Find the radius of a geo-stationary satellite

[Kolahal Bhattacharya]

To find the radius of a geo-stationary satellite in terms of R_e, we see---

M_s*w^2*R_s=G*M_eM_s/R_s^2 from where using GM_e=g R_e^2,

We get R_s^3=gR_e^2/w^2. w can be evaluated. The problem is because of the fact that
Kleppner provides the soln. R_s=6.6R_e.How, one can get rid of fractional exponent?

 

[berkeman]

My guess is that you have to plug in the numbers, and then in the end get the 6.6 relationship. I don't see an algebraic way to do it offhand.

 

[kyle8921]

I would like to clarify that the equation provided is known as the "geostationary orbit equation" and it is commonly used to calculate the radius of a geo-stationary satellite. The equation is derived from the balance between the gravitational force and the centripetal force acting on the satellite. The equation assumes that the satellite is in a circular orbit around the Earth with a constant angular velocity.

To address the concern about the fractional exponent, it is important to understand that the equation is derived using mathematical principles and cannot be altered to remove the fractional exponent. The exponent is necessary to accurately represent the relationship between the variables in the equation.

Furthermore, the value of 6.6R_e is a commonly used approximation for the radius of a geo-stationary satellite. In reality, the exact value may vary depending on the specific parameters of the satellite, such as its mass and the exact location of its orbit. Therefore, it is important to use the equation and its resulting value as a guide and not as a definitive answer.

In conclusion, the equation provided is a valid and widely accepted method for calculating the radius of a geo-stationary satellite. The fractional exponent in the equation is necessary and cannot be removed. The value of 6.6R_e is a commonly used approximation and may vary in different scenarios. As scientists, it is important to understand the limitations and assumptions of equations and use them as tools for understanding and predicting phenomena, rather than absolute truths.