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Sunday, January 30, 2011

The Size and Shape of Earth

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We know that the Earth is not a perfect sphere but has a slight bulge at equator due to centrifugal force exerted on Earth's crust by the rotation of earth on its North-South axis.The earth makes one rotation in 24 hours causing the Equatorial diameter to be 12,756.274 km while its Polar diameter is 12,713.5046282 km. ( Just for reference; this makes the circumferences to be 40,075.1669 km as equatorial circumference and 39940.653 kms as Polar circumference.)

To the naked eye the shape appears to be a neat sphere and we can simply ignore its existence for most of the time.

However, this small difference in the two diameters plays a major role in rotating the orbit ellipse  around earth.  An example of such a rotation of ellipse is shown in the figure on right wherein you can seen 3 consecutive orbits. This is the effect of varying gravitational force ( caused by varying local radii of Earth )  as the satellite rotates in each orbit. Notice how the ellipse has rotated from orbit to orbit, but we will cover that in some details in the latter part of this post.

Adjacent figure shows the two diameters of Earth viz Polar dia of 12713.5 Kms and Equatorial dia of 12756.3 Kms.

So the Polar dia has flattened from 12756.3 kms to 12713.5 Kms. ( i.e. by  42.8 Kms )

This gives a flattening ratio of
 42.8 Kms / 12756.3 Kms = .003355 or nearly 1/298.3

For scientific definition this flattening factor is defined by following relation:

Flattening factor = f = (Semi-major-axis    -   semi-minor-axis)/ semi-major-axis

which gives the flattening factor for earth = 1/298.257.

This is the reference value used in  calculations pertaining to geodesy.

We now try to define the shape of Earth in mathematical terms.

For very accurate definition of Earth's shape modelling higher Harmonics are used.

What is meant by higher harmonics? In my other blog we have seen that a complex  wave-shape like a square wave or a triangular wave can be defined by combining simple sinewaves  of multiples of basic frequency.  In a similar fshion we can define the shape of earth  by combining several simple mathematical figures like a Sphere + an ellipse,which is a second harmonic of a circle.

If we use a second harmonic superimposed on 1st Harmonic we get a shape nearer to Earth's shape and to get a more accurate shape we can use higher harmonics.

This figure shows how 1st and 2nd harmonics ( green and pink coloured shapes respectively ) combination gives a near match to real Earth shape shown in a dotted line.

A standard for geodesy that is intensively used in GIS applications is referred to as WGS84 (World Geodetic System  as defined in 1984) wherein the parameters used are :

DE  =  Equatorial Diameter                                                       =  12756.274 kms
DP  =  Polar Diameter                                                                =  12713.505 kms
CE  =  Equatorial Circumference  ( =   DE  * p  )                     =  40075.017 kms
CE  =  Polar Circumference  ( =   DP  * p  )                              =  39940.654  kms
RE  =  Equatorial Radius ( Semi-major axis ) ( =   DE  / 2  )  =  6378.137 kms
RP  =  Polar Radius ( Semi-minor axis ) ( =   DP  / 2  )           =  6356.752 kms
R  =  Mean Radius   ( =  [ 2* RE  +  RP ]/3 )                              =  6371.009 kms
F  =  Flattening Factor   ( =  [  RE  -  RP ]/ RE  )                         =  1/298.2572

These values define the linear measurements or Size/Shape of Earth.

The other important values associated with Earth are its gravitational properties and the rotation of Earth around North-South axis.

We have learnt in our schools that the acceleration due to gravity, g = 9.80665 m/s^2 ( or 32.174 ft/s^2 in FPS system) but that is sufficient only for the calculations pertaining to objects falling within Earth's gravitational field.

Note that this constant g, is only related to Earth's Gravitational field and is true only at the surface of Earth and so it can't be used for every calculation pertaining  to Earth's gravity and so a more universal quantity is required.

 G, the  Universal Gravitational Constant , is such a quantity  which can be used for any calculations pertaining to gravity in general.

This Gravitational Constant (  also known as Newton's constant or universal gravitational constant )  has a value of
G = 6.67384 × 10-11  N m2 kg-2  in terms of Newton-Meter .  

Earth's acceleration due to gravity, g, is derived using this Universal Constant G, as follows:

g = GM / r2     where M is the Mass of Earth, and r is the radius of Earth.

The Mass of Earth M = 5.9737 × 10^24 kgs. But this value of M has a large uncertainty ( 1 part in 10^4 )

Therefore generally the two parameters G and M  are not used independently. Instead a product G*M is used as this is independently known more accurately than the accuracy of G or M.

GM = 398600.4418 meter cube /sec sq  for Earth.

Let's now try to compute g using Earth's mean radius r  which is
r = (polar radius +equatorial radius )/2 = (6378.137+6356.752)/2 =  6367.445
g = GM/r^2

Hooray, we use these values and compute g. which turns out to be  g = 9.83313457085332 m/s^2

But this is wrong , g should be 9.80665 m/s^2?  why this discrepancy?

It is because the value 9.80665 m/s^2 that we use is actually defined as g(0) and is arrived at by making adjustments for the centrifugal forces generated due to rotation of Earth at standard latitude of 45deg.
We summarize below the various parameters:
ME =  Mass of Earth = 5.9737 * 1024    Kgs
G = 6.67384 × 10-11 m3 kg-1 s-2   in MKS system or  
G = 6.67384 × 10-11  N m2 kg-2  in terms of Newton-Meter
GM = G * M = 398600.4418 m3/r2
E  =    g at equator    =    G * ME  / R E  = 9.800194  m/s 2
gp  =    g at Poles    =    G * ME  / R p  = 9.866242  m/s 2
gh  =    g at any height  h    g(0) * ( R / ( R +h ) 2   m/s 2 ,  R is the mean radius defined above.

= = = = =
Coming back to the subject of effect of the flattening of Earth on satellite orbits:

This deviation causes the  ellipse of the satellite orbit to rotate i.e. the point nearest to Earth ( called Perigee ) does not remain stationary in one place but it shifts in every orbit causing the ellipse rotation as shown in first figure above.

The effect of this bulge is so prominent that the direction in which this ellipse rotates is decided by the inclination that the orbit makes with equator. This deviation is known in technical parlance as Perigee Perturbation and is plotted in the adjacent figure.

As seen the perigee rotates

  - If the inclination is < 63.4 degrees or > 116.5 deg the perigee in orbit rotates forward in the orbital plane
  - If the inclination is  between 63.4 degrees and 116.5 deg  the perigee in orbit  rotates rearwards in the orbital plane
  - If the inclination is  = 63.4 degrees or = 116.5 deg the perigee in orbit  remains fixed at one latitude.    Such an orbit is called a Critically inclined orbit. 

We have prepared a video depicting this Perturbation ( Orbit Rotation ) for several satellites inclined at various inclinations here .

Molniya orbits are used very effectively by Russians  for their high latitude stations to remain in contact with the satellites for more time.
Being at high latitudes, these stations do not get a good visibility to equator based Geosynchronous satellites.
  So they placed   their Molniya series of satellites in a highly elliptical orbit. Major portion of orbit is over North Hemisphere centered over Russia ( A typical orbit has 40000 kms Apogee fixed over Russia and inclined at 63.4 degrees so that the ellipse does not rotate and the major portion of orbit would be visible over northern part of Russia.

In the  adjacent figure we have plotted the track of a Molniya  ( Molniya 1-93 to be more specific ) satellite as it falls on the surface of Earth, a rather weird shape to look at, isn't it .

But it serves the purpose ..  it remains visible over Northern Russia for about 8 hours out of its orbital period of  11Hrs 57 Min 36 sec.

Why has this shape been generated?

To understand this satellite let's look at the orbit in detail as shown from two directions in these two images. Right side image shows entire orbit with markers placed at 30 minutes interval. Notice the distance traveled in each interval ..  the satellite moves fast when near the Earth and slow when away from Earth. This is a simple Kepler's law.

Also it is seen that the satellite crosses Earth  ( the south most portion of orbit  near Perigee )  in just 3 intervals of 30 minutes each ..  i.e. in 1.5 hrs while the remaining full 10.5 hrs it is visible over  Northern hemisphere.

Left figure shows the same orbit but in a different perspective such that Russia is visible directly in front of the viewer.
Same view of Earth is magnified in the figure on right.

Notice the dots which represent the subsatellite point every 90 minutes.

Molniya Video :

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1 comment:

  1. Added the concept gravitational constants G,M,g and g(0)