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Friday, June 13, 2014

A three dimensional view of orbit

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Satellites in Polar orbit move from pole to pole in every orbit. For example 3 orbits of such a satellite are shown in this figure.

 This view is actually true ONLY IF we are sitting on a platform outside Earth ( but which is rigidly tied to Earth ) and looking to Earth. So such a view is a boring static scene wherein the Earth is stationary and the Satellite is moving around it in almost a circular pattern. Boring ..  isn’t it seeing the same circular pattern repeating over and over again.

But such a platform is impossible to be built. So let’s try to have a more realistic ( and a dynamic ) view. Let’s visualize  these satellite orbits from a stand alone platform  fixed to solar system.

So..  what happens now? The Earth is moving around Sun at about 14Kms/sec. Therefore the scene that we were seeing from earlier mentioned platform is now moving with that speed. In short now the satellite motion  gets superimposed with Earth’s movement to around Sun. So the movement of Satellite gets vectorially added with Earth’s speed.
Points 1 to 5 are the positions of sunsynchronous satellite at equal interval. As the Earth moves  towards right the satellite system which is a part of Earth also moves along and the Satellite movement  gets vectorialy added with Earth’s speed and point 2 in adjacent figure appears at 2' instead of at 2 when Earth has moved to position B. Similarly 3,4 5 ...  etc appear at 3', 4' ,5'  and so on. In this way the circular orbit appears to have been stretched and the resultant orbit shape is elongated ( like a cycloid ) as depicted by the path in green line. But there is one more complexity. The plane of the ellipse is also rotating because it has to keep a constant angle with sun ( we have taken an example of a Sunsynchronous satellite ). This motion is best viewed when looking from top ( i.e. looking into Earth through N Pole .) as seen in the adjacent figure.
The points 2,3,4,5 ..  etc are actually not in the same plane ( say plane of paper ) but are sequentially behind the plane of paper. So the points 2', 3'  etc are also placed behind the paper. But since the Erth speed is affecting in vertical plane the points do not get moved in this plane.

So overall the circular path of satellite around Earth gets stretched like a spiral as shown in subsequent diagrams.  Also the  orbits appeared to be crossing over poles as seen in 1st diagram above but in reality they never cross each other in space because the Earth is not stationary as is always shown to be in all the diagrams.

We use the orbits of ‘Chandra X Ray observatory’  spacecraft to generate these graphics. Chandra orbit as drawn in conventional diagrams is on left.  ( ‘Chandra’ is a NASA spacecraft for cosmic studies named after Subramanian Chandrasekhar, 1910 -1995 ). Chandra has 133000 kms Apogee and at perigee it is as near to Earth as 16000 Kms. Inclination is 60 deg.
With 64.47 Hrs period the subsequent perigees ( or apogees ) are separated in space by (14kms/sec)*(64.47 Hrs) = 3200000 kms along the Earth’s path around Sun ( ecliptic ).
So how does the orbit look from space? 

Here is a view from a point just outside the ecliptic path and we look towards the satellite.The graphic is generated for 6 days.  Earth position at 00:00 GMT everyday is shown as a blue dot while the Earth path  is seen as a pink line. Notice the curvature caused due to Earth’s orbit around Sun along the ecliptic.

Green line is the path that Chandra Probe has taken during this period. Notice that twice the green line has gone below the pink line. These are 2 consecutive perigees that the spacecraft has undergone in the 6 day period. The transit of spacecraft from one side of Earth to the other is clearly visible as a ‘twist’ below Earth path. ( If you closely observe this 'twist' is actually not a twist it is optical illusion.

Red dots on the green line are the positions of Chandra at 10 minute interval. Notice that the dots are crowded on top and are well separated at bottom. This is due to unequal speed of spacecraft while it traverses the orbit. When it is near to Earth ( near perigee ) its speed is maximum and so the distance that it travels in 10 minutes is much more than the 10 minute distance while it is away from perigee. The speed is minimum ( so the dots are crowded ) near apogee i.e. the top edge of orbit seen in diagram.

As the spacecraft travels around Earth in an elliptical path it is also travelling along with the Earth with Earth speed. The combination of these speeds results in a spiral like shape as previously explained. 

This is seen in the two subsequent drawings when viewed from different angles ( chosen to bring out the features prominently ). Left side figure is how the orbit looks when you look 'end on ' along the path of Earth. Next is when look from Top perpendicular to Ecliptic.
The curved pink line clearly brings out the Ecliptic motion of Earth around Sun.
A much fascinating view is seen when we generate similar drawings for Mangalyan( MOM ) because there are several orbit manuvers over a period of time.
Orbit raising around Earth in several steps and a final escape from Earth’s gravity gives an interesting view. 

The suddenly jetting out blue line is MAVEN spacecraft of NASA which was also launched within same time slot. Notice that both are finally moving parallel towards MARS but MAVEN was directly launched into transfer orbit while the MOM was placed in Transfer orbit through several sequentially increasing orbits around Earth.
Still more breathtaking is the insertion of MOM  and MAVEN into their respective MARS orbit. The Red line is MARS orboit around Sun. Blue line is MAVEN spacecraft arriving from  Left and going into MARS orbits around 22nd Sept . 2 orbits are shown subsequent to insertion seen as a sharp turning when it approaches the MARS orbit. 

Green line is the MOM arriving from top left corner. Its insertion is seen as a shrp change near MARS orbit. Due to lack of clarity the actual orbit around Mars is not shown. 

These are preliminary in nature ( based as per original mission plans of respective spacecrafts ) and with available information upto  last correction. ( i.e. after 1st TCM of MOM ..  TCM2 data is not yet available ).  More on these mission orbit views will be posted soon in my ISRO related blog.

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