perpendicular to it. (Fig.2)
Then we prove Kepler’s second law before the other ones.
Suppose radial vector ������ changed into ������ + ∆������, (fig.3) shows that ������ sweeps an area which is
=
[������̈
−
������������̇ 2]������̂
+
[������������̈
+
2������̇ ������̇ ]���̂���
Consider that the acceleration vector pointing origin point O, knowing ������������̈ + 2������̇������̇ is 0, from
Assume
������̂ = (cos������)������̂ + (sin������)������̂ ���̂��� = −(sin������)������̂ + (cos������)������̂
Then
Fig.4
d������̂ d������
=
−(sin������)������̂
semi-major axis d, period of motion T and distance to focus r, then the value of acceleration is ������ = ������������2, where ������ = 4������������22������3. Through this method Newton proved that the value of a planet can be derived from Kepler’s first law.
������
=
−
������������ |������|2
∙
������̂
=
[������̈
−
������������̇ 2]������̂,
we have
������̈
−
������������̇ 2
=
−
������������ ������2
When planet on perihelion,(Fig.5) assume ������ = 0, obtain the initial condition, ������|������=0 =
+
������
×
������̈
=
������
×
������̈
=
������
×
������
=
������,
������ × ������ = ������.
That tells us that the motion of planet is restricted to a plane which has sun fixed on it and ������
������
=
d������ d������
=
d d������
(������
∙
������̂ )
=
������̇ ������̂
+
������������̇ ���̂���
Acceleration
������
=
d������ d������
=
d d������
(������̇ ������̂
+
���������������̇̂���)
=
d������̂ d������
∙
d������ d������
=
���̂���
∙
d������ d������
d���̂��� d���̂��� d������
d������
d������ = d������ ∙ d������ = −������̂ ∙ d������
So the velocity
similar to a triangle, gives
1
1
1
1
∆������ = 2 ������ × (������ + ∆������) = 2 ������ × ������ + 2 ������ × ∆������ = 2 ������ × ∆������,
Divide the upper equation by ∆������ ,
demonstration to explain Newton’s work on demonstrating Kepler’s Laws.
1. Demonstration by Newton: Kepler’s Laws simply described the phenomenon of planets’ motion, which were definitely meaningful in the development of Physics and Astronomy, however lacked with some supports from theorem, which lately became the work of one of the great man in history of humanity, Newton, who mathematically find the essential of Kepler’s Law. In terms of Kepler’s second law, Newton demonstrated that it is equivalent between the second law and central motion, i.e., supposing a particle moving on a plane where the segment between it and a fixed point sweeps a fixed area, then the particle is moving under the central force, and vice versa. From the equivalence of the second law and central force, we know the acceleration vector of a planet always pointing the
Then discuss
Байду номын сангаас
������ = ������ × ������ = ������������̂ × (������̇������̂ + ���������������̇̂���) = ������(������������̇)���̂���
Make ������ = 0,
������
=
������(������������̇)| ���̂���
=
������������̇ 2
−
������������ ������2
=
������02������02 ������3
−
������������ ������2
∆������ 1 ∆������ ∆������ ≈ 2 ������ × ∆������,
make ∆������ → 0,
d������ 1 d������ = 2 ������ × ������.
Already knows, ������ × ������ = ������, which is constant. Thus we can conclude the second law, areal
Report
Abstract:
Demonstration of Kepler’s Laws
HAN Fang-Zhou
The mathematical explanation about planet’s motion given by Newton and Kepler are two of
the most important achievement in science history. In this article we have a modern
2. Demonstrating Kepler’s Laws with modern method:
Suppose ������ to be radial vector emitting from sun pointing a planet with mass of ������.
Universal gravitational law told us that the attraction between planet and sun is ������������������
������=0
=
������0������0���̂���
Fig.5
or
������2������̇ = ������0������0,
������̇
=
������0������0 ������2
Substitute the result into the equation above
������̈
������0,������|������=0 = 0. ������0 = |������|������=0 = |������̇������̂ + ������������̇ ���̂���|������=0 = |������������̇ ���̂���|������=0 = |������������̇ ||���̂���|������=0 = (������������̇ )������=0