- #1
Leo Liu
- 353
- 156
The author of my textbook writes that a spacecraft 's "thrust in the radial direction at perigee changes the energy but not the angular momentum". Such a thrust increases the eccentricity of the elliptical orbit of the spacecraft because ##\epsilon \equiv \sqrt{1+2EL^2/\mu C^2}##, where epsilon determines the eccentricity. When L is constant, an increase in E will give rise to a more prolate ellipse.
While I understand that the angular momentum does not change because the pseudo force or the added velocity is central, which implies ##\vec\tau=\vec r\times\vec F=0## or ##\vec L_f=\vec r\times \vec p=\vec r_p\times m(\vec v_t+\vec v_r)=L_i+\vec 0##, the reason the energy increases when the thrust is turned on puzzles me. It is true that the thrust increases the velocity in the radial direction, and therefore increases the mechanical energy of the spacecraft , according to##E=1/2mv^2-C/r_p## and ##\Delta v_r=\text{the rocket equation}##. However, from
##\vec F_r\cdot d\vec r=0=m\vec v\cdot d\vec v## we know that the pseudo force of the thrust, which is tangential to the trajectory, should not cause any change in the velocity.
I can't really see why my reasoning is wrong. I guess the contradiction is the result of viewing momentum change along the radial direction as force. Could you point out the reason(s)?
While I understand that the angular momentum does not change because the pseudo force or the added velocity is central, which implies ##\vec\tau=\vec r\times\vec F=0## or ##\vec L_f=\vec r\times \vec p=\vec r_p\times m(\vec v_t+\vec v_r)=L_i+\vec 0##, the reason the energy increases when the thrust is turned on puzzles me. It is true that the thrust increases the velocity in the radial direction, and therefore increases the mechanical energy of the spacecraft , according to##E=1/2mv^2-C/r_p## and ##\Delta v_r=\text{the rocket equation}##. However, from
##\vec F_r\cdot d\vec r=0=m\vec v\cdot d\vec v## we know that the pseudo force of the thrust, which is tangential to the trajectory, should not cause any change in the velocity.
I can't really see why my reasoning is wrong. I guess the contradiction is the result of viewing momentum change along the radial direction as force. Could you point out the reason(s)?