Motion of Orbiting Satellites

Lesson ~11 min read 8 MCQs

In simple terms: In simple terms, this topic is about how satellites move in circles or ovals around planets, and how the laws of conservation of energy and angular momentum explain their speed and path.

Why this matters

But what keeps them up there? Why don't they just fall back down or fly off into space? It's not magic; it's physics. A satellite's motion is a delicate dance between its forward speed and Earth's relentless gravitational pull. In this lesson, we'll explore the physics that governs these orbits, from perfect circles to long ellipses. We'll see how fundamental principles like conservation of energy and angular momentum are the cosmic choreographers of this dance.

Diagram

This diagram shows a central mass with two types of satellite orbits. A solid blue line depicts a circular orbit where all quantities are constant. A dashed yellow line shows an elliptical orbit, highlighting the perihelion (closest point, max speed) and aphelion (farthest point, min speed) to illustrate how energy and speed vary.

Concept map

flowchart TD
    A[Analyze Satellite Motion] --> B{Is orbital radius 'r' constant?};
    B -- Yes --> C[Circular Orbit];
    C --> D[K, U_g, E_total, and L are all constant];
    B -- No --> E[Elliptical Orbit];
    E --> F[E_total and L are constant];
    F --> G[K and U_g vary];
    G --> H[Perihelion (min r): max v, max K, min U_g];
    G --> I[Aphelion (max r): min v, min K, max U_g];

Core explanation

When we analyze the motion of satellites, we're looking at a system of two objects: a massive central body (like the Earth or the Sun) and a much, much smaller satellite. Because the central body is so massive, its own movement is tiny, so we can treat it as stationary. The satellite is the one doing all the interesting moving.

The force holding it all together is gravity. But when we talk about energy in space, we need a new perspective on gravitational potential energy.

The Universal Gravitational Potential Energy

You're used to U_g = mgh. That formula is great for things near Earth's surface, like a basketball at the top of its arc. But for satellites, where the distance r from the center of the Earth changes significantly, we need the universal formula:

U_g = -G(m₁m₂)/r

Think of it like being in a "gravity well." The satellite is trapped by the planet's gravity. To escape the well and get infinitely far away, you have to add energy to the system (like firing thrusters). We define the potential energy to be zero when the satellite is infinitely far away (r = ∞), free from the gravitational pull. Since you have to add energy to get to zero, your starting energy in the well must be negative.

A lower (more negative) potential energy means the satellite is deeper in the well—closer to the planet and more tightly bound by its gravity.

Two Types of Orbits: Circular and Elliptical

The path a satellite takes depends on the speed and direction it has when it enters orbit. All orbits are governed by conservation laws, but what's conserved depends on the shape.

1. Circular Orbits

This is the simplest case. A satellite in a perfect circular orbit moves at a constant distance r from the center of the central body.

Constant Speed and Kinetic Energy (K)
Since gravity F_g = G(Mm)/r² provides the constant centripetal force F_c = mv²/r, and r is constant, the speed v must also be constant. If speed is constant, the kinetic energy K = ½mv² is also constant.
Constant Potential Energy (U_g)
Since the distance r never changes, the gravitational potential energy U_g = -G(Mm)/r is also constant.
Constant Total Mechanical Energy (E)
Total energy is E = K + U_g. Since both K and U_g are constant, E is constant.
Constant Angular Momentum (L)
Angular momentum for a point mass is L = mvr (or more precisely, r x p). Since m, v, and r are all constant, L is constant.

In a circular orbit, everything is constant and stable. It's a smooth, steady ride.

2. Elliptical Orbits

Most orbits, like those of comets or many spy satellites, are elliptical. The satellite's distance r from the central body changes continuously. The central body sits at one of the two foci of the ellipse.

The two most important points in an elliptical orbit are:

Perihelion
The point of closest approach (r is minimum).
Aphelion
The point of farthest distance (r is maximum).

Here's how conservation laws apply:

Conserved: Total Mechanical Energy (E)
Gravity is a conservative force, so as long as it's the only force doing work (no air drag, no thrusters), the total energy E = K + U_g of the system is constant.
Conserved: Angular Momentum (L)
Gravity always pulls the satellite toward the central body. This force is "central," meaning it exerts no torque on the satellite. With zero net torque, angular momentum L = mvr is conserved.

This is the key! Since L is constant, the product v * r must be constant.

At perihelion, r is at its minimum. To keep L constant, the speed v must be at its maximum.
At aphelion, r is at its maximum. To keep L constant, the speed v must be at its minimum.

Now think about energy. Since E = K + U_g is constant:

At perihelion (max speed), kinetic energy K is at its maximum. To keep E constant, potential energy U_g must be at its minimum (most negative). This makes sense, as r is smallest.
At aphelion (min speed), kinetic energy K is at its minimum. To keep E constant, potential energy U_g must be at its maximum (least negative). This also makes sense, as r is largest.

Think of it like a cosmic skateboarder in a valley. The satellite "falls" toward the planet, picking up speed (trading potential for kinetic energy) until it whips around at perihelion. Then it coasts "uphill" away from the planet, slowing down (trading kinetic for potential energy) until it reaches its high point at aphelion, and the cycle repeats.

Escaping the Well: Escape Velocity

What if you give a satellite so much kinetic energy that it can climb all the way out of the gravity well and never come back? The minimum speed to do this is called escape velocity.

To "just escape" means the satellite will get infinitely far away (r → ∞) and have just run out of speed (v → 0).

Let's use energy conservation to find the formula. The total energy required to be "free" is zero.

At r = ∞, U_g_final = -G(Mm)/∞ = 0.
At r = ∞, K_final = ½m(0)² = 0.
So, the total final energy is E_final = 0.

By conservation of energy, the initial energy at launch must also be zero: E_initial = K_initial + U_g_initial = 0

Let's say we launch from a distance r from the center of the planet with mass M. The initial speed is the escape velocity, v_esc.

½mv_esc² + (-G(Mm)/r) = 0

Now, solve for v_esc: ½mv_esc² = G(Mm)/r v_esc² = 2GM/r v_esc = √(2GM/r)

Worked examples

Example 1

The International Space Station (Circular Orbit)

The International Space Station (ISS) orbits at an altitude of approximately 408 km above the Earth's surface. Assuming a circular orbit, what is its orbital speed? (Mass of Earth M_E = 5.97 x 10²⁴ kg, Radius of Earth R_E = 6.37 x 10⁶ m, G = 6.67 x 10⁻¹¹ N·m²/kg²).

1. State the Goal: We need to find the speed v of the ISS.

2. Identify the Physics: For a circular orbit, the gravitational force provides the centripetal force. F_g = F_c G(M_E * m_ISS) / r² = (m_ISS * v²) / r

3. Find the Orbital Radius (r): This is a classic trap! The r in the equation is the distance to the center of the Earth, not the altitude above the surface.

Altitude h = 408 km = 408,000 m = 4.08 x 10⁵ m
Orbital radius r = R_E + h
r = (6.37 x 10⁶ m) + (0.408 x 10⁶ m) = 6.778 x 10⁶ m

4. Solve for v: Let's go back to our force equation. Notice the mass of the ISS, m_ISS, cancels out! G * M_E / r² = v² / r Multiply both sides by r: v² = G * M_E / r v = √(G * M_E / r)

Now, plug in the numbers: v = √((6.67 x 10⁻¹¹ * 5.97 x 10²⁴) / (6.778 x 10⁶)) v = √(3.98 x 10¹⁴ / 6.778 x 10⁶) v = √(5.87 x 10⁷) v ≈ 7660 m/s

That's over 17,000 miles per hour!

Example 2

A Comet's Journey (Elliptical Orbit)

A comet is in an elliptical orbit around the Sun. At its aphelion (farthest point), it is 5.0 x 10¹² m from the Sun and moving at 900 m/s. At its perihelion (closest point), it is 8.0 x 10¹⁰ m from the Sun. What is its speed at perihelion?

1. State the Goal: Find the speed v_p at perihelion.

2. Identify the Physics: In an elliptical orbit, angular momentum is conserved. The mass of the comet m is constant. L_aphelion = L_perihelion m * v_a * r_a = m * v_p * r_p

3. Solve for v_p: The comet's mass m cancels out. v_a * r_a = v_p * r_p v_p = (v_a * r_a) / r_p

4. Plug in the numbers:

v_a = 900 m/s
r_a = 5.0 x 10¹² m
r_p = 8.0 x 10¹⁰ m

v_p = (900 m/s * 5.0 x 10¹² m) / (8.0 x 10¹⁰ m) v_p = (4.5 x 10¹⁵) / (8.0 x 10¹⁰) v_p = 56,250 m/s

Why it makes sense: The comet is much closer to the Sun at perihelion, so to conserve angular momentum, it must be moving much, much faster. It whips around the Sun at high speed before heading back out to the far reaches of the solar system.

Try it yourself

Practice Problem 1

Satellite A is in a circular orbit of radius R around a planet. Satellite B is in a circular orbit of radius 2R around the same planet. Both satellites have the same mass. How does the total mechanical energy of Satellite A (E_A) compare to that of Satellite B (E_B)?

Practice Problem 2

A probe is in a highly elliptical orbit around Jupiter. Point P is its perihelion (closest point) and Point A is its aphelion (farthest point). At which point is the probe's angular momentum greater? At which point is the Jupiter-probe system's total mechanical energy greater?

TL;DR

In simple terms, this topic is about how satellites move in circles or ovals around planets, and how the laws of conservation of energy and angular momentum explain their speed and path.

Key terms

AP Physics 1 Satellite motion Circular orbit Elliptical orbit Conservation of energy Angular momentum Escape velocity Gravitational potential energy Perihelion Aphelion ===END===

You can now…

6.6.A: Describe the motions of a system consisting of two objects interacting only via gravitational forces.

Essential knowledge (exam-tested)

6.6.A.1: In a system consisting only of a massive central object and an orbiting satellite with mass that is negligible in comparison to the central object's mass, the motion of the central object itself is negligible.
6.6.A.2: The motion of satellites in orbits is constrained by conservation laws.
6.6.A.2.i: In circular orbits, the system's total mechanical energy, the system's gravitational potential energy, and the satellite's angular momentum and kinetic energy are constant.
6.6.A.2.ii: In elliptical orbits, the system's total mechanical energy and the satellite's angular momentum are constant, but the system's gravitational potential energy and the satellite's kinetic energy can each change.
6.6.A.2.iii: The gravitational potential energy of a system consisting of a satellite and a massive central object is defined to be zero when the satellite is an infinite distance from the central object. Relevant equation: U_g = -G(m₁m₂)/r
6.6.A.3: The escape velocity of a satellite is the satellite's velocity such that the mechanical energy of the satellite-central-object system is equal to zero.
6.6.A.3.i: When the only force exerted on a satellite is gravity from a central object, a satellite that reaches escape velocity will move away from the central body until its speed reaches zero at an infinite distance from the central body.
6.6.A.3.ii: The escape velocity of a satellite from a central body of mass M can be derived using conservation of energy laws. Derived equation: v_esc = √(2GM/r)

Concept map

flowchart TD
    A[Analyze Satellite Motion] --> B{Is orbital radius 'r' constant?};
    B -- Yes --> C[Circular Orbit];
    C --> D[K, U_g, E_total, and L are all constant];
    B -- No --> E[Elliptical Orbit];
    E --> F[E_total and L are constant];
    F --> G[K and U_g vary];
    G --> H[Perihelion (min r): max v, max K, min U_g];
    G --> I[Aphelion (max r): min v, min K, max U_g];

Read what Saavi narrates

Have you ever used your phone's GPS to get to a friend's house in a new part of town, say, navigating the suburbs of Atlanta? Or maybe you've watched a live soccer game being broadcast from halfway across the world. Both of these everyday marvels rely on satellites orbiting high above the Earth. But what keeps them up there? It's a delicate dance between its forward speed and Earth's relentless gravitational pull.

In this lesson, we're going to explore the physics that governs these orbits. We'll see that it all comes down to two main ideas. First, gravity provides the inward pull, the centripetal force, that bends the satellite's path into an orbit. Second, the shape of that orbit, whether it's a perfect circle or a stretched-out ellipse, tells us which important quantities, like energy and angular momentum, stay constant.

Let's take a look at a worked example. Imagine a comet in an elliptical orbit around the Sun. At its farthest point, called aphelion, it's five times ten to the twelfth meters away, and it's moving at a relatively slow 900 meters per second. The question is, how fast is it moving when it gets to its closest point, or perihelion, which is eight times ten to the tenth meters from the Sun?

The key here is that for any elliptical orbit, angular momentum is conserved. The formula for angular momentum is L equals m v r. So, the angular momentum at aphelion must equal the angular momentum at perihelion. That means m times v-aphelion times r-aphelion equals m times v-perihelion times r-perihelion.

The comet's mass, m, is on both sides, so we can cancel it out. Now we just rearrange the equation to solve for the speed at perihelion. v-perihelion equals v-aphelion times r-aphelion, all divided by r-perihelion.

Plugging in the numbers, we get 900 meters per second, times five times ten to the twelfth meters, divided by eight times ten to the tenth meters. When you run the calculation, you get fifty-six thousand, two hundred and fifty meters per second. It's moving so much faster! And that makes perfect sense. To keep angular momentum constant, when the radius gets smaller, the velocity has to get bigger. The comet whips around the sun at incredible speed.

One of the most common mistakes I see on this topic is forgetting that gravitational potential energy is negative. The formula is U-g equals negative G times m1 times m2, all over r. That negative sign is so important. It means the satellite is "trapped" in a gravity well. A more negative energy means it's deeper in the well, and more tightly bound. A less negative energy, which is a mathematically larger number, means it's farther away. If you forget that negative sign, all your energy comparisons will be backward.

So as you study orbits, remember to focus on those conservation laws. They are your guide to understanding the beautiful, predictable motion of everything from tiny satellites to giant comets. You've got this.

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Settings & Accessibility

Motion of Orbiting Satellites

Why this matters

Diagram

Concept map

Core explanation

The Universal Gravitational Potential Energy

Two Types of Orbits: Circular and Elliptical

1. Circular Orbits

2. Elliptical Orbits

Escaping the Well: Escape Velocity

Worked examples

The International Space Station (Circular Orbit)

A Comet's Journey (Elliptical Orbit)

Try it yourself

Practice Problem 1

Practice Problem 2

Practice — 8 questions