This set of Spaceflight Mechanics Multiple Choice Questions & Answers (MCQs) focuses on “N Body Problem”.

1. Which of these problems helps us in analyzing motion of multiple celestial objects?

a) Two-body problem

b) N-body problem

c) Euler problem

d) Kepler problem

View Answer

Explanation: When we have o predict and analyze the motion of multiple celestial objects in the sky as a result of their gravitational forces acting on each other, we use N-body problem to solve the dynamics.

2. When there are multiple celestial bodies in the sky, each of their gravitational forces affect the orbital properties.

a) True

b) False

View Answer

Explanation: A celestial body’s motion and orbital parameters are affected by the presence of the gravitational force from the other planets and other objects. This is a result of N-body problem.

3. What is the formula to compute N-body problem?

a) \(\frac{dR_i}{dt}\) = G\(\Sigma_{i\ne 1}^3 \frac{m_j}{R_{ij}^3}\)(R_{j} – R_{i})

b) \(\frac{dR_i}{dt^2}\) = G\(\Sigma_{j\ne i}^3 \frac{m_j}{R_{ij}^3}\)(R_{i} – R_{j})

c) \(\frac{d^2R_i}{dt^2}\) = G\(\Sigma_{j\ne 0}^N \frac{m_j}{R_{ij}^3}\)(R_{i} – R_{j})

d) \(\frac{d^2R_i}{dt^2}\) = G\(\Sigma_{j\ne i}^N \frac{m_j}{R_{ij}^3}\)(R_{j} – R_{i})

View Answer

Explanation: The equation of motion for a N-body problem is computed using the formula:

\(\frac{d^2R_i}{dt^2}\) = G\(\Sigma_{j\ne i}^N \frac{m_j}{R_{ij}^3}\)(R

_{j}– R

_{i})

Where, G is the universal gravitational constant

R

_{i}is the position of center of mass of body i

|R

_{j}– R

_{i}| is relative separation between the center of mass of i and j bodies

4. What is formula to find the motion of center of mass of the N-body system?

a) r_{c} = \(\frac{\Sigma_{i=1}^N m_i R_i}{\Sigma_{i=1}^N m_i}\) = v_{co}t + r_{co}

b) r_{c} = \(\frac{\Sigma_{i=1}^N m_i R_i}{\Sigma_{i=1}^N m_i}\) = v_{co} + r_{co}

c) r_{c} = \(\Sigma_{i=1}^N\)m_{i} R_{i} = v_{co}t

d) r_{c} = \(\Sigma_{i=1}^N\)m_{i} R_{i} = r_{co}

View Answer

Explanation: While analyzing the N-body problem, the center of mass is seen to follow straight line with a constant velocity v

_{co}which begins from a constant initial position r

_{co}. The formula to compute its motion is: r

_{c}= \(\frac{\Sigma_{i=1}^N m_i R_i}{\Sigma_{i=1}^N m_i}\) = v

_{co}t + r

_{co}

R

_{i}is the position of mass of body i

m

_{i}is the mass of body i

5. What order of magnitude does the earth’s oblateness affect the gravitational forces?

a) 10^{-2} g’s

b) 10^{-3} g’s

c) 10^{-4} g’s

d) 10^{-5} g’s

View Answer

Explanation: There are certain variations introduced due to the earth’ oblateness effect. This force is of the order 10

^{-3}g’s. Newton’s law is only applicable for spherical objects.

6. The mass of a rocket is not considered among the system of N-bodies for analyzing the motion.

a) True

b) False

View Answer

Explanation: While examining the N-body system, each and every small celestial objects plays a role in gravitational forces acting on each other. Even the effects of mass of rocket has to be incorporated while determining the equations of motion.

7. What is the formula for vector sum of all the gravitational forces acting on the i^{th} body?

a) F_{g} = Gm_{i}\(\Sigma_{j=1,j\ne i}^n \frac{m_j}{r_{ji}^3} \vec{r_{ji}}\)

b) F_{g} = -Gm_{i}\(\Sigma_{j=1,j\ne i}^n \frac{m_j}{r_{ji}^2} \vec{r_{ji}}\)

c) F_{g} = -Gm_{i}\(\Sigma_{j=1,j\ne i}^n \frac{m_j}{r_{ji}^3} \vec{r_{ji}}\)

d) F_{g} = -Gm_{i}\(\Sigma_{j=1,j\ne i}^n \frac{m_j}{r_{ji}^2}\)

View Answer

Explanation: If position vectors of n masses are given in the coordinate system, then Newton’s law of gravitation can be used to compute the force exerted by m

_{n}on body m

_{i}. Thus the formula for vector sum of all the gravitational forces acting on the i

^{th}body is given by:

F

_{g}= -Gm

_{i}\(\Sigma_{j=1,j\ne i}^n \frac{m_j}{r_{ji}^3} \vec{r_{ji}}\)

Where, m

_{1}, m

_{2}….m

_{n}are the masses of the body

r

_{1},r

_{2}….r

_{n}are the position of n masses.

8. Which of these is the simplest case of N-body problem?

a) One-body problem

b) Two-body problem

c) Three-body problem

d) Five-body problem

View Answer

Explanation: The simplest case of the N-body problem is a case which has only one body. In the entire space, there lies only one body with mass m. It experiences no force, acceleration and moves with a constant velocity.

9. Why is it not possible to solve N-body problems analytically?

a) Closed form solution does not exist

b) It is time consuming

c) Lack of initial condition data

d) Result converge

View Answer

Explanation: N-body problems are usually chaotic for initial conditions and require numerical method to compute the result. Analytical results cannot be obtained wince there no closed form solution existing. So far, two body problems and restricted 3-body problem can be solved.

10. What is the assumption while solving restricted three-body problem?

a) Their positions don’t change over time t

b) Mass of one of the bodies is negligible

c) Orbits are circular

d) The gravitational force exerted on each other cancels out

View Answer

Explanation: In case of restricted three-body problem, it is a simplified version of three-body problem where it is assumed that one of the masses is almost negligible compared to the other two. This m1 and m2 move in Keplerian orbit which is unaffected by m3.

11. What is the degree of freedom of N-body problem?

a) N

b) 2N

c) 3N

d) 6N

View Answer

Explanation: The N-body problem has 3N degrees of freedom which makes use of 3N second order differential equations for finding the position of the particle.

12. How many motion variables are there in N-body problem?

a) N

b) 2N

c) 3N

d) 6N

View Answer

Explanation: In case of N-body problem, there are 3N second order differential equations which results in 6N motion variables.

13. In case of N-body problem. The net torque on the system in non-zero.

a) True

b) False

View Answer

Explanation: In N-body problems, where there’s mutual forces between several celestial bodies, the net external force and torque are both zero due to the Newton’s third law.

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