I What does this description mean? (two bodies orbiting each other)

[n3pix]

Hello, I'm reading a textbook named "Introduction to Rocket Science and Engineering" by Travis S. Taylor.

I'm now in a general cosepts about orbits. I have a question, what the book says in here?

From Equation 2.24, it is clear that as the distance, r, between the rocket and Earth gets larger and larger, the potential energy between them decreases. At a certain distance, the effect of the gravitational potential well of the Earth becomes neglible compared to other forces acting on it (such as gravitational pull of the Moon or the Sun) and the rocket is said to be out of the Earth's "sphere of influence". The term is only applicable for three or more body systems and can be dscribed as,

r_SOI = α_p(m_smallerbody/M_largerbody)^2/5

Here, α_p is the length of the semimajor axis of the smaller body's orbit relative to the larger body.

I can't understand the bold formatted sentence. What the author try to mean? If you can, can you describe this sentence by using images, datas or diagrams, please?

Thanks to everybody, have a good day!

 

[haruspex]

This is my interpretation...
M is the mass of the body whose sphere of influence we are determining.
m is the mass of a smaller body orbiting the larger body, and which may serve to limit the larger's sphere of influence. I.e., when a third body is sufficiently far from M (>r_SOI) its orbit may be significantly affected by m.
a_p is the semimajor axis of m's orbit about M.

 

[gleem]

I see it differently. Since the sphere of influence decreases as the mass of the larger body increases I am lead to believe that it refers to the distance from the smaller mass. Thus the larger the big mass gets the closer you have to get to the smaller mass for it to have a larger affect than the big mass.

 

[n3pix]

This is my interpretation...
M is the mass of the body whose sphere of influence we are determining.
m is the mass of a smaller body orbiting the larger body, and which may serve to limit the larger's sphere of influence. I.e., when a third body is sufficiently far from M (>r_SOI) its orbit may be significantly affected by m.
a_p is the semimajor axis of m's orbit about M.

Well this description is good but, I can't image what is going on, sadly. Can you describe this with some paper & pen drawings or an paint image, please? My English and Google Translator's Turkish is not enough to describe three things' relations :P

I see it differently. Since the sphere of influence decreases as the mass of the larger body increases I am lead to believe that it refers to the distance from the smaller mass. Thus the larger the big mass gets the closer you have to get to the smaller mass for it to have a larger affect than the big mass.

Thanks for your reply, I got what you tried to mean, thanks!

 

[haruspex]

I see it differently. Since the sphere of influence decreases as the mass of the larger body increases I am lead to believe that it refers to the distance from the smaller mass. Thus the larger the big mass gets the closer you have to get to the smaller mass for it to have a larger affect than the big mass.

It specifically mentions the sphere of influence of the Earth being constrained by the sun and the moon, but you are right that the form of equation only makes sense if the sphere of influence in question is that of the smaller body.
Thanks for the correction.