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Issue 52

08 Flash Archive · Members Only

Jack’s Astro Corner: Inclination – The Tilted Element (Part III)

Jack's Astro Corner: Inclination - The Tilted Element (Part III) Over the summer, Jack Anthony will break down each of the six orbital elements required to uniquely identify a specific orbit and satellite in that orbit. This week we examine inclination. For those who can't wait the entire summer, please visit Jack's “…

Author Minhas Published July 17, 2022

Jack's Astro Corner: Inclination – The Tilted Element (Part III) Over the summer, Jack Anthony will break down each of the six orbital elements required to

uniquely identify a specific orbit and satellite in that orbit. This week we examine inclination.

For those who can't wait the entire summer, please visit Jack's “Orbit Element Dance” on

YouTube” and you'll find a 1:02 video featuring Jack in his driveway demonstrating this highly

effective way to learn about the 6 classical orbital elements (COE). Each movement ties to an

important astrodynamics principle. Below is a screen grab from this video. As you can see,

Jack uses the STP method of remembering the 6 COEs. Size, Shape, Tilt, Twist, Position of

Perigee and Position of the Satellite at a particular time. Boogie Down. –

You are HERE

We are now moving into the “T’s” of the STP method of remembering the Classical Orbital Elements. TILT is for orbit Inclination. I’m sure by now you have learned the Orbit Element Dance, the dance move for TILT is a crisp bend forward (please mind the hamstrings). To get us started, I’m going to introduce some inclination definitions. I will define several inclination buzz words so you can use them and dazzle friends, family and Generals. We will learn about the significance of the inclination 63.4° as well the many satellites in the vicinity of 98° (no relation to the 1990s Boy Band…which apparently is still going strong).

Most Astro enthusiasts will tell you “Inclination measures the tilt of the orbit away from the equatorial plane.” This is about right, but begs the question, “which way do you measure from the equatorial plane to the orbit plane?” As long as you make that clear, you should be OK. Let’s look at the REAL definition of inclination and how to solve for its value. We’re going to talk vector math….don’t panic! I just love vectors (inclination and the remaining 3 orbital elements involve angles between vectors!) Inclination is defined as the angle between the K axis of the Earth Centered Inertial (ECI) coordinate frame and the orbit angular momentum vector, denoted the h-vector. Let me show you in a simple illustration I made where the IJK red coordinate frame is the ECI coordinate frame and the h-vector is shown in green, it’s perpendicular to the orbit plane. Do you see the angle between K axis and the H-vector?

Jack's Astro Corner: Inclination - The Tilted Element (Part III) - page 8

Jack's Astro Corner: Tilt (Cont)

Here are two illustrations from my favorite astrodynamics book by Bate, Mueller and White

“Fundamentals of Astrodynamics”, fondly called BMW (it was

recently updated, Bill Saylor is the 4th author now). You’ll see

the ever popular Earth Centered Inertial coordinate frame.

Look close at the I and J or X and Y axis. They form the

fundamental plane, it’s the equatorial plane. The K axis

is the Earth’s spin axis, it is perpendicular to the equatorial

Right Hand Rule

plane. There’s a lot more science and details to this, but this is

the basic stuff of ECI. Remember the K axis, because we are

now going to introduce the h-vector.

What’s this h-vector? It’s the angular momentum of the orbit. Here’s where you get to

demonstrate your right-hand rule “astro gangster” sign. If you take your right hand and form a

two finger and thumb coordinate frame you are cool! Then if you curve your fingers in the

direction of orbit travel, your thumb will point in the direction of the h-vector. Here below is an

illustration from BMW that shows angle between h-vector and k-vector.

Inclination = Angle between h-vector and K-vector

You’ll see the Astro smarties doing this all the time. It shows the right-hand coordinate system.

Here I am a few years back with several Weapons School/Astro experts showing off the right-

hand rule also known as the “Astro gangster” sign.

I can see you are itching to learn how to calculate the angle between

2 vectors. The dot product of two vectors equals the Cosine of the

angle between them. Let’s try it. In vector notation, K-vector is

simply [0, 0, 1), h-vector of the orbit is [-1, +2, -0.3]. What’s the

"Astro Gangster Sign"

in the Wild

angle between them? (Checkout this online tool. What answer did

you get? (answer: 97°)

Jack's Astro Corner: Inclination - The Tilted Element (Part III) - page 9

Jack's Astro Corner: Tilt (Cont)

If you ask around or surf the internet you’ll probably get

a few different explanations for inclination. I’ll use the

NASA drawing to show this. It shows Inclination is the

angle between an equator’s plane the Earth Centered

Inertial (ECI) coordinate frame and the orbit’s plane. Here

we see a straight on view of the ascending node of the orbit.

To me this looks like a 45° inclined orbit. The maximum latitude it will fly directly over is 45° North

and South. Make sure you keep track of the ascending node passage and which way the satellite is

heading; 45° and 135° can look alike. With the h-vector definition previously explained this is

clearer. No worry, you’ll figure it out, right Bueller?

Now that you know what inclination is, let’s learn the “lingo” of this “tilted” orbit element called

inclination. For orbits with an inclination of more than 90° we call them retrograde. If less than 90

degrees, we have a prograde or direct orbit. If the inclination is

zero degrees, we call that an equatorial orbit. Very popular for

the geosynchronous satellites. I mentioned 90 degrees earlier,

that’s a Polar orbit. The polar orbit has a ground track that flies

over the poles. Here’s a weird one nobody has tried, an

inclination of 180 degrees. It is called a retrograde equatorial

orbit. Have you ever heard of a sun-synchronous orbit? Well,

hang on, we first must learn about some orbit perturbations

that affect an orbit and that can be used to create an orbit of great operational benefit.

The central gravitational force of the Earth is what makes satellites orbit the Earth. We assume a

spherical Earth and that’s pretty good for starters. But in reality, it’s not a spherical point source of

gravity. The Earth has oblateness, that is it's squished a little and football shaped, wider at equator

and shorter to the poles. Can’t really see it, but this is true. This causes some additional forces to

be put on an orbiting satellite. The biggy Earth oblateness effect is called the J2 effect. It has an

effect mostly the right ascension of the ascending node (RAAN) (twist) and the Argument of

Perigee (defines where the closest approach to Earth is). You can get a constant change over time

in those orbit elements; it’s called a secular perturbation. The equation that determines the

motion of perigee due to J2 can be solved to find the inclination where perigee will not drift. There

are two specific inclinations where the rate of change of perigee movement due to J2 is zero. Holy

cow, now that can be helpful. The answer is 63.4 and 116.6 degrees and it’s called the critical

inclination. Remember the HEO orbit? Well, if you launched into 63.4-degree inclined orbit and

had perigee positioned in the southernmost part of your orbit, then it would not drift away from

there and rotate away from this southernmost point. That’s helpful for keeping the “hang time”

part of the orbit (apogee) over the northern hemisphere. Remember that from the Eccentricity

article?

Let’s look at another source of forces on a satellite in its orbit: The Sun and the Moon gravity. For

the GEO satellites, the Sun and Moon’s gravity play a big role in what happens to their inclination.

Jack's Astro Corner: Inclination - The Tilted Element (Part III) - page 10

Jack's Astro Corner: Tilt (Cont)

These gravitational bodies are far away, but do “pull” on the GEO birds and thus create something

called the 54-year cycle. If the orbit inclination is left to its own, that is not “north-south”

controlled, the inclination will go through a 15-degree span in 54 years. Again, it can get really

complicated as to how this plays out, but inclination goes for a 15-degree cycle in 54 years. Most

GEO satellite operators don’t allow this to happen. They activate their propulsion systems to keep

the inclination at zero or within certain limits.

Last thing, remember I mentioned

sun-synchronous orbit? Well, let’s learn what

inclinations support that orbit. It is a VERY

popular orbit for Earth observing space

systems (imagery). Here’s superb illustration

courtesy of Technobytes.org, it shows what’s

going on with the sun-synch orbit.

The orbit plane maintains it’s orientation to

the Sun. This is helpful since you can fly over

a place of interest every day at the same local

time and have the same Sun illumination

orientation of the place below, photo analysts

dig that. This is orbit plane movement caused by the J2 perturbation. Let’s look at an example

below.

What are the sun-synchronous orbit inclinations? Here’s a

plot that helps answer that. Thank you SpaceDaily.com for

this plot. Let’s say you are in an 800 Km circular altitude

orbit, well, if you launched into a 98.6 degrees inclined

orbit, your orbit plane will remain in the same orientation

to the Sun. You set your launch time to get the timing you

want. Why does this happen? Well, you get the RAAN to

drift about +1 deg/day. This is about how much the Sun

moves about the Earth per day. So, the angular

relationship holds steady as the earth orbits the Sun and

your orbit plane moves +1 deg/day. Now you know about

the sun-synchronous orbit. Pop quiz, what is the inclination

needed for a 400 km circular orbit altitude? “Anyone, anyone? Bueller?"

Next time we’ll learn about TWIST, the Right Ascension of the Ascending Node (RAAN). Be sure to

practice your orbit element dance. I’m sure your Guardian shipmates will value you demonstrating

it, maybe during a senior office visit to your unit. I want to thank 3rd Test & Evaluation Squadron

Commander Anna “Sumo” Gunn-Golkin for reviewing a draft of this article. She was a great Astro

teacher at USAF Academy and today leads the premier test squadron in Space Force. Yay Anna!

Finally, General Willie Shelton also took a look and helped tweak this essay.

Jack's Astro Corner: Inclination - The Tilted Element (Part III) - page 11

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