Course note of "Introduction to Astrophysics"

Introduction to Astrophysics


Instructor-
Golam Dastegir Al-Quaderi
Professor
Department of Physics, DU


1

, Kepler’s Laws
• Orbital Angular Momentum of a System of
Particles:
• The orbital angular momentum of a particle is
defined with respect to some reference point,
from where the position vector 𝑟Ԧ𝑖 of the particle
is calculated:
𝑙Ԧ𝑖 = 𝑟Ԧ𝑖 × 𝑝Ԧ𝑖
• The total angular momentum of a system of
particles is the sum of the angular momenta of
the particles, about some coordinate system (say
the laboratory system).
2

, Kepler’s Laws
• Thus, we define
𝑛 𝑛
𝐿=෍ 𝑙Ԧ𝑖 = ෍ 𝑟Ԧ𝑖 × 𝑝Ԧ𝑖
𝑖=1 𝑖=1
• Now, the force on each (say on the 𝑖-th) particle has
two parts, external force (𝐹Ԧ𝑖𝑒𝑥𝑡 ) and internal force (𝐹Ԧ𝑖𝑗 ):
𝑛
𝐹Ԧ𝑖 = 𝐹Ԧ𝑖𝑒𝑥𝑡 + ෍ 𝐹Ԧ𝑖𝑗
𝑗=1,𝑗≠𝑖
• The time rate of change of the momentum of a particle
is the net force on it (Newton’s 2nd law):
𝐹Ԧ𝑖 = 𝑝Ԧሶ𝑖

3

, Kepler’s Laws
• Hence the time derivative of the total angular
momentum of a system of particles becomes:
𝑑𝐿 𝑑 𝑛
= ෍ 𝑟Ԧ𝑖 × 𝑝Ԧ𝑖
𝑑𝑡 𝑑𝑡 𝑖=1
𝑛 𝑑 𝑟Ԧ𝑖 𝑛 𝑑𝑝Ԧ𝑖
=෍ × 𝑝Ԧ𝑖 + ෍ 𝑟Ԧ𝑖 ×
𝑖=1 𝑑𝑡 𝑖=1 𝑑𝑡
𝑛 𝑛 𝑑 𝑝Ԧ𝑖
ሶ
⇒𝐿=෍ 𝑣Ԧ𝑖 × 𝑝Ԧ𝑖 + ෍ 𝑟Ԧ𝑖 ×
𝑖=1 𝑖=1 𝑑𝑡
4