4. Let's look in more detail at how a satellite is moved from one circular orbit to another. The figure shows two circular orbits, of radii ₁ and r2, and an elliptical orbit that connects them. Points 1 (perigee) and 2 (apogee) are at the ends of the semimajor axis of the ellipse. (a) A satellite moving along the elliptical orbit has to satisfy two conservation laws. Use these two laws to prove that the velocities at points 1 and 2 are v₁ = 2GM¹2 7₁+72 and V₂ = 2GM¹1 T₁+T2 (b) Consider a 1000 kg communications satellite that needs to be boosted from an orbit 300 km above the earth to a geosynchronous orbit 35,900 km above the earth. Find the velocity v₁ on the inner circular orbit and the velocity v₁' at the perigee on the elliptical orbit that spans the two circular orbits. (c) How much work must the rocket motor do to transfer the satellite from the circular orbit to the elliptical orbit? (d) Now find the velocity v₂' at the apogee of the elliptical orbit and the velocity v₂ of the outer circular orbit. (e) How much work must the rocket motor do to transfer the satellite from the elliptical orbit to the outer circular orbit?
4. Let's look in more detail at how a satellite is moved from one circular orbit to another. The figure shows two circular orbits, of radii ₁ and r2, and an elliptical orbit that connects them. Points 1 (perigee) and 2 (apogee) are at the ends of the semimajor axis of the ellipse. (a) A satellite moving along the elliptical orbit has to satisfy two conservation laws. Use these two laws to prove that the velocities at points 1 and 2 are v₁ = 2GM¹2 7₁+72 and V₂ = 2GM¹1 T₁+T2 (b) Consider a 1000 kg communications satellite that needs to be boosted from an orbit 300 km above the earth to a geosynchronous orbit 35,900 km above the earth. Find the velocity v₁ on the inner circular orbit and the velocity v₁' at the perigee on the elliptical orbit that spans the two circular orbits. (c) How much work must the rocket motor do to transfer the satellite from the circular orbit to the elliptical orbit? (d) Now find the velocity v₂' at the apogee of the elliptical orbit and the velocity v₂ of the outer circular orbit. (e) How much work must the rocket motor do to transfer the satellite from the elliptical orbit to the outer circular orbit?
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![**How to Transfer a Satellite Between Circular Orbits**
This section explores the detailed process of transferring a satellite from one circular orbit to another. The accompanying diagram illustrates two circular orbits with radii \( r_1 \) and \( r_2 \), along with an intervening elliptical orbit. The points marking the closest (perigee) and furthest (apogee) sections of the elliptical path are labeled as points 1 and 2, respectively, and are positioned at the ends of the semimajor axis.
**Diagram Explanation:**
- The inner and outer circular orbits are shown with radii \( r_1 \) and \( r_2 \).
- The transfer ellipse connects these two orbits, with points 1 (perigee) and 2 (apogee).
**Steps to Transfer a Satellite:**
(a) **Elliptical Orbit Velocities:**
- A satellite in an elliptical orbit must satisfy conservation laws. The velocities at points 1 and 2 are defined by:
\[
v_1 = \sqrt{\frac{2GM r_2}{r_1 (r_1 + r_2)}}
\]
\[
v_2 = \sqrt{\frac{2GM r_1}{r_2 (r_1 + r_2)}}
\]
(b) **Example Scenario:**
- For a 1000 kg communications satellite moving from a 300 km orbit to a geosynchronous orbit 35,900 km above Earth, compute \( v_1 \) on the inner orbit and \( v_1' \) at the perigee of the elliptical path.
(c) **Work Done by Rocket Motor (Inner to Elliptical):**
- Calculate the work required to transition the satellite from the inner orbit to the transfer ellipse.
(d) **Finding Velocity \( v_2' \) and \( v_2 \):**
- Determine the velocity \( v_2' \) at the apogee of the elliptical orbit and \( v_2 \) on the outer circular orbit.
(e) **Work Done by Rocket Motor (Elliptical to Outer):**
- Calculate the work required to move the satellite from the elliptical path to the outer orbit.
This process illustrates the application of orbital mechanics to effectively change a satellite's trajectory and altitude for desired positioning in space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbf0bff4d-5282-48d4-ae5d-254768574db3%2F248233e1-1e46-4426-be92-86ea00a6b43e%2F5ag2rt_processed.png&w=3840&q=75)
Transcribed Image Text:**How to Transfer a Satellite Between Circular Orbits**
This section explores the detailed process of transferring a satellite from one circular orbit to another. The accompanying diagram illustrates two circular orbits with radii \( r_1 \) and \( r_2 \), along with an intervening elliptical orbit. The points marking the closest (perigee) and furthest (apogee) sections of the elliptical path are labeled as points 1 and 2, respectively, and are positioned at the ends of the semimajor axis.
**Diagram Explanation:**
- The inner and outer circular orbits are shown with radii \( r_1 \) and \( r_2 \).
- The transfer ellipse connects these two orbits, with points 1 (perigee) and 2 (apogee).
**Steps to Transfer a Satellite:**
(a) **Elliptical Orbit Velocities:**
- A satellite in an elliptical orbit must satisfy conservation laws. The velocities at points 1 and 2 are defined by:
\[
v_1 = \sqrt{\frac{2GM r_2}{r_1 (r_1 + r_2)}}
\]
\[
v_2 = \sqrt{\frac{2GM r_1}{r_2 (r_1 + r_2)}}
\]
(b) **Example Scenario:**
- For a 1000 kg communications satellite moving from a 300 km orbit to a geosynchronous orbit 35,900 km above Earth, compute \( v_1 \) on the inner orbit and \( v_1' \) at the perigee of the elliptical path.
(c) **Work Done by Rocket Motor (Inner to Elliptical):**
- Calculate the work required to transition the satellite from the inner orbit to the transfer ellipse.
(d) **Finding Velocity \( v_2' \) and \( v_2 \):**
- Determine the velocity \( v_2' \) at the apogee of the elliptical orbit and \( v_2 \) on the outer circular orbit.
(e) **Work Done by Rocket Motor (Elliptical to Outer):**
- Calculate the work required to move the satellite from the elliptical path to the outer orbit.
This process illustrates the application of orbital mechanics to effectively change a satellite's trajectory and altitude for desired positioning in space.
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