Transit satellite navigation

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Problem 7.3a

Determine the acceleration of gravity at the orbit of a Transit satellite 1070 km above the Earth, knowing that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{g} at the surface of the Earth is 9.81 m/sFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): ^{2} , and that the gravitational force varies inversely as the square of the distance between the centers of gravity of the masses. The radius of the Earth is 6370 km.

Background

A satellite is in a stable orbit around the Earth when the gravitational force Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): (\textit{mg}) pulling it earthward equals the centrifugal force Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): mV^{2}/R , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g is the acceleration of gravity, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V the satellite’s mass and velocity, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): R the radius of its orbit about the center of the Earth.

Solution

The radius of the satellite’s orbit is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(6370+1070\right)=7440 km. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): g is proportional to the force of gravity, at the satellite’s orbit,

$ {\begin{aligned}g=9.81(6370/7440)^{2}=7.19\ \mathrm {m/s} ^{2}.\end{aligned}} $

Problem 7.3b

What is the satellite’s velocity if its orbit is stable?

Solution

For a stable orbit, the gravitational acceleration is balanced by the centripetal acceleration.

Thus,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} V^{2} /R=g=7.19\ \mathrm{m/s}^{2},\\ V=(7.19\times 10^{-3} \times 7440)^{1/2} =7.31\ \mathrm{km/s}. \end{align}

Problem 7.3c

How long does it take for one orbit?

Solution

The length of the nearly circular orbit is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2\pi \times 7440 km, so the time for one orbit is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} T=2\pi \times 7440/7.31=6395\ \mathrm{s} =106\ \mathrm{minutes}, 35\ \mathrm{seconds}. \end{align}

Problem 7.3d

How far away is the satellite when it first emerges over the horizon?

Solution

In Figure 7.3a, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{O} is the point of observation. The satellite first becomes visible when it reaches the tangent to the Earth at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{O} . The tangent is normal to the radius at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{O} , so

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} x=(7440^{2} -6370^{2} )^{1/2} =3840\, {\rm km.} \end{align}

Figure 7.3a  First visibility of satellite.

Problem 7.3e

What is the maximum time of visibility on a single satellite pass?

Solution

In Figure 7.3a the angle subtended at the center of the Earth by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{x} is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \theta =\cos^{-1} \left(6370/7440\right)=31.1^{\circ} . \end{align}

The satellite is visible while it traverses an arc subtending Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2\theta =62.2^{\circ} . Since the entire orbit is traversed in 6395 s, the time of visibility is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 6395\left(62.2/360\right)=1105\ \mathrm{s}=18 minutes, 25 seconds.

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