# A satellite is in Earth's orbit with an altitude at the perigee of 550.0 km and an altitude at...

## Question:

A satellite is in Earth's orbit with an altitude at the perigee of 550.0 km and an altitude at the apogee of 5000.0 km. If the true anomaly is 195 degrees, what is the satellite's altitude?

## True Anomaly:

When a satellite revolves round a planet in elliptical path, its angular position is specified by an angular parameter called True Anomaly. The angle between the direction of periapsis and the current position of satellite as seen from main focus of ellipse is called the angle of 'true anomaly'.

## Answer and Explanation:

Given:

Radius of apogee, {eq}R_a=5000\ km \\ {/eq}

Radius of perigee, {eq}R_p=550\ km \\ {/eq}

True anomaly, {eq}\theta = 195^\circ \\ {/eq}

Semi Major axis, {eq}a= \dfrac{R_a+R_b}{2} \\ a=9150\ km {/eq}

Eccentricity {eq}e=\dfrac{R_a+R_b}{R_a-R_p} =0.243 {/eq}

Radius {eq}r= \dfrac{a(1-e^2)}{1+e \cos195} \\ r= \dfrac{9150(1-0.243^2)}{1+ 0.243 \cos195} \\ r =1125\ km {/eq}

Therefore

Altitude, {eq}h=r=R_E \\ h=1125-6375 \\ h=4876\ km \\ {/eq}

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