FRAMES OF REFERENCE
Part II (B)

Toronto, Ontario, Canada
© J.L. Gaasenbeek, B.Sc., P.Eng. 1990

Numerical Example

Let us suppose that d = l5C light year meters and V = 2C light year meters per year or a = 2 then:

The observer will first see the star at

At t_o = l7.3205 years when the star first becomes visible the distance
Next when we substitute t_o in equation [39] X_{l,2 OBS} should be equal to zero.

or X_l,2 = l.333C (8.6603 - 8.6603 + (300-150-150)^1/2]

or x_l,2 = 0

Next let us calculate V_{1 OBS} and V_{2 OBS} at time t_o = l7.3205 years.

Therefore V_{1 OBS} = infinitely large in the positive direction.

and V_{2 OBS} = infinitely large in the negative direction.

No wonder a gamma-ray burster is so intense, of such high frequency and of such short duration!

Immediately after it first appears, the star image splits into two historic star images. One follows the actual star in space and time at a progressively slower observed velocity whereas the second historic star image travels back in space and time at a progressively slower observed velocity.

One side effect of the above phenomenon is that as the historic star image first splits into two images, optical interference may result, causing the intensity of the image to oscillate over a period of seconds.

This phenomenon has in fact been observed⁽³⁾.

STELLAR ABERRATION

When the altitude of a star is measured in relation to the earth's orbit the angle between the earth's orbital plane and the line of sight of the star is not constant but increases when the earth travels away from the star and decreases when the telescope travels towards the star as shown in Figure 4⁽⁴⁾.

Figure 4 Basis of stellar aberration

(a) A stationary telescope is aligned on a star. The star's altitude is equal to the angle .

(b) A moving telescope, which travels at velocity V, is aligned on the same star. The star's altitude angle is increased by a small angle

This phenomenon, referred to as stellar aberration, is readily explained when we look at equation [10] which gives the observed distance (X_OBS) an object appears to have travelled as compared to the actual distance (X_ACT) it has travelled at time t, as derived in the Appendix to this paper.

That is to say, when the observer starts to move to the left at velocity V it will appear to him as if the star he is looking at began to move to the right at the same velocity V with the result that he will see the star where it used to be rather than where it actually is as per the equation:

Also see Figure 5.

Figure 5 Schematic of a moving telescope which makes it appear as if the star is moving in the opposite direction

It follows from the above that the altitude angle of a stationary star can be calculated as follows: when the observer is in motion the altitude angle can be calculated as follows:

After substitution in [10] we get:

Numerical Example

Let us calculate the minimum and maximum altitude of the star Draconis who's stationary altitude = 75^o, The orbital speed of the earth V = 0.0001 C or a = 0.0001.

These values are in close agreement with Bradley's data⁽⁴⁾ on the north-south component of the aberration of Draconis (1727-1728).

CONCLUSIONS

It is concluded that every observer should consider himself stationary, even though he may be in motion in relation to a frame of reference which he has in common with the object he is observing, in which case he must vectorially subtract his velocity from that of the actual object. In addition the EFOR(s) through which the light image travels on its way to the observer, must be taken into account.

Since the universe presents us with a multitude of cosmological phenomena which should obey Equations l to 44, including several man made space probes, it should not be difficult to verify them by astronomical means.

Moreover, by comparison, the descriptive power of Equations l to 44 far exceeds the Lorentz-Einstein transformation equations⁽²⁾.

APPENDIX

Below follows the derivation of a typical equation [10] which gives the observed distance (X_OBS) an object appears to have travelled as compared to the actual distance (X_ACT) it has travelled at time t.

C is the speed of light in a vacuum relative to the stationary observer. That is to say, it is assumed that the moving object is small as compared to the observer's platform, the earth. Consequently the strength or k-factor of the object's EFOR is insignificant compared to the observer's EFOR. This means that the historic image of the moving object will travel towards the observer at the speed of light C since the observer's EFOR dominates and consequently is the DEFOR.

Similarly, all distances are measured in light-seconds, for example 10 light-seconds equal l0C or 2.998 x l0⁹ meters.

The shortest distance between the flightpath of the moving object and the observer is (d).

x_ACT = y is equal to the actual distance the object has travelled, from (2) to (4), at time t.

x_OBS = x is equal to the observed distance the object appears to have travelled, from (2) to (3), at time t.

Both x and y are negative if the object has not yet reached (2).

As the object approaches (2) the length of time required to reach (2) is measured in negative seconds. When the object reaches (2) time t = 0. As the object moves beyond (2) the time travelled is measured in positive seconds.

The actual velocity of the moving object is given as V_ACT or V.

Next follows the derivation of equation [10].

Firstly, by the time the historic image of the object has travelled from (3) to the observer at (l) the object will have travelled a further distance from (3) to (4) or,

Numerical Example

Suppose d = 5c meters and v = .2c meters

What is the actual and observed distance the object has travelled at

t = 10 and t = -10 seconds?

For t = 10 seconds:

X_ACT = Vt = .2c x 10 = 2c meters.

X_OBS = 0.9809c meters

For t = -10 seconds:

X_ACT = Vt = (.2c) (-10) = -2c meters

X_OBS = - 3.l857c meters

REFERENCES

(l) J.L. Gaasenbeek, Frames of Reference,

(2) A.P. French, Special Relativity, Pages 76 to 85.

(3) Bradley E. Schaefer, Scientific American, February 1985, Volume 252, No. 2, Gamma-Ray Bursters, Pages 52-58.

(4) A.P. French, Special Relativity, Pages 41-44.

End of Paper 4 of 5