2. The Different Frames and the Keplerian Elements

All Good Things Start With Frames

For a satellite to control its attitude, it needs to know its position along with that of some other stuff wandering in space. This can include a star, sun, moon or other heavenly bodies in our galaxy. Sensors on-board the satellites use these heavenly bodies as a reference to provide vector observations . Don’t worry if you could not get hold of the last line, things will be clearer as we progress through future posts. Apart from this, we can also determine the various physical disturbances acting on the satellite in space because of the different heavenly bodies, by accurately knowing their locations.

So, the obvious question which hits our curious mind is, how do we make vector observations of things which are zipping around the universe? To record a vector, we need a co-ordinate system, a frame of reference. Now, which frame should we be using to make some larger than life calculations?

As always, the key to simplify these vector calculations is to take the right frame of reference. And in this situation, where everything, from sun to moon, earth to satellite, is not at all stationary, we need to be extremely careful while selecting the frames and this selection will keep on varying with the observer and the subject.

Considering the non-inertial nature of satellites and the heavenly bodies, it will be a respite for us if we observe them from an inertial frame of reference. This stationary nature of observer can help us in easing out our calculations by getting rid of an additional velocity component that could have been imparted from the non-inertial frame.

Few of the important things which describe any co-ordinate system are, the fundamental plane(i.e. the X-Y place), the principal direction (i.e. the direction of X axis) and the direction of Z axis. Since the Z-axis must be perpendicular to the fundamental plane. The Y axis is chosen to form the right handed set of co-ordinate axes. Sounds a bit complicated? Don’t mind! You will develop a better idea about these things once we start reading specifically about the different frames.

Now, let’s talk about our first inertial frame of reference! The Earth Cantered Inertial (ECI) frame!

The Earth Cantered Inertial (ECI) frame

The Geocentric-Equatorial Coordinate System a.k.a. the Earth Centred Inertial frame has its origin right at the centre of the earth, however it is not fixed to the earth.

Although this frame has its origin at the centre of the earth, but it does not rotate with the earth.

this video can help you better visualize the ECI frame. The red lines in the video denote the three axes of the ECI frame. The fundamental plane contains the equator and the positive X-axis points in the vernal equinox direction. The Z-axis points in the direction of the geographical North Pole and the Y axis consequently completes the right hand set of co-ordinate axes. Here is a picture of this,

a1

Figure 1. The Earth Centred Inertial(ECI) frame

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Now you might have been wondering, what do we mean by Vernal Equinox? Well, to make things a bit more interesting and to understand other inertial frames we need to throw some light on vernal equinox and something called the Keplerian Elements.

Vernal Equinox

You might have heard this strange term “vernal equinox” before, and in most cases we have been told that it is “the place in the sky where the sun rises on the first day of Spring”. This definition is not just vague and confusing, but it is terrible. Most of us don’t have any idea what is the first day of spring and why the sun should be in the same place in the sky on that date every year.

To get a better picture of vernal equinox, imagine the Sun’s orbit around the earth. Yes, right, the earth orbits the sun, but if we start observing from the earth, things will look the other way but the math is equally valid this way too (remember the concept of relative motion?). The plane formed by the hypothetical orbit of the sun around the earth is called the ecliptic. Similar to the ecliptic, we have the equatorial plane, the plane formed by the equator. Now, the sun will intersect the equatorial plane at two points in one orbit, one where the sun crosses the equator while ascending, the point where the sun pops out the equatorial plane and goes up, and the other one when the sun is descending, it punches the equator and dives down before coming up again. If you join these two points, you will get the nodal line for the sun’s orbit. Well, well, well, forgot to tell you something, the sun’s ascending node is called the Vernal Equinox!

Similar to the nodal line of the sun, we have nodal line for the satellites too. These lines are formed by the intersection of the plane formed by satellite’s orbit with the equatorial frame.

Since we have understood the concept of nodal line, let’s move forward and understand the Keplerian Elements.

Keplerian Elements, also referred as the orbital elements or simply the elements are a set of parameters used to define an elliptical orbit.

The first in the list is orbital eccentricity, sometimes referred as eccentricity

Eccentricity (e),

Eccentricity of an elliptical orbit lies between zero and one. In this domain of zero to one, higher the eccentricity, more elliptical is the orbit.

Semi major axis (a)

Imagine an ellipse traced by the satellite moving around the earth as its focus. The longest possible straight line in this ellipse is its major axis (2a, Figure 2), so half of this is the semi major axis. Cool?

Inclination (i),

The orbital inclination is the angle between the satellite’s orbital plane and the equatorial plane. Orbits having inclination lesser than 90 degrees are called prograde orbit and the orbits with inclination greater than 90 degrees are called retrograde orbit. Satellites in a prograde orbit rotate in the same direction as that of the earth, while those in retrograde, rotate in direction opposite to that of the earth.

Next comes the Right Ascension of Ascending node (RAAN)

 

Right Ascension of Ascending node (Ω)

Too weird a name, right? Never mind! This is the angle subtended between the nodal line of the sun (vernal equinox side) and that of the satellite (nodal line of the satellite from ascending node side).

Well, RAAN is not the only one with a horrible name, one another element with comparatively lesser weird name is the “Argument of Perigee”

Argument of Perigee

To visualise this we will have to dig a bit deeper into the satellite’s orbit. A satellite in an elliptical orbit will have the earth at one of its focus. So the satellite following the elliptical path around the earth will be closest to the earth at one point of time and farthest at the other. The point where the satellite and our planet maintains the minimum distance is known as perigee, on the other hand, the point where satellite is at the highest possible distance from the earth is called the apogee.

a2Figure 2. Elliptical Orbit

The vector in the direction from the earth’s centre to the perigee defines the eccentricity vector, whose magnitude equals the eccentricity of the orbit.   The angle subtended between the eccentricity vector and the nodal line of the satellite marks the “Argument of Perigee”.

True anomaly

Yet another orbital element with another weird name!

Consider a vector from earth’s origin to the satellite, let’s call this vector as the position vector. Now imagine an angle subtended by this vector and the eccentricity vector. This angle is what we call the “True anomaly “

Mean motion

Number of revolution the satellite completes per day. As simple as that!

Mean Anomaly

For a satellite moving in an elliptical orbit, if we draw a circle which passes through the apogee and perigee of the ellipse and has its centre at the centre of the ellipse, and consider a hypothetical satellite to be moving along that circle with angular velocity equal to the average angular velocity of the satellite moving in the elliptical orbit, then the angle subtended between the position vector of the satellite moving on the newly made circle and the eccentricity vector of the elliptical orbit is called the mean anomaly.

a3Figure 3. Orbital Elements

Now, since we are done with the orbital elements, we can peacefully move on and understand the other reference frames

Perifocal Frame

This is popularly known as the “natural frame” for an orbit. This frame is centred at the centre of the earth, the orbital plane is the fundamental plane (XY plane) itself, X axis is directed to the eccentricity vector, Z axis is in the direction of the satellite’s angular momentum which lies perpendicular to the orbital plane, and the Y axis completes the right hand set of co-ordinate axis.

a4Figure 4. Perifocal Frame

Let’s now move towards understanding the non-inertial frame. We previously said that an inertial frame helps us in simplifying the calculations, so why are we now bringing the non-inertial frames into picture? One way of answering this question is by looking at the Earth’s magnetism. The magnetic field produced by earth differs with the location. So if we have a frame which rotates with the earth, then it will be very convenient to locate different points on earth relative to this moving frame as all points on the earth will be stationary to this frame. This fact is crucial for a satellite having a magnetic field sensor (magnetometer) which needs to know its position with respect to different points of the earth. This non inertial frame is not just important for predicting magnetic field at different points in the orbit but it is also important for any satellite which is bothered about the different points on earth, be it for taking pictures of different parts of the world or for locating the different ground station on the home planet. And last but not the least, the non-inertial frame is the key for analysing the dynamics of a satellite!

Earth Centred Earth Fixed (ECEF) Frame

This frame keeps on rotating with the Earth. Centred at the equator, it has the equatorial plane as its fundamental plane (XY plane), the X axis can be traced by joining a line starting from the centre of the earth, to the point of intersection of the prime meridian and the equator, the Z axis points towards the geographical north pole and the Y axis completes the right hand set of co-ordinate axes.

The green axes in the video represent the ECEF frame.

a5Figure 5. Earth Centred Earth Fixed (ECEF) frame

Since the magnetic field produced by the satellite at different points in space in known, the ECEF frame can be used to locate the position of the satellite with respect to the moving earth, and consequently find the magnetic field at that point.

The frames we have talked about so far are the ones used to observe the satellite and other things in space from the earth, but then we cannot observe everything from our planet, to stabilise the satellite, we need to change our perspective, we need to look at the things from the satellite.

Orbit Reference Frame

The orbit reference frame has its origin at the centre of mass of the satellite, the Z axis points towards the centre of mass of the earth, the X axis which is perpendicular to the Z axis, and is in the direction of the velocity of the satellite, and the Y axis completes the right hand set of co-ordinate axis. In this article, the axes of orbit reference frame are denoted with a subscript R (XR, YR and ZR). Irrespective of the satellite’s orientation in the space, the orbit reference frame will always have its Z axis pointing towards the centre of the Earth, X axis pointing towards the velocity and the Y axis will be completing the right hand set of co-ordinate axes.

a6Figure 6. Inertial frame, Orbit Reference Frame and the Satellite Body frame

The inertial frame has been denoted by the axes set XI, YI, and ZI.

Satellite Body frame

Satellite body frame is fixed to the satellite’s body, with its origin at the centre of mass of the satellite. This frame is used to represent the actual satellite in space. The X, Y and Z axis need to be perpendicular to each other and should be popping out of the different faces of the satellite. An example of Satellite Body frame has been given in figure 7.

a7Figure 7. Satellite Body Frame

 

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