ASTRONOMY - Subjects

ASTRONOMY

Subjects

1. Kepler's Laws.

2. Newton's theorem regarding the attraction of an empty homogenous sphere.

3. Determine the differential equation for the movement for the two bodies problem.

4. The analitycal solution of the two bodies problem.

5. Enounce and prove the cosine formula in spherical trigonometry.

6. Show that in a spherical triangle ABC, the next relationships holds:

7. If D is on BC, a side in any spherical triangle ABC, prove that:

8. Prove 21121b15v that belongs to a spherical issosscelus triangle or to a spherical triangle in which A = B + C.

9. Show that in any spherical, rightangle triangle ABC (A = 90^o) we have:

10. Prove that in a spherical equilateral triangle ABC, we have:

ASTRONOMY

Resolves

1. Kepler's Laws.

a.      The orbits of the planets are ellipses with the Sun in one of the focal points.

b.     The vector that joins the Sun with a Planet sweeps equal areas in equal time.

c.     The square of the period of a Planet is proportional with the cube of the great semiaxis of the orbit.

a.     We scalar add to the Laplace moving integer

(1)

On the left we reduce it to and so (1) becomes where is the angle between the position vector and Laplace's vector and is called true anomaly.

and having in mind the standard ecuation of a conic, (2)

the parameter of the conic

the excentricity

so that shows us that the great semiaxis of the conic depends only on energy

If , the distance reaches minimum value known as the distance at the pericenter.

The form of the conic is determined by the sign of based upon (2)

b.     In the sistem of polar coordinates for the vectors of position and speed we have the following representations:

, , ,

The versors form an orthogonal system with the cinetic moment vector:

The area swept by the position vector in time is:

but

c.     We will consider the case

(1)

(2)

Using (1), (2) becomes:

, where T represents the period of an orbit or the time it takes E to reach from 0.

2. Newton's theorem regarding the attraction of an empty homogenous sphere.

The attraction of an empty homogenous sphere is the same with the attraction of its center where the entire mass of the sphere is concentrated when the material point is outside the spehere or equals 0 when the material point is inside the spehere.

We consider a spehere of radius of equation with the center in the origin of the axis system .

We consider the material point of unitary mass situated on and a point of the sphere.

The potential of the forces:

where is the constant density of the surface, an element of the surface.

3. Determine the differential equation for the movement for the two bodies problem.

We will make two assumptions to simplify the model:

a)     the bodies are uniformly spherical and their entire mass is concentrated in the center

b)    no other internal or external forces act upon the sistem except the gravitational forces that act along the line that bind the centers of the two bodies

Now we have the following problem: "Study the relative motion of the material point of mass in the gravitational field created by the material point of mass

The two masses interract through a force that depends upon the relative distance between the two masses and has the orientation from to .

is called the reduced mass

4. The analitycal solution of the two bodies problem.

5. Enounce and prove the cosine formula in spherical trigonometry.

6. Show that in a spherical triangle ABC, the next relationships holds:

(3)

(4)

(3) + (4) (1)

,

(2)

7. If D is on BC, a side in any spherical triangle ABC, prove that:

The cos formula in ADB and ADC

But the relation

8. Prove 21121b15v that belongs to a spherical issosscelus triangle or to a spherical triangle in which A = B + C.

A=B+C

B=A-C

C=A-B

We use the cos formula for the angles B and C

| :

, , the relation

9. Show that in any spherical, rightangle triangle ABC (A = 90^o) we have:

, because applying Neper's rule to side a, we get:

or

and the relation was proven

10. Prove that in a spherical equilateral triangle ABC, we have:

but because a=b=c we have

or or

This is the relation that exists between a side and the opposite angle in a equilateral triangle and we can but it in the requested form: , and

or so we get the requeste formula because the memebers are positive.

DISCRETE MATHEMATICS

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