JED - February 2017 - 45

```E W

the threat is calculated by determining
side c in Figure 2 from the formula:
cos c = cos a x cos b
Where side a is the latitude of the
threat and side b is the difference in
longitude between the ascending node
and the threat.
The time it takes the satellite subvehicle point to reach the threat is determined from
satellite period x (side c/360°)
For example, if the satellite period
is 120 minutes, the latitude of the
threat is 45°, and the longitude of the
threat is 100 degrees from the ascending node. The geocentric angle from
the ascending node to the threat is:
arccos[ cos(45°) cos(100°)] = 97°
The satellite arrives over the threat
120 minutes x (97/360) = 32.33 minutes after passing the ascending node.

TIME DURING WHICH A SATELLITE
CAN SEE A POINT ON THE EARTH
The Earth surface distance to the
horizon from a satellite is the distance
from the threat to either point F or
point G. Thus, the satellite can see
the threat during the time it takes
the sub-vehicle point to travel from
F to G. The geocentric angle between
points F and G is twice the arccos
axis of the orbit). Note, that you can
see this relationship in Figure 3 of the
December 2016 column, and the angle
is shown as angle H in Figure 4 this
month.

Fig 4: The area of the Earth's surface seen by a satellite is a function of
the subvehicle point and the altitude of the satellite.

The time that it takes the sub-vehicle
point to travel from G to H can be calculated from:
period of satellite (angle H/360°)
For example, if the satellite period is
180 minutes, the semi-major axis is 10,560
km. The radius of the Earth is 6,371 km,
so angle H is
2 x arccos (6371/10560) = 105.78°
Therefore, the observation time is:
180 minutes x (105.78/360) =
52.8 minutes

rotates during that time. The observation time over the rotating earth is
longer than the observation time over a
non-rotating Earth because the satellite
goes over a fixed latitude and longitude
point at a slower net speed. The satellite's speed over the Earth is the actual
satellite speed reduced by the speed at
which the Earth is moving under the satellite. This is defined by the following
(rather ugly) formula:

THE IMPACT OF THE
MOVEMENT OF THE EARTH
In the interest of avoiding spending
a whole column on some rather abstract
spherical trig, we will make a "pretty
close" estimate of the amount of additional look time the satellite gets because of
the Earth's rotation.
The satellite's sub-vehicle point moves
to the East over the Earth according to the
relationship:
(2π RE / P) x cos i
where R E is the radius of the Earth, P is
the satellite's period, and i is the inclination of the orbit.
But the Earth rotates to the east 366
times per year. This means that a spot on the
Earth (like the threat location) moves east
at 0.25068 equatorial degrees per minute.
This means the eastward speed of a
point on the Earth is 27.90 km/minute reduced by the cosine of its latitude.
The time it takes the satellite to move
from point F to point G (in Figure 4) is
increased by the amount that the Earth

Where a is the satellite semi-major
axis (i.e., the radius of a circular orbit).
For example: Let's make the period of
a circular orbit satellite 180 minutes, the
inclination of the orbit 60° and the latitude of the threat 45° North.
We calculated the observation time
over a non-rotating Earth to be 52.8
minutes. The observation time over the
rotating Earth is found by plugging into
the above formula. The result of this
calculation is 57.9 minutes. This is 5.14
minutes longer than it would be over a
non-rotating Earth.

WHAT'S NEXT
Next month, we will put the information from the last few columns
together to look at the problem of intercepting a ground based threat from
dave@lynxpub.com. a

The Journal of Electronic Defense | February 2017

Fig 3: This figure shows the great circle plane of the satellite orbit. The
angle from the ascending node increases at a constant rate.

101

45

```

JED - February 2017

The View From Here
Conferences Calendar
Courses Calendar
From the President
The Monitor
World Report
Asia-Pacific SIGINT Programs
Technology Survey: Analog-to-Digital Converters
Operator 101
EW 101
AOC News
JED Quick LookThe
JED - February 2017 - cover1
JED - February 2017 - cover2
JED - February 2017 - 3
JED - February 2017 - 4
JED - February 2017 - 5
JED - February 2017 - The View From Here
JED - February 2017 - 7
JED - February 2017 - Conferences Calendar
JED - February 2017 - 9
JED - February 2017 - Courses Calendar
JED - February 2017 - 11
JED - February 2017 - From the President
JED - February 2017 - 13
JED - February 2017 - 14
JED - February 2017 - The Monitor
JED - February 2017 - insert1
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