 - Report from the 2016 AOC CEMA Conference
JED - December 2016 - 91
JED - December 2016 - 92
JED - December 2016 - 2016 AOC Award Winners
JED - December 2016 - 94
JED - December 2016 - 95
JED - December 2016 - 96
JED - December 2016 - 97
JED - December 2016 - 98
JED - December 2016 - 99
JED - December 2016 - AOC News
JED - December 2016 - 101
JED - December 2016 - 102
JED - December 2016 - 103
JED - December 2016 - 104
JED - December 2016 - Index of Advertisers
JED - December 2016 - JED Quick Look
JED - December 2016 - cover3
JED - December 2016 - cover4
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