THE AETHER & THE GALILEAN TRANSFORMATION

Isaac Newton adopted a constant mass for the particle:

m2  m1  const
According to Newton’s second law of motion, the equations of motion for particle P have the

following form:   m1    ,     m2   m2  2 , hence
(3.1)  d (m1V1 ) dV1 m1a1 F2 d (m2V2 ) dV2 a
F1  dt dt
dt dt

F2  F1

Therefore Newton’s second law of motion is invariant with respect to the Galilean

transformation. This means that Newton’s laws of mechanics are the same for both inertial

systems 1 and 2.

III.1 VARIABLE MASS OF PARTICLE CONSIDERED IN NEWTON’S SECOND LAW

OF MOTION
The existence of the aether and the applicability of the Galilean transformation have been
described in Chapter I. Experimental data indicate that the mass of a particle depends upon its
speed. Then let us consider the variability of the particle mass in Newton’s second law of
motion.
INERTIAL SYSTEM 1 (motionless with respect to the aether)

The expression given by H. A. Lorentz for  is defined by:

(3.2)  1
1 (V / C ) 2

1o

where: V the speed of particle P in the inertial system 1,

1

C the speed of light in a vacuum with respect to the aether.

o

(3.3) V =C , V <V , V V
1max o
1 1max 1 1max

The speed V = C is the limit speed of the particle P in the inertial system 1. That speed
1max o

is identical in all directions.

The condition (3.3) limits the speed of particle P with respect to the aether.

We assume: m1  m1 (V1 )  mo1 rest mass of particle P in the inertial system 1,
(3.4) mo1  m1(V1  0) the mass of moving particle P in the inertial system 1,
the Lorentz factor (3.2).
where: m1 (V1 )


Then let us introduce the variable mass of particle P into Newton’s second law of motion

(3.1). The mass can bedefined by (3.4):
(3.5a)  where:
(3.5b) F1  d (m1V1 ) m1  m1 (V1 )  mo1 relationship (3.4)
(3.5c) dt  which after differentiation takes the following form:
  d (mo1 V1 ) d 
F1 dt  dt V1
dV1
  mo1 dt  mo1
F1

(3.5d)   mo1   mo1 d 
F1  a1 dt V1

Relationships (3.5a - d) express Newton’s second law of motion in the inertial system 1 after

the variable mass of particle P has been introduced.

51

INERTIAL SYSTEM 2 (O’EQW system)

The limit speed V of the accelerating particle P depends upon the angle  0,2 between
 2max
vectors V0 and V2 .

(3.6)  0,2  ∡ (V0 ,V2 ) so V2 max  V2 max ( 0,2 )

DETERMINING THE SPEED V2 max

The speed V is the limit speed of the particle P in the system 2 which moves at a fixed
2 max

speed in a given direction (angle  0,2 ) with respect to the velocity V.
V
0
0

ASSUMPTIONS. 

The particle P is accelerated in any given direction in relation to the V velocity (Fig.14).
0
1) The velocity V is parallel to the OXo axis.
0

The coordinate of the velocity: V , where: 0  V0  C0 .

0
2) The force F acts on the particle in any given direction. The angle  0,2 represents
2

any angle. 

3) The velocity V (t  0)  0 .
2

Fig.14 
The force F acts on the particle in any given direction (angle  0,2 ) in relation to the
2
velocity V .
0

 
Coordinates of the velocity V : Coordinates of the velocity V :
 2 max 0
V2 max  [V2 max cos  0,2 , V2 max sin 0,2 , 0] V  [V , 0 , 0]
00

V 0 .
2 max
 
According to the Galilean transformation: V V  V so the following equations can be
0 2 max 1max

written: (V0 V2max cos  0,2 ) 2  (V2max sin 0,2 )2  V12max , V1max  C0  const (3.3).

Hence the V2max (0,2 ) is obtained as follows:

52

(3.7) V2 max  C0[ 1 (V0 / C0 )2 sin2  0,2  (V0 / C0 ) cos  0,2 ] , hence

(3.7a) V2 max ( 0,2  0)  C0 V0 ,
(3.7b) V2 max ( 0,2  1800 )  C0 V0 .

From the equation (3.7) we obtain the inequality: C0 V0 ≤ V2max (0,2 ) ≤ C0 V0 .

If the particle P is accelerated in any given direction (3.6) then the Lorentz relation (3.2)
takes the form as follows:

(3.8) a  1 , where: V2 max relationship (3.7).

1 (V2 / V2 max ) 2

V2  V2 max , V2  V2 max

When the particle P is accelerating along in the direction of the vector  (  0 ), then the
V
0,2
o

relation (3.8) takes the form of  b :

(3.9) b  1 according to the equation (3.7a).
1[V2 /(Co Vo )]2

 (  0,2  180o ),
When particle P is accelerating in the direction opposite to that of the vector V

o

then the relation (3.8) takes the form of  c :

(3.10) c  1 according to the equation (3.7b).

1[V2 /(Co Vo )]2

Let us assume:

(3.11) m2  m2 (V2max ,V2 )  mo2 a

where: m2 (V2 max ,V2 ) the mass of the moving particle P in the system 2,
mo2  m2 (V2max ,V2  0) the rest mass of the particle P in the system 2,
formula (3.8).
a

Let us introduce to Newton’s second law of motion (3.1) the variable mass of particle P. Its

mass is determineFFFd222byddmt((hmmo2deo2tdV2art2ead)lVdVat2t2i)onmswhow2hiphedidrc(etha3:V.a12f1tme):2r  m2 (V2max ,V2 )  mo2 a relationship (3.11).
(3.12a) differentiation takes the form of:
(3.12b)
(3.12c)

(3.12d)   mo2 a   mo2 d a 
F2 a2 dt V2

Relationships (3.12 a - d) express Newton’s second law of motion in system 2 after
introducing a variable mass of the particle P.

53

III.1.1 THE VELOCITY OF THE PARTICLE

THE VELOCITY OF THE PARTICLE IN SYSTEM 2, (in the O’ EQW system)

When the particle P is accelerated in system 2, then its speed V2 depends upon the direction
the particle is accelerating towards with respect to the vector V .

o
If we assume F  const then from equation (3.12b) we obtain:

2

d[ aV2 (t)]  F2 dt ,  d[ aV2 (t)]  F2  dt .
mo2 mo2

After integration V2 (t) a  F2 t C4 .
mo2

From the premise that t  0  V  0 we obtain the integration constant C  0 .

24

Hence V2 (t) a  F2 t , V2 (t) a  k4t , F
mo2 where: k  2 .
The V (t)
4m
2
o2
(3.13a)
speed we define as follows:

V (t)  k t when the particle P is being accelerated in any given direction (3.6)
2a4 

with respect to the vector V

o

(3.13b) V (t)  k t when the particle P is being accelerated from rest along the
2b4

direction of the vector V ( 0,2  0) ,

o

(3.13c) V (t)  k t when the particle P is being accelerated in the direction opposite
2c4

to that of the vector V (  0,2  180o ),

o

Where:  formula (3.8),  formula (3.9),  formula (3.10),
a b c
t
time in which a constant force F is acting on the particle P.
2

Fig. 15 

The relationship between the speed V2 (t) and the time in which a constant force F is acting

2

on the particle P.

SYMBOLS: 1 equation (3.13a),

2 equation (3.13b),

3 equation (3.13c).

54

III.1.2 THE ENERGY OF THE PARTICLE

THE ENERGY OF PARTICLE IN SYSTEM 2 ( in the O’EQW system)

We assume that  0,2  ∡ (V0 ,V2 )  const (3.6), V2 (t  0)  0

When a force F acts on particle P in system 2 then the elementary work performed within a
2 
  d (mo2 aV2 )
 dE2  F2  dL , where: F2   dt ( 3.12b)
distance dL is equal to:

dL  V dt
 2
Then dE2  d(mo2 aV2 )    mo2d( a   mo2 (d a   
V2 dt  V2 ) V2 V2 a dV ) V
dt  
22

 mo2 (d a V2 V2 a dV V )  mo2 (d a V22   a V dV ) .

2 2 22

Hence dE2  mo2 (d a V22 a V dV )

22

The differential d a of the formula (3.8) equals:

d a  V2 dV2 so
V22max [1 (V2 / V2 max ) 2 ]3 / 2

( V 3dV V dV )
dE2  mo2 V 2 22  22
) 2 ]3/ 2 [1 (V / V ) 2 ]1/ 2
[1 (V / V 2 2 max
2 max 2 2 max

Total work which needs to be performed in order to m ove the particle P from rest point A in
system 2 to point B over the distance L at velocity V (Fig. 13) equals:

2

 (E2  V 3dV  V dV )
m
22 22
o2
V 2 [1  (V / V ) 2 ]3/ 2 [1  (V / V ) 2 ]1/ 2
2 max 2 2 max 2 2 max

After integration we obtain: E2  mo2V22max  C6
1 (V2 / V2 max ) 2

From the assumption that V 0  E2  0 we obtain the equation:

2

0  m V 2  C so the integration constant C  m V 2 . Hence
o2 2 max 6 6 o2 2 max

(3.14) E2  mo2V22max  mo2V22max , E2  mo2V22max  a  mo2V22max
1 (V2 / V2 max ) 2

Work E2 equals the kinetic energy E of the particle P.

k

(3.15) E2  Ek  mo2V22max  a  mo2V22max

The speed V is defined by relationships (3.7).

2 max

The expression m V 2 in (3.15) represents the rest energy E of the particle P for a given
02 2 max o

direction (3.6).

(3.16) E m V2

o o2 2 max

The expression mo2V22max  a in (3.15) represents total energy E of the particle P in system 2.

s

(3.17) Es  mo2V22max  a .

Hence (3.15) takes the following form: Ek  Es  Eo  mo2V22max ( a 1)

(3.18) Ek  mo2V22max ( a 1) is the kinetic energy of the particle P in system 2.

After expanding the formula for  a (3.8) in a power series we obtain:

a 1 1 (V2 / V2 max ) 2  1 3 (V2 / V2 max ) 4 ...
2 24

55

For small speeds V2 of the particle P: a 1 1 (V2 / V2 max )2 , hence the kinetic energy E
2
k

specified by the formula (3.18) equals:

E  m V 2 [1 1 (V / V )2 1]  1 m V 2
k o2 2 max 2 2 2 max 2 o2 2

(3.19) E 1m V2
k 2 o2 2

The formula (3.19) defines kinetic energy of the particle P, which results from Newton’s

second law of motion when the mass of the particle P is constant.

The experiments with particles are carried out in laboratories that are located on the Earth

and it is where system 2 (O’EQW) is also located. Despite Earth’s rotary and orbital motion

round the Sun, for adequately small time intervals it can be assumed that system 2is inertial
and it moves with respect to system 1 ( OXoYoZo system) at a constant velocity V which

o

modulus is defined by the inequality (1.124):

104 ≤ V / C  1.5104 .

oo

Hence 104 C ≤ V  1.5104 C o
oo

The value of the V speed is small when compared with C and therefore it can be omitted in

oo

formulae (3.8), (3.9) and (3.10). Having done that, the speed V  C and consequently the
2 max o

formulae (3.8), (3.9) and (3.10) take the following form:

(3.20)     1 .

a b c 1 (V / C ) 2
2o

From relationships (3.16), (3.17) and (3.18) we obtain relationships that give approximate

values of energies of the particle P in system 2:

(3.21) E m C2 rest energy,
(3.22)
(3.23) o o2 o

Es  mo2Co2 a  m2Co2 total energy,

Ek  mo2Co2 ( a 1) kinetic energy.

Where:  a formula (3.20).

Now the relationship between total energy E of the particle and its momentum  needs to
p
s2

be expressed.

From the equation (3.22) we obtain: E 2  m022C04 2
s a

Then the following can be written: Es2  m022C04 2  ( p22C02  p22C02 )
a

The modulus of the particle’s momentum is: p2  m02 aV2 , so

E 2  m022C04 2  m022 a2V22C02  p2C2 .
s a
20

After transforming this equation, we obtain the following:

Es2  m022C04 2 (1V22 / C02 )  p 2 C02 , where:  a relationship (3.20).
a 2

Hence E2  m2 C4  p2C2 , because  2 (1V22 / C02 ) 1.
s 02 0 20 a

The ultimate relationship between total energy E of the particle and its momentum  takes
p
s2

the form as follows:

(3.22a) E  m2 C4  p2C2 .

s 02 0 2 0

56

PARTICLE’S ENERGY IN SYSTEM 1 (in the OXoYoZo system)
The energies of the particle P in system 1 can be determined in the same manner as those in
system 2, with the use of formula (3.5b).

The following relationships determine the energies of the particle P:

(3.24) E  m C2 rest energy,

o o1 o

(3.25) Es  mo1Co2  m1Co2 total energy,
(3.26) E  m C 2 ( 1) kinetic energy,

k o1 o

where:  formula (3.2).

III.1.3 REST MASS OF THE PARTICLE WITH RESPECT TO THE AETHER

Let us consider the mass of the particle P in systems 1 and 2:

m (V )  m o1 so m (V  V )  m o1

1 1 1 (V / C ) 2 11 o 1 (V / C ) 2

1o oo

m (V ,V )  m so m (V ,V  0)  m o2
2 2 max 2
2 2 max 2 o2

1 (V / V ) 2

2 2 max

m (V  V )  m (V ,V  0) , hence
11 o 2 2 max 2

mo2  mo1 , then the rest mass of the particle P with respect to system 1
1 (Vo / Co ) 2

(with respect to the aether) equals:

(3.27) m  m 1 (V / C )2  m [1 1 (V / C )2 ] because V / C  1
o1 o2 oo o2 2 o o
oo

The quotient V / C is defined by the relationship (1.124):

oo

104 ≤ V / C  1.5104 . Hence

oo

[1 1 (1.5104 )2 ]m  m ≤ [1 1 (104 ) 2 ]m and after reduction
2 o2 o1 2 o2

(11.125108 )  m  m ≤ (1 0.5108 )  m .
o2 o1 o2

III.1.4 THE LAWS OF MECHANICS

Velocities and accelerations of the particle P in inertial systems 1 and 2 are:
V V V ,  
a2  a1 , t2  t1  t
1 o2

The mass of the particle P in systems 1 and 2 are respectively:

m m o1 , m m

1 1 (V / C ) 2 2 o2

1o 1 (V / V ) 2

2 2 max

m2  m1

The forces acting upon particle P in systems 1 and 2 are:

 m    m d  (3.5d),   mo2   mo2 d a  (3.12d),
F a V F2 a2 dt V2
1 o1 1 o1 dt 1 a

  d   ). 
F m   a m (V V Hence F2  F1 .
1 o1 1 o1 dt o 2

After including the variable mass of the particle P, Newton’s second law of motion (3.5a-d),

(3.12a-d) has the form which is non-invariant with respect to the Galilean transformation.

Hence Newton’s laws of mechanics are different in inertial systems 1 and 2.

57

III.1.5 DETERMINING THE  FORCE
 F1

We determine the F force acting on the particle P in the system 1 when the same particle is
1
acted on by the force F  const in system 2.
2

ASSUMPTIONS A. 

The particle P is accelerated in the direction of the V velocity.
0

The angle  0,2 = ∡ (V0 ,V2 )  0 relationship (3.6).

1) The absolute velocity V is parallel to the OXo axis (Fig.13).
0

The coordinate of the velocity: V , where: 0  V0  C0 .
0
2) The force F  const acting on the particle P is parallel to the O’E axis (Fig.13).
2

The coordinate of the force: F ; F  0 .
 22

3) The velocity V (t  0)  0 .
  2

The vectors F , V , V are parallel to those axes, which also results from these assumptions.

11 2

The coordinates of forces: The coordinates of velocities:

F [ F , 0, 0 ] V [ V , 0, 0 ]
2 2 0 0
F  [ F (t), 0 , 0 ] V  [ V (t), 0 , 0 ]
11 2 2

F  const V  [ V (t), 0 , 0 ]
11
2

V  C V relationship (3.7a).
2 max 00
 
V1  V2  Vo so V1  [ V2 (t) V0 , 0, 0]

 V1(t)  V2 (t) V0
The V (t) coordinate of the V velocity is defined by the relationship (3.13b).

22

, ,Fig.16 The coordinates V1  V1(t  t p ) , V1k  V1(t  tk ) V2  V2(t  tp ) V2k  V2 (t  tk )

of the V , V velocities of the accelerated particle P.
12

According to the Galilean transformation V  V V , V  V V .
1 20 1k 2k 0

58

From the (3.5b) equation we obtain:
F
d (V  )  1 dt where:  relationship (3.2). Hence
1m
01

F (t)
(3.28) )  1
 d[V (t ]  dt
m
1

01

Let us take any given time t of the particle motion under consideration (Fig. 16).

k

We set a time interval:

(3.29) t  t  t  t , t  0

kk 
F
If the set time interval is very small, it can be assumed that the coordinate value of the
1

.force which is acting on the particle within this interval is constant: F  const
1

Then the equation (3.28) takes the form as follows:

 d[V (t)  1 .F After integration we obtain:
1 ]  dt we obtain the following equation:
m
01

F
(3.30) V (t)  1 t  C
From the condition: 1m k

01

t t , V1(t  tk )  V1k  
1k
k

F
V   1 t  C hence the integration constant C equals:
1k 1k m k k k

01

(3.30a) F where:
C V   1 t

k 1k 1k m k

01

(3.31)  1

1k 1 (V / C ) 2
1k 0

(3.32) V (t  t )  V , where: t is within the time interval (3.29) t  t  t  t .

1 p1 p k pk

From the equations (3.30), (3.32) at t  t :

p

(3.33) F
V   1 t C

1 mp k

01

And from the equations (3.33) and (3.30a):

(3.34) F
V  V   1 t

1 1k 1k m k
01  t

Fp

1

m

01

From the equation (3.13b) at t  t :

p

(3.35) V2 b tp where: V2  V2 (t  t p ) ,  b relationship (3.9).
F2

m02

From the equation (3.13b) at t  t :

k

(3.36) tk  V2k bk where: V2k  V2 (t  tk ) ,
F2

m02

(3.37)  bk  1
1[V2k /(C0 V0 )]2

59

Because the Galilean transformation is in operation, the times in both frames of reference 1

and 2 are equal: t2  t1  t  t p . After comparing the left-hand sides of the equations (3.34)
and (3.35) the following is obtained:

V1 V1k  1k  F1 tk  V2 b
m01
(3.38) , where: m  m 1 (V / C )2 relationship (3.27).
F1 F2 01 02 00

m01 m02

From the equations (3.36) and (3.38) we obtain:

(3.39) F1  (V1 V1k  1k ) 1 (V0 / C0 ) 2 , where: F  const ,
(3.40) F2 V2 b V2k  bk 2
V V V ,

1 20

(3.41) V V V .

1k 2k 0

By assumption, the time interval in (3.29) is very small and the inequality V  V is fulfilled

2 2k

within, therefore the value of the V / V quotient is virtually equal to 1 and is less than 1.

2k 2

If we define: V / V  a then

2k 2 k

(3.42) V a V ; a  0.999999 was adopted for calculations.

2k k 2 k

The quotient (3.39) F / F is the function of the V and V coordinate values:
12 02

F1  f A (V0 , V) relationship (3.39)
F2 
2

For a given coordinate value V of the V velocity, the value of the quotient F / F
00 12

determined from the relationship (3.39) corresponds with every V coordinate value of the V
22

velocity. Table 13 presents the values of the F / F quotient for different values of V / C
12 00

and V / C .

20

V /C V /C

00 20

1 2 3 4 5
0.00001 0.2 0.49 0.69 0.97

1.5 104 0,99999998 1.0000737 1.0001475 1.0001833 1.0002215
10 3 0.99991068 1.0005030 1.0009883 1.0012262 1.0014788
10 2 1.00039406 1.0051134 1.0100151 1.0124266 1.0149834
1.01805269 1.0623422 1.1156201 1.1426304
0.1 1.08450322 1.2925644 1.5075330 1.6235630 -
1.18146817 1.8305992 2.4310503 -
0.3 - -
0.5

F1  f A (V0 / C0 , V /C ) (3.39), The angle  0,2  0
F2 a  0.999999
20
k
F  const

2

TABLE 13 The values of the F / F quotient, F  const

12 2

F1  F2 f A (V0 / C0 , V /C ).
20

Following (3.13b) V2 (t) is known, then consequently F1  f (V0 / C0 , t ) is known too.

From the results of calculations show in Table 13, it can be concluded that the quotient

F / F takes different values.

12

60

ASSUMPTIONS B. 

The particle P is accelerated in the direction opposite to the V velocity.
0

The angle  0,2  ∡ (V0 , V2 )  180o relationship (3.6).

1) The absolute velocity V is parallel to the OXo axis (Fig.13).
0

The coordinate of the velocity: V , where: 0  V0  C0 .
0
2) The force F  const acting on the particle P is parallel to the O’E axis (Fig.13).
2

The coordinate of the force: F , F  0 .
 22
3) The velocity V (t  0)  0 .
2  

Following the above assumptions B, the vectors F , V and V are parallel to these axes.
11 2

Coordinates of forces: Coordinates of velocities:

F [ F , 0, 0 ] V [-V , 0, 0 ]
2 2 ] 0 0
F  [ F (t), 0 , 0 V  [ V (t), 0 , 0 ]
11 2 2

F  const V  [ V (t), 0 , 0 ]
11
2

  V2 max  C0 V0 relationship (3.7b).
V1  V2  Vo , so V [ V2 (t) V0 , 0, 0 ]

1

V1(t)  V2 (t) V0


The V (t) coordinate of the V velocity is defined by the relationship (3.13c).

22

Fig.17 The coordinate V1  V1(t  t p ) , V1k  V1 (t  tk ) , ,V2  V2 (t  t p ) V2k  V2 (t  tk )

of the V , V velocities of the accelerated particle P.
12

According to the Galilean transformation V  V V , V  V V .
 1 2 0 1k 2k 0

The quotient F / F of coordinate values of the F , F forces can be determined as shown
12 12

under assumptions A.

61

Under assumptions B, the quotient F / F is defined by the equation (3.43):

12

(3.43) F1  (V1 V1k  1k ) 1 (V0 / C0 ) 2 , where: F  const ,
F2 V2 c V2k  ck
2

 relationship (3.2),  relationship (3.31),  c relationship (3.10),
1k

(3.44)  ck  1 ,
1[V2k /(C0 V0 )]2

(3.45) V V V ,

1 20

(3.46) V V V ,

1k 2k 0

V a V relationship (3.42)

2k k 2

a  0.999999 was adopted for calculations.

k

F1 / F2  f B (V0 , C) relationship (3.43).

0

For a given value of the coordinate -V of the V velocity, the quotient F / F determined
00 1 2

from the relationship (3.43) corresponds with every V coordinate of the V velocity.
22

Table 14 presents the F / F quotients for different values of V0 / C0 and V /C .

12 20

V /C 1 2 V /C 4 5 6
0.00001 0.2 0.69 0.97 1.49
00 20
0.99999998 0.9999283 0.9998164 0.9997786 -
1.5 104 3 0.9987770 0.9985242 -
10 3 0.99991068 0.9995032 0.49 0.9879245 0.9854337 -
10 2 0.9998536 0.8929439 0.8706278 -
1.00039406 0.9951119 0.9990161 0.7437849 0.6881060 -
0.1 0.9902803 0.6406240 0.5588903 0.4725897
0.3 1.01805269 0.9604290 0.9145631
0.5 1.08450322 0.9342512 0.8010324
1.18146817 0.9711417 0.7312806

F1  f B (V0 / C0 , V /C ) (3.43) The angle  0,2  1800
F2 a  0.999999
20
k
F  const
F  const
2
2
TABLE 14 The values of the F / F quotient,

12

62

III.2 TIME MEASURED BY ATOMIC CLOCKS

Let us introduce the following notations:

 time measured by the clock in the inertial system 1,
1

 time measured by the identical clock in the inertial system 2.
2

Then we adopt the assumptions:
1) The clocks are located in the origins O and O’of the systems 1 and 2 respectively.
2) The origin O of system 2 is in motion with constant velocity V along a straight

o

line running through the origin O of system 1 (Fig 13).

3) The clocks were synchronized     0 when the origins of the two systems
12
overlapped.

Time measured by the atomic clock depends on the rest mass of its particles, therefore the
following equations can be written:

(3.48)  2   A2  m01  ( m01 )1/ 2 ,
where:  1  A1 m02 m02

 A1, A2 are atom vibration frequencies in systems 1and 2 respectively

and m  m 1 (V / C )2 relationship (3.27).
01 02 oo

From the equations (3.48) and the relationship (3.27):

   ( 1 (V / C )2 ) 1/ 2   [1 (V / C )2 ]1/ 4 , then
21 oo 1 oo

(3.49)    [1 (V / C )2 ]1/ 4
2 1 oo

There is a dilation in the times measured by the clocks (3.49). The clock in system 2 is

delayed with respect to the clock in system 1.

The time measured by the clock in the inertial system 1, which presents a preferred absolute
system, defines the absolute time t .

(3.50) t  
1

Then applying (3.49) and (3.50) we obtain:

(3.51) t     2

1 [1 (V / C ) 2 ]1/ 4
oo

Hence knowing the time  that has been measured by the clock in the inertial system 2
2

and the value of the system’s absolute speed V , the absolute time can be calculated from the

o

relationship (3.51).


And as the values of the modulus of clock’s velocity Vo vary (relationship (1.120)), the times
measured by the clocks on the Earth’s surface are subject to continuous changes.

63

III.3 DECAY OF PARTICLES

An unstable particle is subject to a decay process which course can be described by the
following equations:

(3.52) m (t)  m N exp( t ) ,
1 01 0 

1

(3.53) m (t)  m N exp( t ) , where:
2 02 0 

2

m , m rest masses of the particle in inertial systems 1 and 2,

01 02

N initial number of particles (at t  0 ), which is identical in inertial

0

systems 1and 2,

m (t) , m (t) masses of particles undecayed during t period

12

in inertial systems 1 and 2,

 ,  average life of particles in inertial systems 1 and 2.
12

Let us write equations: m 1 ,   const
2  02  1
m
1 01 1 (V / C ) 2

oo

Hence average life  of particles in the inertial system 2:

2

(3.54)   1

2 1 (V / C ) 2
oo

The equations that define the number of undecayed particles during the decay time are:

(3.56) N (t)  N exp( t ) ,
(3.57) 1 0

where: 1

N (t)  N exp( t ) ,
2 0

2

N (t) , N (t) number of particles undecayed during t period in the inertial

12

systems 1 and 2,
 ,  relationship (3.54).

12

Fig.18 Graphic representation of equations (3.56) and (3.57).
64

A laboratory can be regarded as the reference system 1, resulting from the absolute speed

of the Earth being very small (1.126).

The average life time  1 of mezons   that are motionless in relation to the laboratory is:

1  2.603108 s .
When the relative speed of mezons   reaches value V0 / C0  0.99 , their average life time  2
in system 2 i.e. where these particles actually are, can be calculated from the equation

(3.54):  2  1  2.603108  7.088  1.845107 s , then 2 1 .

1 0.992

Experimental results found in subject literature ([3], [5]) are in agreement with the average

life time  2 of mezons   as calculated above. A compliance with relationship (3.54) is also
confirmed by experiments with other unstable particles [1].

Equations (3.52), (3.53), (3.56) and (3.57) imply that the decay process of particles in the
inertial system 2 is slower than the decay of identical particles in the inertial system 1.
The life time of particles in an inertial system that is in motion in relation to the aether is
longer than the life time of identical particles in a preferred reference system which is
motionless in relation to the aether.

III.4 DETERMINING A SIDEREAL DAY WITH ATOMIC CLOCKS

We start with the following equation: J  J  which implies that
11 2 2
(3.58) J m 1 , where:
1  2  02 
 Jm
21 01 1 (V / C ) 2

00

m  m 1 (V / C )2 relationship (3.27),
01 02 oo

J ,J Earth’s moment of inertia in systems 1 and 2 respectively,
angular speed with which the Earth rotates in systems 1 and 2,
12 rest mass of particles on the Earth in systems 1 and 2,
the speed at which the Earth’s center travels with respect to the
 , aether (1.126).
12

m ,m

01 02

Vze  V0

From the relationship: T2  1 , we have  where:
T1 2 T  1T ,

21
2

T ,T Earth’s sidereal day in systems 1 and 2.

12

By applying equations (3.58) and inequality V / C  1 , we obtain:

00

(3.59) T  1 T  [1 1 (V / C )2 ]T ,
2 1 (V / C ) 2 1 20 0 1

00

65

The time measured by an atomic clock on the Earth’s surface i.e. in system 2 is:

  [1 (V / C )2 ]1/ 4    [1 1 (V / C )2 ]  relationship (3.49).
2 0o 1 40 0 1

Time  that is measured by the clock at   T is:
2 11

(3.60)   [1 1 (V / C )2 ]T
2(T1) 40 o 1

The difference R of the duration of the two times:

T

R  T   ,
T2 2(T1)

which after taking into consideration equations (3.59) and (3.60) becomes:

(3.60a) R  3 (V / C )2 T . From equation (3.59) we obtain:
T 40 0 1

(3.60b): T  T 1 (V / C )2
12 00

Hence the R of the time between the duration of the Earth’s sidereal day and the time

T

measured by the atomic clock after the day elapsed:

(3.61) R  3 (V / C )2 1 (V / C )2 T ; from the equations (3.60a) and (3.60b), where:
T 40 0 00

T2  T  86164.091 s

The R of the time between the duration of the Earth’s stellar year and the time measured

Trg

by the atomic clock after the year elapsed:

(3.62) R  3 (V / C )2 1 (V / C )2 T , where:
Trg 4 0 0 00 rg

Trg  365.256366 days

V /C R (3.61) R (3.62)

00 T Trg

10 4 s s
1,24 104 0.646 10 3 0.236
1.5 10 4
10 3 0.364
2 104 1.454 10 3 0.532
5 10 4 2.584 10 3
TABLE 15 16.155 10 3 0.946
5.917

The inequality T   results from equations (3.59) and (3.60). Hence the elongation of
2 2(T1)

the Earth’s sidereal day as well as the stellar year with respect to the time measured by an
atomic clock is only apparent (see Table 15). In reality the time measured by an atomic clock
is delayed with respect to the time defined by Earth’s rotation which angular speed varies
slightly not only due to the movement of masses such as water, snow and lava but also due to
the fact that the Earth’s speed on its orbit constantly changes.

66

III.5 DETERMINING THE ABSOLUTE VELOCITIES OF THE EARTH

AND THE SUN WITH ATOMIC CLOCKS
There are two methods for determining the absolute velocities of the Earth and the Sun. Both
of them involve the use of atomic clocks.

METHOD I:

In which the difference in times that have been measured by two identical atomic clocks

ZA , ZA is exploited.

ap

Assumptions: 1) Clock ZA is situated along any given Earth’s parallel.

a

2) Clock ZA is situated at the South Pole.
p
Clock’s velocity V on Earth’s surface with respect to the aether is the sum of three vectors:
0   
(3.63) V  V V V relationships (2.1), (2.2).
 0 ra zs se
Vector V is the velocity of the ZA clock on the plane of Earth’s parallel.
ra a

The Earth’s center travels with respect to the aether with velocity:
(3.64) V V V so
ze  zs  se
(3.65) V V V
0 ra ze

(3.66) V  V 2 V 2

ze zs se

 plane. The
In the coordinate system OX2Y2Z2 (Fig. 19) vector V is located on the Y2Z2

ze

Earth’s parallel with clock ZA coincides with the X2Y2 plane. Thus vector V is located on

a ra

the plane X2Y2 .


Fig.19 The position of vector V with respect to V vector .

ra ze

SYMBOLS: 

 an angle between vector V and the Earth’s parallel (plane X2Y2),

ze

 an angle between OX2 axis and vector V ,

ra
 
 an angle between vectors V and V ,   ∡(V ,V )
V ra ze V ra ze

67


The direction of vector V varies as a result of changes in the values of angle     

ra  2 2 
(Fig.19). Clocks are synchronized at the time when vector V is perpendicular to vector V

ra ze

i.e.  0. On the clocks’ synchronization day the UT needs to be determined when the

vectors V and V become perpendicular to each other.
ra ze

The angles in Fig.20 follow the equation:

  3600  (GHA   )
 ZAa sR
 
Vector V is perpendicular to both vector V and V when angle   1800 ;
ra zs ze ZAa

thus 1800  3600  (GHA   ) . Hence
sR

(3.67) GHA  1800   
sR

If the expression 1800    takes a negative value then GHA :

Rs

(3.68) GHAs  3600 1800  R  

(3.69)     , where:
R s zs

 right ascension of the Sun at 12 o’clock of the UT on the day when clocks
s

were synchronized, 

 right ascension of the V velocity determined from (2.26) or (2.27).
zs zs

True anomaly can be obtained from relationship (2.20) or (2.21) adopting for calculations the

12 o’clock of the UT on the day when clocks were synchronized.

Knowing a synchronization day and the value of the GHA angle obtained from relationships

s

(3.67) or (3.68), the UT of clocks synchronization time can be found in The Nautical Almanac.

The coordinates of vectors V and V in the OX2Y2Z2 system (Fig.19) are as follows:
 ra
ze

V  [V cos  , V sin , 0]
ra ra ra

V  [0 , V cos  , V sin  ]
ze  ze ze

Scalar product of vectors V and V implies:
ra ze

V V V cos  V sin
cos   ra ze  ze ra  cos  sin . Therefore
V VV VV
ra ze ra ze

(3.70) cos   cos sin
V

The absolute speed V of the clock located on a parallel can be obtained from the following

0ra

expression:

V 2  (V V cos  )2  (V sin )2  V 2 V 2  2V V cos  .
0ra ze ra V ra V ze ra ra ze V

Applying (3.70) we have:

(3.71) V 2  V 2 V 2  2V V cos sin
0ra ze ra ra ze

The absolute speed V of the clock located at the South Pole:

0p

(3.72) V V

0 p ze

68

Fig.20 Angles in the equatorial coordinate system.

SYMBOLS:  
P the North Pole, V, projection of vector V on the equator’s plane,
ra
N V, ra
projection of vector V on the equator’s plane,
1 equator, zs
zs
2 parallel (of altitude) 
G Greenwich,  the longitude of clock’s position,
S the Sun, the  angle in equatorial system (Fig.10),
R
ZA atomic clock, Greenwich Hour Angle of the Sun,
GHA 
a
s Greenwich Hour Angle of the V vector.

GHA zs

zs

Relationship (3.49) determines the times measured by the clocks in systems 1 and 2

  [1 (V / C )2 ]1/ 4   [1 1 (V / C )2 ] 
2 00 1 40 0 1

The time measured by the ZA clock The time measured by the ZA clock

a p

at a selected point on the parallel: at the North Pole:

  [1 1 (V / C )2 ]    [1 1 (V / C )2 ] 
2ra 4 0ra 0 1 2p 4 0p 0 1

The difference in times measured by the clocks Rpa   2 p   2ra  1 (V02ra V02p )  1
4C02

which, after applying relationships (3.71) and (3.72), takes the following form:

R  1 [V 2  2V V cos  sin] 
pa 4C 2 ra ra ze 1

0

The value of the  angle varies, hence very small values of time increments  should be
1

considered.

As a result: dR  1 [V 2  2V V cos  sin] d( ) ,     .
Earth’s sidereal day pa 4C 2 ra ra ze 1 22
According to (3.51)
0

T  T  86164.091 s ,   2  2 . So   2  .
2T T T2
2
2

   2   , since the value V / C is very small.

1 [1 (V / C ) 2 ]1/ 4 2 00

00

69

We now have the following equation: dR  1 [V 2  2V V cos  sin(2  )] d( ) .
pa 4C 2 ra ra ze T2 2

0 ze

The difference in times that have been measured by the clocks during a sidereal half-a-day

which commenced at the time of their synchronization:

1 T /2 T / 2 2

 [V 2 d( )  2V V   )  d( and after integration
 R cos sin( )]
pa(T / 2) 4C 2 ra 2 ra ze 0 T2 2
0
0

(3.73) V 2T V T
R  ra  ra V cos 

pa(T / 2) 8C 2 2  C 2 ze
00

The difference in times that have been measured by the clocks during one sidereal day

which commenced at the time of their synchronization:

 1 T T 2  )d ( . After integration

[V 2 d( )  2V V 
 R cos sin( )]
pa(T ) 4C 2 ra 2 ra ze 0 T2 2
0
0

(3.74) V2
R  ra T

pa(T ) 4C 2

0

Half-a-day fluctuations of difference in times that have been measured by atomic clocks are

observed.

After equation (3.73) has been transformed and relationship (3.66) introduced the following

equation appears:

(3.75) 2  C02Rpa(T / 2)   Vra  Vz2s  Vs2e cos 
VraT 4

Now the value of cos  that appears in equation (3.75) needs to be determined. It can be done

by following this procedure:
Vector V is the sum of two vectors perpendicular to each other:
ze   
V  V V relationship (3.64).
 ze zs se
Vector V is situated on the plane of the ecliptic (Fig.9).
zs  
Vectors V and V are both perpendicular to the plane of the ecliptic (Fig.11).
se se

In the OX1Y1Z1 system, the coordinates of vector V are (Fig.9):

ze

(3.76) V  [V cos , V sin , V ] , where:
ze zs zs se

(3.77)   1800  when 0   1800 
2 0

(3.78)   1800  when 1800    1800
2 0

(3.79)   when 1800   3600 , where:
3

 relationship (2.4),  relationship (2.6),
3 0

 relationship (2.5),  true anomaly.
2
Let W be a unit vector situated along the Earth’s axis and pointing north. This vector is

therefore perpendicular to the plane of the parallel.
The coordinates of vector W in system OX1Y1Z1 are (Fig.9):
W  [cos(900   )  cos( ) , cos(900   ) sin( ) , sin(900   )] . After reduction
1 1

(3.80) W  [sin   cos ,  sin  sin , cos  ] , where:
11

 the inclination of the ecliptic to the equator,

 an angle obtained from equation (2.17), (Fig.9).
1

70

   
Fig.21a Position of vector V  V V Fig.21b Position of vector V  V V

ze zs se ze zs se

SYMBOLS:

1 parallel plane i.e. its projection,

2 Earth’s axis, 
W a unit vector.   ∡ (W ,Vze )
W

 the angle between vectors W and V ,

W ze

Scalar product of vectors (3.7 6) and (3.80) gives:
 W Vze
cos W WVze , W 1. This implies that

cos   1 (V cos sin   cos V sin sin  sin V cos  )
WV zs 1 zs 1 se

ze

which, after transformation and with relationship (3.66) included, makes:

V sin   cos(  ) V cos 
(3.81) cos   zs 1 se

W V 2 V 2
zs se

According to Figures 21a and 21b the following expressions can be written respectively:

  900   ,
W

  900   . Hence
W

cos   cos(900  )   sin  . In this way sin    cos  . So
W W

  arcsin( cos  )   arcsin(cos  ) . Hence
WW

cos   cos[ arcsin(cos  )]  cos[arcsin (cos  )] .
WW

Then after applying equation (3.81):

V sin   cos(  ) V cos 
(3.82) cos   cos[arcsin zs 1 se
]

V 2 V 2
zs se

If relationship (3.82) is used in equation (3.75), the following expression appears:

(3.83) 2  C02Rpa(T / 2)   Vra  Vz2s  Vs2e  cos[arcsin Vzs sin   cos( 1)  Vse cos  ]
VraT 4 Vz2s  Vs2e

Now we have two equations for calculating the speed V of the Sun with respect to the

se

aether:

(3.84) 2  C02Rpa(T / 2)   Vra  Vz2s  Vs2e  cos[arcsin Vzs sin  cos( 1) Vse cos  ] ,
VraT 4   Vz2s  Vs2e

when V V V , or
ze zs se

71

(3.85) 2  C02Rpa(T / 2)   Vra  Vz2s  Vs2e  cos[arcsin Vzs sin   cos( 1)  Vse cos  ] ,
   Vz2s  Vs2e
VraT 4

when V V V
ze zs se

Knowing R , the absolute speed V of the Sun can be calculated from equations (3.84)
pa(T / 2) se

or (3.85) by the method of successive approximations. The R pa(T / 2) is the absolute value of

the difference in times that have been measured by atomic clocks after half a sidereal day

has elapsed since the time of their synchronization.

Having calculated V , the absolute speed V of the Earth can be obtained as follows:

se ze

V  V 2 V 2 relationship (3.66), where:

ze zs se

V relationship (2.35).

zs

The speed of clock: V  R cos  relationship (2.3).
ra

Thus two solutions are possible.

Solution (1): V, Vse(1) from equations (3.84) and (3.66),

ze(1)

Solution (2): V, V from equations (3.85) and (3.66).

ze(2) se(2)

Solution (1) Solution (2)
values obtained from values obtained from
equations (3.84) & (3.66). equations (3.85) & (3.66).

No. R V /C V /C V /C V /C

pa(T / 2) ze(1) 0 se(1) 0 ze(2) 0 se(2) 0

s-- --

1 0.798 10 6 2.0255 10 4 1.7619 10 4

2 0.898 10 6 1.4890 10 4 1.1040 10 4 No values. No values.

3 1.50110 6 4.349110 4 4.2327 104 1.1104 10 4 4.8444 10 5
1.499110 4 1.1174 10 4
4 1.799 10 6 4.9672 10 4 4.8657 104 1.6877 104 1.360110 4
2.000110 4 1.7327 104
5 1.917 106 5.2052 10 4 5.1084 10 4

6 2.10110 6 5.5713 10 4 5.4810 10 4

R value adopted for calculations,

pa(T / 2)

Vze(1) ,Vze(2) Earth’s absolute speeds,

Vse(1) ,Vse(2) Sun’s absolute speeds.

TABLE 16.

Table 16 provides the results of calculations of the V / C and V / C values which
ze 0 se 0

correspond to the R pa(T / 2) values adopted for calculations.

The values of the Solution (1) nos. 3, 4, 5, 6 cannot be accepted for two reasons:

1. If Vze(1) / C0 took values given in solution (1) nos. 3. 4, 5, 6, the shifts of interference fringes

in the Michelson’s interferometer would be clearly visible (Tables 2 &3).

2. Apparent elongation of the Earth’s sidereal day would take more than a few milliseconds

and apparent elongation of the stellar year - a few seconds (Table 15).

Given R pa(T / 2) , the value calculated from the experiment, the absolute speeds of the Earth

and the Sun can be obtained with the use of method I .

PROGRAM VzeVse was applied for calculations (for results - see Table 16).

72

METHOD II.

In which the difference in times that have been measured by two identical atomic clocks
ZA , ZA , that are located along any given Earth’s parallel, is exploited.

ab

Assumptions: 1) The distance clock-Earth’s center is identical.
2) Clock ZA is placed in location of  longitude.

a

Clock ZA is placed in location of  1800 longitude.
b

The clocks are synchronized at the time when velocity V of the ZA clock is perpendicular to
 ra a

vector V (Fig.20). Method I discussed above describes procedures for determining the UT

ze

of synchronization time.
The absolute speed V of the ZAa clock:

0ra

V 2  V 2 V 2  2V V cos sin equation (3.71).
0ra ze ra ra ze

Hence the absolute speed V of the ZA clock:

0rb b

(3.86) V 2  V 2 V 2  2V V cos sin(   ) , V V
0rb ze rb rb ze
rb ra

Time measured by the ZAa clock: Time measured by the ZA clock:

b

  [1 1 (V / C )2 ]  1   [1 1 (V / C )2 ]  1
2ra 4 0ra 0 2rb 4 0rb 0

The difference in times measured by the clocks: Rba   2rb   2ra  1 (V02ra V02rb )  1
4C02

After applying equations (3.71) and (3.86): R  1 V V cos  [sin  sin(   )] 
ba 2C 2 ra ze 1

0

If  are the values of very small time increments, then:
1

dR  1 V V cos  [sin   sin(   )] d( ) .
ba 2C 2 ra ze 1

0

According to relationship (3.51)    2   as the V / C value is very small.

1 [1 (V / C ) 2 ]1/ 4 2 00

00

The angles   2  ,     2 (  T ) .
T2 T 22

Hence dR  1 V V cos  {sin(2  )  sin[2 (  T )]}  d( )
ba 2C 2 ra ze T2 T 22 2

0

Difference in times measured by the clocks during a sidereal half-a-day that commenced at

the synchronization time:

1 T / 2 2 2 T )]}  d( )
2C 2 T T 22
R / 2)  VV cos    {sin(  2 )  sin[ (  . After integration
0 2
ba(T ra ze 0

(3.87) VT
R  ra V cos  .

ba(T / 2)   C 2 ze
0

Difference in times measured by the clocks during a sidereal day that commenced at the

synchronization time:

R  1 VV cos   T {sin( 2   )  sin[2 (  T )]}  d( ) . After integration
2C 2 T 2T 2 22
ba(T ) ra ze 
0
0

(3.88) R  0

ba(T )

After equation (3.87) has been transformed and relationships (3.66) and (3.82) implemented:

 C2R V sin   cos(  ) V cos 
0 ba(T / 2)  V 2 V 2  cos[arcsin zs 1 se
VT ]
zs se
ra V 2 V 2
zs se

73

That provides two equations for calculating the speed V of the Sun with respect to the

se

aether:

 C2R V sin   cos(  ) V cos 
(3.89) 0 ba(T / 2)  V 2 V 2  cos[arcsin zs 1 se ],

VT zs se V 2 V 2

ra  zs se


when V  V V , or
ze zs se

 C2R V sin   cos(  ) V cos 
(3.90) 0 ba(T / 2)  V 2 V 2  cos[arcsin zs 1 se ],

VT zs se V 2 V 2

ra  zs se

when  
V V V
ze zs se

Knowing R , the absolute speed V of the Sun can be calculated from equations (3.89)
ba(T / 2) se

or (3.90) by the method of successive approximations. The R ba(T / 2) is the absolute value of

the difference in times that have been measured by the atomic clocks after half a sidereal

day elapsed since the synchronization time.

Having V , the absolute speed V of the Earth can be obtained as follows:

se ze

V  V 2 V 2 relationship (3.66), where:

ze zs se

V relationship (2.35),

zs

The speed of the clock : V  R cos  relationship (2.3).
ra

Thus two solutions are possible.

Solution (1): V ,V from equations (3.89) and (3.66)

ze(1) se(1)

Solution (2): V ,V se(2) from equations (3.90) and (3.66).
ze(2)

Solution (1) Solution (2)
values obtained from values obtained from
equations (3.89) & (3.66). equations (3.90) & (3.66).

No . R V /C V /C V /C V /C

ba(T / 2) ze(1) 0 se(1) 0 ze(2) 0 se(2) 0

s - - --

1 1.822106 1.4238 10 4 1.0143 10 4 No values. No values.

2 2.164106 1.1121104 4.8837 105

3 2.956106 4.3222104 4.2052104 1.0986 10 4 4.5688 10 5
1.5004 10 4 1.1193 10 4
4 3.579106 4.9692104 4.8677 104 1.6876 10 4 1.3600 10 4
2.0087 104 1.7426 10 4
5 3.813106 5.2051104 5.1083104

6 4.191106 5.5811104 5.4910104

R value adopted for calculations,

ba(T / 2)

Vze(1) ,Vze(2) Earth’s absolute speeds ,

Vse(1) ,Vse(2) Sun’s absolute speeds.

TABLE 17.

Table 17 provides the results of calculations of the V / C , V / C values which
ze 0 se 0

correspond to the R ba(T / 2) values that were adopted for calculations.

The values of the solution (1) nos. 3, 4, 5, 6 cannot be accepted due to reasons described in

method I.

Given R ba(T / 2) , the value calculated from the experiment, the absolute speeds of the Earth

and the Sun can be obtained with the use of method II .

PROGRAM VzeVse was applied for calculations (for results – see Table 17).

74

III.5.1 CALCULATING ABSOLUTE VELOCITIES OF THE EARTH AND THE SUN

(Example)

Assumptions:

1) Atomic clock ZA is located in a place with geographical coordinates:

a

  50o34' ,   21o41' (Tarnobrzeg city, Poland)

2) Experiment begins on 28th October 2011 with the aim to obtain the difference in

times that have been measured by the atomic clocks.

First, the synchronization time of atomic clocks needs to be calculated as follows:

Year 2010. Astronomical winter starts on 21st December 23h38m.5 of the UT.

Year 2011. Astronomical spring starts on 20th March, 23h20m.7 of the UT.
From that it can be inferred that the duration of astronomical winter in 2010-2011:

Tz  88d 23h42m.2  88.9876388 days .
Precession in longitude during astronomical winter (2.16) is:

p  Tz 50''.292  5.9404049 105 rad
Trz

From equation (2.17) i.e.: 88.9876388  t( / 2  5.9404049105 1)  t(2 1) the value of the
angle  (Fig.9) can be calculated by the method of successive approximations:

1

1  0.2295109 rad  13o.1501154
From relationship (2.18) we have: Ta  t(1)  12.9054684 days .
The period of time that elapsed from the start of astronomical winter of 2010 until the end of

the calendar year: Tb  10d 21m.5  10.0149305  days .

Difference of the two times: Ta Tb  2.8905379 days .

Time t ( ) which elapsed from the start of the calendar year of 2011 until 12 o’clock UT
4

on 28th October 2011 is t4 ( )  300.5 days . Given the inequality 180o   360o and the

equation (2.21), in which 300.5  Trg  t( )  2.8905379 , the value of true anomaly at 12 o’clock

UT can be calculated by the method of successive approximations:

  5.08857577  rad  291o.5539157 .
In order to determine the value of right ascension of the Sun on 28th October 2011 at

12 o’clock of UT, we need to revert to The Nautical Almanac in which we find that:

 s  GHAaries  GHAsun  216o29'.4  4o2'.8  212o.4433333
PROGRAM GHA, detailed in Chapter IV of this work, was applied to calculate the values of

the GHA angle (see equations (3.67) and (3.68)). After the value of the Sun’s right ascension

s

 s  212o.4433333 and the value of true anomaly   291o.5539157 were applied in the

PROGRAM the value of the GHA angle was obtained: GHAs  73o.8401234 .

s

According to The Nautical Almanac, the time that corresponds with that angle is: 16h39m21s
of UT and it is when the clocks need to be synchronized on 28th October. Then after half a

sidereal day has elapsed since the synchronization time of the clocks i.e. at 4h37m23s of UT
on 29th October, the difference in times that had been measured by the atomic clocks has to

be taken and used in calculations.

PROGRAM VzeVse, detailed in this work, was used to calculate the absolute speed values of

the Earth and the Sun. After the values R pa(T / 2) or R ba(T / 2) were applied to the program

together with the value of true anomaly   291o.5539157 , the absolute speed values of the
Earth and the Sun were obtained (see method I and method II).

From the values of the solutions (1) and (2) it is possible to determine the direction of the
Sun’s absolute velocity, whether it is V or V , (Fig. 11).

se se

The results for values of R pa(T / 2) ,Rba(T / 2) are presented in tables 16 and 17.

75

CHAPTER IV

PROGRAMS

IV.1 PROGRAMS: abIntM, baIntM, IntM

The following symbols were adopted and used in the programs:

Vw  V / C , ew  | e  e |/  , L0  
oo a5 b5 o o

g thickness of the semi-transparent PP plate,

g1 angle  ,
1

g2 angle  ,

2

ap values of the  angle,
bp values of the  angle,

a angle  ,
b angle  ,

h increment of  ,  angles,

F angle  ,
de a very small positive number used for calculations.

Angles given in radian measure.

In PROGRAM abIntM and PROGRAM baIntM the following values were used:

h = 1014  rad , de = 107 .

Shifts of interference fringes are determined with respect to point Mo with a coordinate

e0  0.1508323849500 m .
After the values of ap, bp angles have been entered to the program, the calculations end when

the condition of mutual approximation of points A and B is satisfied i.e. when ew ≤de.

55

With the method of successive approximations the values of ap, bp angles are selected and

then successively entered into programs until the condition of mutual approximation of

approximated points A5 and B to point Mo is satisfied, namely when:

5

| e  e |  1011  m , | e  e |  1011  m
o a5 o b5

Consequently the calculated values of the pair of angles (,  ) and of the relative difference

R of light rays distances correspond now to the approximation of points A , B and Mo

w 55

specified above.

Following values were used in calculations:

1) Basic dimensions of the Michelson’s interferometer.

L1  1.20 m , L  0.15 m , g  0.5103  m (thickness of PP plate).

3

L  1.05 m , L  0.10 m .

2 4

2) The wavelength of light in a vacuum   5.9107  m .
o

3) The PP plate refractive index with respect to a vacuum n2  1.52 .

Programs are written in TURBO PASCAL 7.

76

IV.1.1 PROGRAM abIntM;
Var
g=0.5E-3;
a, ap, bp, b,Vw, h, de, ew, Rw, g1, g2, g11, g22,
a1, a2, a3, a4, a5, b1, b2, b3, b4, b5,
xa1, xa2, xa3, xa4, xa5, xa21, xa31, xa41, xa51,
xb1, xb2, xb3, xb4, xb5, xb21, xb31, xb41, xb51,
ya1, ya2, ya3, ya4, ya5, ya21, ya31, ya41, ya51,
yb1, yb2, yb3, yb4, yb5, yb21, yb31, yb41, yb51,
xya1, xya31, xya4, xya41,
xyb1, xyb2, xyb4, xyb21, xyb41,
r21, r22, r221, r23, r31, r32, r321, r33,
r41, r411, r42, r421, r43, r51, r52, r521, r53,
s21, s211, s22, s221, s23, s31, s32, s321, s33,
s41, s411, s42, s421, s43, s51, s52, s521, s53,
ea1, ea2, ea3, ea4, ea5, eb1, eb2, eb3, eb4, eb5,
qa1, qa2, qa3, qa4, qa5, qb1, qb2, qb3, qb4, qb5,
a1u, a2u, a3u, a4u, a5u, b1u, b2u, b3u, b4u, b5u: real;

Const
L1=1.2; L2=1.05; L3=0.15; L4=0.1; L0=5.9E-7;
Pi=3.14159265358;

BEGIN write(‘ap=’); read(ap);
write(‘h=’); read(h);
write(‘de=’); read(de); a:=ap; b:=a; ew:=0;
write(‘F=’); read(F);
write(‘Vw=’); read(Vw);

REPEAT b:=b-(ew/de)*h;
if b<-10*ABS(a) then begin a:=a-h; b:=a; end;

g11:=sin(Pi/4-a)/n2;
g1:=arctan(g11/sqrt(1-g11*g11));
g22:=sin(Pi/4+b)/n2;
g2:=arctan(g22/sqrt(1-g22*g22));
a1:=L3/(cos(a)-sin(a)-Vw*(cos(F)-sin(F)));

xya1:=L3+a1*Vw*(cos(F)-sin(F));
xa1:=xya1*cos(a)/(cos(a)-sin(a));
ya1:=xya1*sin(a)/(cos(a)-sin(a));

xa21:=(L2-ya1+a1*Vw*sin(F))*sin(a)/cos(a);
ya21:=L2+a1*Vw*sin(F)-ya1;

r21:=Vw*sin(F)*(xa21*sin(a)/cos(a)+ya21);
r221:=Vw*sin(F)/cos(a);

r22:=1-r221*r221;
r23:=r21*r21+r22*(xa21*xa21+ya21*ya21);
a2:=(r21+sqrt(r23))/r22;
xa2:=xa1+(L2-ya1+(a1+a2)*Vw*sin(F))*sin(a)/cos(a);
ya2:=L2+(a1+a2)*Vw*sin(F);

xya31:=L3+ya2+(a1+a2)*Vw*(cos(F)-sin(F));
xa31:=sin(a)*xya31/(sin(a)+cos(a))+cos(a)*xa2/(sin(a)+cos(a))-xa2;
ya31:=sin(a)*xya31/(sin(a)+cos(a))+cos(a)*xa2/(sin(a)+cos(a))+
-L3-(a1+a2)*Vw*(cos(F)-sin(F))-ya2;
r31:=(xa31*sin(a)-ya31*cos(a))*Vw*(cos(F)-sin(F))/(sin(a)+cos(a));
r321:=Vw*(cos(F)-sin(F))/(sin(a)+cos(a));
r32:=1-r321*r321;

77

r33:=r31*r31+r32*(xa31*xa31+ya31*ya31);
a3:=(r31+sqrt(r33))/r32;
xa3:=(sin(a)/(sin(a)+cos(a)))*(L3+ya2+(a1+a2+a3)*Vw*(cos(F)-sin(F)))+

+cos(a)*xa2/(sin(a)+cos(a));
ya3:=(sin(a)/(sin(a)+cos(a)))*xya31+cos(a)*xa2/(sin(a)+cos(a))+

-L3-(a1+a2)*Vw*(cos(F)-sin(F))+
-(cos(a)/(sin(a)+cos(a)))*a3*Vw*(cos(F)-sin(F));

xya41:=L3+sqrt(2)*g+(a1+a2+a3)*Vw*(cos(F)-sin(F))+
+sin(Pi/4+g1)*xa3/cos(Pi/4+g1)+ya3;

xa41:=(cos(Pi/4+g1)/(sin(Pi/4+g1)+cos(Pi/4+g1)))*xya41-xa3;
ya41:=-(sin(Pi/4+g1)/(sin(Pi/4+g1)+cos(Pi/4+g1)))*xya41+

+sin(Pi/4+g1)*xa3/cos(Pi/4+g1);
r411:=xa41*cos(Pi/4+g1)-ya41*sin(Pi/4+g1);

r41:=r411*n2*Vw*(cos(F)-sin(F))/(sin(Pi/4+g1)+cos(Pi/4+g1));
r421:=n2*Vw*(cos(F)-sin(F))/(sin(Pi/4+g1)+cos(Pi/4+g1));

r42:=1-r421*r421;
r43:=r41*r41+r42*(xa41*xa41+ya41*ya41);
a4:=(r41+sqrt(r43))/r42;
xya4:=L3+sqrt(2)*g+(a1+a2+a3+n2*a4)*Vw*(cos(F)-sin(F))+

+sin(Pi/4+g1)*xa3/cos(Pi/4+g1)+ya3;
xa4:=cos(Pi/4+g1)*xya4/(sin(Pi/4+g1)+cos(Pi/4+g1));
ya4:=-sin(Pi/4+g1)*xya4/(sin(Pi/4+g1)+cos(Pi/4+g1))+

+ya3+sin(Pi/4+g1)*xa3/cos(Pi/4+g1);

xa51:=(L4-(a1+a2+a3+n2*a4)*Vw*sin(F)+ya4)*sin(a)/cos(a);
ya51:=-L4+(a1+a2+a3+n2*a4)*Vw*sin(F)-ya4;

r51:=(ya51-xa51*sin(a)/cos(a))*Vw*sin(F);
r521:=Vw*sin(F)/cos(a);

r52:=1-r521*r521;
r53:=r51*r51+r52*(xa51*xa51+ya51*ya51);
a5:=(r51+sqrt(r53))/r52;
xa5:=(L4-(a1+a2+a3+n2*a4+a5)*Vw*sin(F)+ya4)*sin(a)/cos(a)+xa4;
ya5:=-L4+(a1+a2+a3+n2*a4+a5)*Vw*sin(F);

b1:=L3/(cos(b)-sin(b)-Vw*(cos(F)-sin(F)));
xyb1:=L3+b1*Vw*(cos(F)-sin(F));

xb1:=xyb1*cos(b)/(cos(b)-sin(b));
yb1:=xyb1*sin(b)/(cos(b)-sin(b));

xyb21:=L3+sqrt(2)*g+b1*Vw*(cos(F)-sin(F))+yb1+
+sin(Pi/4-g2)*xb1/cos(Pi/4-g2);

xb21:=cos(Pi/4-g2)*xyb21/(sin(Pi/4-g2)+cos(Pi/4-g2))-xb1;
yb21:=-sin(Pi/4-g2)*xyb21/(sin(Pi/4-g2)+cos(Pi/4-g2))+

+sin(Pi/4-g2)*xb1/cos(Pi/4-g2);
s211:=xb21*cos(Pi/4-g2)-yb21*sin(Pi/4-g2);
s21:=s211*n2*Vw*(cos(F)-sin(F))/(sin(Pi/4-g2)+cos(Pi/4-g2));
s221:=n2*Vw*(cos(F)-sin(F))/(sin(Pi/4-g2)+cos(Pi/4-g2));
s22:=1-s221*s221;

s23:=s21*s21+s22*(xb21*xb21+yb21*yb21);
b2:=(s21+sqrt(s23))/s22;

xyb2:=L3+sqrt(2)*g+(b1+n2*b2)*Vw*(cos(F)-sin(F))+yb1+
+sin(Pi/4-g2)*xb1/cos(Pi/4-g2);

xb2:= cos(Pi/4-g2)*xyb2/(sin(Pi/4-g2)+cos(Pi/4-g2));
yb2:=-sin(Pi/4-g2)*xyb2/(sin(Pi/4-g2)+cos(Pi/4-g2))+yb1+

+sin(Pi/4-g2)*xb1/cos(Pi/4-g2);

78

xb31:=L1+(b1+n2*b2))*Vw*cos(F)-xb2;
yb31:=(L1+(b1+n2*b2)*Vw*cos(F))*sin(b)/cos(b)-sin(b)*xb2/cos(b);

s31:=(xb31+yb31*sin(b)/cos(b))*Vw*cos(F);
s321:=Vw*cos(F)/cos(b);

s32:=1-s321*s321;
s33:= s31*s31+s32*(xb31*xb31+yb31*yb31);
b3:=(s31+sqrt(s33))/s32;
xb3:=L1+(b1+n2*b2+b3)*Vw*cos(F);
yb3:=(sin(b)/cos(b))*(L1+(b1+n2*b2+b3)*Vw*cos(F))+yb2+sin(b)*xb2/cos(b);

xyb41:=L3+sqrt(2)*g+(b1+n2*b2+b3)*Vw*(cos(F)-sin(F))+yb3+
+sin(b)*xb3/cos(b);

xb41:= (cos(b)/(sin(b)+cos(b)))*xyb41-xb3;
yb41:=-(sin(b)/(sin(b)+cos(b)))*xyb41+sin(b)*xb3/cos(b);
s 411:=xb41*cos(b)-yb41*sin(b);

s41:=s411*Vw*(cos(F)-sin(F))/(sin(b)+cos(b));
s421:=Vw*(cos(F)-sin(F))/(sin(b)+cos(b));

s42:=1-s421*s421;
s43:=s41*s41+s42*(xb41*xb41+yb41*yb41);
b4:=(s41+sqrt(s43))/s42;
xyb4:=L3+sqrt(2)*g+(b1+n2*b2+b3+b4)*Vw*(cos(F)-sin(F))+yb3+
+sin(b)*xb3/cos(b);
xb4:= cos(b)*xyb4/(sin(b)+cos(b));
yb4:=-sin(b)*xyb4/(sin(b)+cos(b))+yb3+sin(b)*xb3/cos(b);

xb51:=(L4-(b1+n2*b2+b3+b4)*Vw*sin(F)+yb4)*sin(b)/cos(b);
yb51:=-L4+(b1+n2*b2+b3+b4)*Vw*sin(F)-yb4;

s51:=(yb51-xb51*sin(b)/cos(b))*Vw*sin(F);
s521:=Vw*sin(F)/cos(b);

s52:=1-s521*s521;
s53:=s51*s51+s52*(xb51*xb51+yb51*yb51);
b5:=(s51+sqrt(s53))/s52;
xb5:=(L4-(b1+n2*b2+b3+b4+b5)*Vw*sin(F)+yb4)*(sin(b)/cos(b))+xb4;
yb5:=-L4+(b1+n2*b2+b3+b4+b5)*Vw*sin(F);

ea5:=xa5-(a1+a3+a3+n2*a4+a5)*Vw*cos(F);
eb5:=xb5-(b1+n2*b2+b3+b4+b5)*Vw*cos(F);

ew:=ABS(ea5-eb5)/L0;
if a<-0.4 then ew:=de;
UNTIL ew<=de;

ea1:=xa1-a1*Vw*cos(F);
qa1:=ya1-a1*Vw*sin(F);

ea2:=xa2-(a1+a2)*Vw*cos(F);
qa2:=ya2-(a1+a2)*Vw*sin(F);
ea3:=xa3-(a1+a2+a3)*Vw*cos(F);
qa3:=ya3-(a1+a2+a3)*Vw*sin(F);

ea4:=xa4-(a1+a2+a3+n2*a4)*Vw*cos(F);
qa4:=ya4-(a1+a2+a3+n2*a4)*Vw*sin(F); qa5:=-L4

eb1:=xb1-b1*Vw*cos(F);
qb1:=yb1-b1*Vw*sin(F);
eb2:=xb2-(b1+n2*b2)*Vw*cos(F);
qb2:=yb2-(b1+n2*b2)*Vw*sin(F);
eb3:=xb3-(b1+n2*b2+b3)*Vw*cos(F);
qb3:=yb3-(b1+n2*b2+b3)*Vw*sin(F);
eb4:=xb4-(b1+n2*b2+b3+b4)*Vw*cos(F);
qb4:=yb4-(b1+n2*b2+b3+b4)*Vw*sin(F); qb5:=-L4;

79

a1u:=sqrt(ea1*ea1+qa1*qa1);

a2u:=sqrt((ea2-ea1)*(ea2-ea1)+(qa2-qa1)*(qa2-qa1));

a3u:=sqrt((ea3-ea2)*(ea3-ea2)+(qa3-qa2)*(qa3-qa2));

a4u:=sqrt((ea4-ea3)*(ea4-ea3)+(qa4-qa3)*(qa4-qa3));

a5u:=sqrt((ea5-ea4)*(ea5-ea4)+(qa5-qa4)*(qa5-qa4));

b1u:=sqrt(eb1*eb1+qb1*qb1);

b2u:=sqrt((eb2-eb1)*(eb2-eb1)+(qb2-qb1)*(qb2-qb1));

b3u:=sqrt((eb3-eb2)*(eb3-eb2)+(qb3-qb2)*(qb3-qb2));

b4u:=sqrt((eb4-eb3)*(eb4-eb3)+(qb4-qb3)*(qb4-qb3));

b5u:=sqrt((eb5-eb4)*(eb5-eb4)+(qb5-qb4)*(qb5-qb4));

Rw:=(a1u+a2u+a3u+n2*a4u+a5u-b1u-n2*b2u-b3u-b4u-b5u)/L0;

write(‘a=’,a); writeln;

write(‘b=’,b); writeln;

write(‘ea5=’,ea5); writeln;

write(‘eb5=’,eb5); writeln;

write(‘ew=’,ew); writeln;

write(‘Rw=’,Rw); writeln;

write(‘frac(Rw)=’,frac(Rw)); writeln;writeln;

END.

Program abIntM is designed to calculate the pairs of (,  ) angles and the relative difference
R of the distances travelled by the rays of light when    .

w

IV.1.2 PROGRAM baIntM;
Var

PROGRAM abIntM

Const

PROGRAM abIntM

BEGIN write(‘bp=’); read(bp);
write(‘h=’); read(h);
write(‘de=’); read(de); b:=bp; a:=b; ew:=0;
write(‘F=’); read(F);
write(‘Vw=’); read(F);

REPEAT a:=a-(ew/de)*h;
If a<-10*ABS(b) then begin b:=b-h; a:=b; end;

PROGRAM abIntM

if b<-0.4 then ew:=de;
UNTIL ew<=de;

PROGRAM abIntM
END.
Program baIntM is designed to calculate pairs of angles (,  ) and the relative difference R

w

of distances travelled by the rays of light when    .

80

IV.1.3 PROGRAM IntM;

Var

PROGRAM abIntM

Const
PROGRAM abIntM

BEGIN write(‘a=’); read(a);
write(‘b=’); read(b);
write(‘F=’); read(F);
write(‘Vw=’); read(Vw);

g11:=sin(Pi/4-a)/n2;
g1:=arctan(g11/sqrt(1-g11*g11));

PROGRAM abIntM

ew:=ABS(ea5-eb5)/L0;
ea1:=xa1-a1*Vw*cos(F);
qa1:=ya1-a1*Vw*sin(F);

PROGRAM abIntM

b5u:=sqrt((eb5-eb4)*(eb5-eb4)+(qb5-qb4)*(qb5-qb4));

Rrw:=(a1u+a2u+a3u+n2*a4u+a5u-b1u-n2*b2u-b3u-b4u-b5u)/L0;

write(‘ea5=’,ea5); writeln;
write(‘eb5=’,eb5); writeln;
write(‘ew=’,ew); writeln;
write(‘Rrw=’,Rrw); writeln;writeln;

END.

PROGRAM IntM is designed to calculate the following (Table 10):

1) The coordinates e , e of non-approximated points A , B .
a5 b5 55

2) Relative distance | e  e |/  of points A , B .
a5 b5 o 55

3) Relative difference R of distances travelled by the light rays reaching

rw

mutually distant points A , B of the screen M.

55

81

IV.2 PROGRAM Vo

Symbols used in the program:

 ALFAs,  NI (true anomaly),
s  FI,
 LAMBDA,
 ALFAse,  PSI,
se  OMEGA.

 ALFAse1,
se1

 DELTAse,
se

 DELTAse1,
se1
 EPSILON,

 ETA0,
o

 ETA1,
1

 ETA2,

2

 ETA3,
3

In this program: V  1.1V was assumed for calculations and angles were given in

se zs

degree ...o measures in decimal system .

Program was written in TURBO PASCAL 7.

PROGRAM Vo;
Var

b, ETA0, ETA1, ETA2, ETA3, NI, PSI, g3, k1, k2, k11, k22, k33,
ALFAs, ALFAzs,
DELTAzs, GHAaries, LHAzs, LHAse, LHAse1,
Hzs, Hse, Hse1, H01, H02,
Azs, Ase, Ase1, A01, A02,
dzs, dse, dse1,
zzs, zse, zse1, z01, z02,
Vzs, Vse,
Vrq, Vzse, Vzsq, Vzsw, Vsee, Vseq, Vsew,
Vse1e, Vse1q, Vse1w,
V01e, V01q, V01w, V02e, V02q, V02w, V01, V02,
h1, h2, h3, h4, h5, az1, az2, az3, az4, az5 : real;

Const Trz=365.242199;
Pi=3.14159265358;
a=149597E3; e=0.01671; EPSILON=0.4090877;
R=6378.1; OMEGA=7.292115E-5; Trg=365.256366;
ALFAse=3*Pi/2; ALFAse1=Pi/2;

BEGIN write (‘FI=’); read (FI);

write(‘LAMBDA=’); read(LAMBDA);

write(‘ALFAs=’); read(ALFAs);

write(‘GHAaries=’); read(GHAaries);

write(‘NI=’); read(NI);

FI:=FI*Pi/180; LAMBDA:=LAMBDA*Pi/180;

ALFAs:=ALFAs*Pi/180; GHAaries:=GHAaries*Pi/180; NI:=NI*Pi/180;

b:=sqrt(a*a-sqr(e*a));

g3:=e*(1+e*cos(NI))/(sin(NI)*(1-e*e));

ETA3:=arctan(-sqr(b/a)*(g3+cos(NI)/sin(NI)));

82

ETA2:=ABS(ETA3); ETA0:=arctan(b/(e*a));
if NI>0 then begin if NI<=Pi-ETA0
if NI>Pi-ETA0 then begin if NI<Pi then PSI:=NI+ETA2; end;
if NI>Pi then begin if NI<=Pi+ETA0
if NI>Pi+ETA0 then begin if NI<2*Pi then PSI:=NI-ETA2; end;

then PSI:=-Pi+NI+ETA2; end;

then PSI:=-Pi+NI-ETA2; end;

k11:=arctan(sin(ALFAs)/(cos(ALFAs)*cos(EPSILON)));

if ALFAs>=0 then begin if ALFAs<Pi/2 then k1:=k11; end;

if ALFAs>Pi/2 then begin if ALFAs<3*Pi/2 then k1:=Pi+k11; end;

if ALFAs>3*Pi/2 then begin if ALFAs<2*Pi then k1:=2*Pi+k11; end;

k2:=k1-PSI;

k22:=arctan(sin(k2)*cos(EPSILON)/cos(k2));

if k2>-Pi/2 then begin if k2<Pi/2 then ALFAzs:=k22; end;

if k2> Pi/2 then begin if k2<3*Pi/2 then ALFAzs:=Pi+k22; end;

k33:=sin(k2)*sin(EPSILON);
DELTAzs:=arctan(k33/sqrt(1-k33*k33));

LHAzs:=GHAaries-ALFAzs+LAMBDA;
h1:=cos(DELTAzs)*cos(FI)*cos(LHAzs)+sin(DELTAzs)*sin(FI);
Hzs:=arctan(h1/sqrt(1-h1*h1));

dzs:=(sin(DELTAzs)-sin(Hzs)*sin(FI))/(cos(Hzs)*cos(FI));
zzs:=dzs/ABS(dzs);
az1:=cos(DELTAzs)*sin(LHAzs)/cos(Hzs);
Azs;=(Pi/2)*(3+zzs)-zzs*arctan(az1/sqrt(1-az1*az1));
Vzs:=2*Pi*a*(1+e*cos(NI))/(Trg*24*3600*sqrt(1-e*e)*sin(PSI));

LHAse:=GHAaries-ALFAse+LAMBDA;
h2:=cos(DELTAse)*cos(FI)*cos(LHAse)+sin(DELTAse)*sin(FI);
Hse:=arctan(h2/sqrt(1-h2*h2);

dse:=(sin(DELTAse)-sin(Hse)*sin(FI))/(cos(Hse)*cos(FI));
zse:=dse/ABS(dse);
az2:=cos(DELTAse)*sin(LHAse)/cos(Hse);
Ase:=(Pi/2)*(3+zse)-zse*arctan(az2/sqrt(1-az2*az2));

LHAse1:=GHAaries-ALFAse1+LAMBDA;
h3:=cos(DELTAse1)*cos(FI)*cos(LHAse1)+sin(DELTAse1)*sin(FI);
Hse1:=arctan(h3/sqrt(1-h3*h3));

dse1:=(sin(DELTAse1)-sin(Hse1)*sin(FI))/(cos(Hse1)*cos(FI));
zse1:=dse1/ABS(dse1);

az3:=cos(DELTAse1)*sin(LHAse1)/cos(Hse1);
Ase1:=(Pi/2)*(3+zse1)-zse1*arctan(az3/sqrt(1-az3*az3));

Vse:=1.1*Vzs;
Vrq:=OMEGA*R*cos(FI);

Vzse:=Vzs*cos(Hzs)*cos(Azs);
Vzsq:=Vzs*cos(Hzs)*sin(Azs);
Vzsw:=Vzs*sin(Hzs);

Vsee:=Vse*cos(Hse)*cos(Ase);
Vseq:=Vse*cos(Hse)*sin(Ase);
Vsew:=Vse*sin(Hse);
Vse1e:=Vse*cos(Hse1)*cos(Ase1);
Vse1q:=Vse*cos(Hse1)*sin(Ase1);
Vse1w:=Vse*sin(Hse1);
V01e:=Vzse+Vsee;

83

V01q:=Vrq+Vzsq+Vseq;
V01w:=Vzsw+Vsew;
V01:=sqrt(sqr(V01e)+sqr(V01q)+sqr(V01w));

h4:=V01w/V01;
H01:=arctan(h4/sqrt(1-h4*h4));

z01:=V01e/ABS(V01e);
az4:=V01q/(V01*cos(H01));
A01:=(Pi/2)*(3+z01)+z01*arctan(az4/sqrt(1-az4*az4));

V02e:=Vzse+Vse1e;
V02q:=Vrq+Vzsq+Vse1q;
V02w:=Vzsw+Vse1w;
V02:=sqrt(sqr(V02e)+sqr(V02q)+sqr(V02w)):

h5:=V02w/V02;
H02:=arctan(h5/sqrt(1-h5*h5));

z02:=V02e/ABS(V02e);
az5:=V02q/(V02*cos(H02));
A02:=(Pi/2)*(3+z02)+z02*arctan(az5/sqrt(1-az5*az5));

H01:=H01*180/Pi; A01:=A01*180/Pi;
H02:=H02*180/Pi; A02:=A02*180/Pi;
if A01>360 then
if A02>360 then A01:=A01-360;
A02:=A02-360;

write(‘Vzs=’,Vzs); writeln;

write(‘Hzs=’,Hzs); writeln;

write(‘Azs=’,Azs); writeln;writeln;

write(‘Vo=V01=’,V01); writeln;
writeln;
write(‘H01=’,H01); writeln;writeln

write(‘A01=’,A01);

write(‘Vo=V02=’,V02); writeln;

write(‘H02=’,H02); writeln;

write(‘A02=’,A02); writeln;writeln;

END.


PROGRAM Vo is designed to calculate the coordinates of velocities V , V (2.1) and
 zs 01
V (2.2) in the horizontal system.

02

84

IV.3 PROGRAM GHA  ETA0,
0
Symbols used in this program: ETA2,
 ALFAs,  ETA3,
NI (true anomaly),
s 2 GHAs.

 ALFAzs, 
zs 3
 EPS,
 LAMBDA, 
GHA
 PSI,
 PSIR, s

R

PROGRAM GHA;
Var

b, g3, ETA0, ETA2, ETA3, k1, k2, k11, k22, ALFAs, ALFAzs,
NI, PSI, PSIR, GHAs : Real;

Const
Pi=3.14159265358;
a=149597E3; e=0.01671; EPS=0.4090877; Trg=365.256366; LAMBDA=0.37844559;

BEGIN write(‘ALFAs=’); read(ALFAs);
write(‘NI=’); read(NI);

ALFAs:=ALFAs*Pi/180; NI:=NI*Pi/180;

b:=sqrt(a*a-sqr(e*a));

g3:=e*(1+e*cos(NI))/(sin(NI)*(1-e*e));

ETA3:=arctan(-sqr(b/a)*(g3+cos(NI)/sin(NI)));

ETA2:=ABS(ETA3); ETA0:=arctan(b/(e*a));

if NI>0 then begin if NI<=Pi-ETA0 then PSI:=NI+ETA2; end;

if NI>Pi-ETA0 then begin if NI<Pi then PSI:=NI-ETA2; end;

if NI>Pi then begin if NI<=Pi+ETA0 then PSI:=-Pi+NI+ETA2; end;

if NI>Pi+ETA0 then begin if NI<2*Pi then PSI:=-Pi+NI-ETA2; end;

k11:=arctan(sin(ALFAs)/(cos(ALFAs)*cos(EPS)));

if ALFAs>=0 then begin if ALFAs<Pi/2 then k1:=k11; end;

if ALFAs>Pi/2 then begin if ALFAs<3*Pi/2 then k1:=Pi+k11; end;

if ALFAs>3*Pi/2 then begin if ALFAs<2*Pi then k1:=2*Pi+k11; end;

k2:=k1-PSI;

k22:=arctan(sin(k2)*cos(EPS)/cos(k2));

if k2>-Pi/2 then begin if k2<Pi/2 then ALFAzs:=k22; end;

if k2> Pi/2 then begin if k2<3*Pi/2 then ALFAzs:=Pi+k22; end;

PSIR:=ALFAs-ALFAzs;

GHAs:=Pi-PSIR-LAMBDA;

If GHAs<0 then GHAs:=2*P+GHAs; GHAs:=GHAs*180/Pi;

write(‘GHAs=’,GHAs); writeln; writeln;

END.

This Program calculates the values of the GHA angle.

s

85

I V.4 PROGRAM VzeVse

Symbols used in this program:

 EPS ,  ETA0 ,  OMEGA,
0

 PSI ,  ETA2 , R Rpa ,

2 pa(T / 2)

 FI ,  ETA3 , R Rba .
3
ba(T / 2)

 NI (true anomaly),

PROGRAM VzeVse;

Var

b, g3, ETA0, ETA, ETA2, ETA3, NI, PSI,

Vzs, Vze, Vse, Vra, d, d1, d0, Rpa, Rba : Real;

Const

Pi=3.14159265358;

a=149597E3; e=0.01671; EPS=0.4090877; R=6378.1; Trg=365.256366; T=86164.1;

OMEGA=7.292115E-5; Co=3E5; ETA1=0.2295132; FI=0.882554825;

BEGIN write(‘Rpa=’); read(Rpa); NI:=NI*Pi/180;
write(‘NI=’); read(NI);

b:=sqrt(a*a-sqr(e*a));

g3:=e*(1+e*cos(NI))/(sin(NI)*(1-e*e));

ETA3:=arctan(-sqr(b/a)*(g3+cos(NI)/sin(NI)));

ETA2:=ABS(ETA3); ETA0:=arctan(b/(e*a));

if NI>0 then begin if NI<=Pi-ETA0 then PSI:=NI+ETA2; end;

if NI>Pi-ETA0 then begin if NI<Pi then PSI:=NI-ETA2; end;

if NI>Pi then begin if NI<=Pi+ETA0 then PSI:=-Pi+NI+ETA2; end;

if NI>Pi+ETA0 then begin if NI<2*Pi then PSI:=-Pi+NI-ETA2; end;

Vzs:=2*Pi*a(1+e*cos(NI))/(Trg*24*3600*sqrt(1-e*e)*sin(PSI));

Vra:=OMEGA*R*cos(FI);

if NI>0 then begin if NI<=Pi-ETA0 then ETA:=Pi-ETA2; end;

if NI>Pi-ETA0 then begin if NI<Pi then ETA:=Pi+ETA2; end;

if NI>Pi then begin if NI<2*Pi then ETA:=ETA3; end;

Vse:=0; d0:=1E-5;
REPEAT Vse:=Vse+d*1E-1;

d1:=(Vzs*sin(EPS)*cos(ETA+ETA1)+Vse*cos(EPS))/sqrt(Vzs*Vzs+Vse*Vse);
d:=ABS(2*Pi*Co*Co*Rpa/(Vra*T)-Pi*Vra/4-sqrt(Vzs*Vzs+Vse*Vse)*

cos(arctan(d1/sqrt(1-d1*d1))));

UNTIL d<d0; Vze:= sqrt(Vzs*Vzs+Vse*Vse);

write(‘ Vze=’,Vze); writeln;
write(‘ Vse=’,Vse); writeln; writeln;

END.

PROGRAM VzeVse calculates the absolute speeds of the Earth and the Sun.
In order to obtain the values of solution (1) or solution (2), the variables d1, d in REPEAT
should correspond to individual equations (3.84), (3.85) or (3.89), (3.90) respectively.
In REPEAT the equations (3.84) and (3.85) were included.
Table 16 contains results obtained from equations (3.84), (3.85) (calculated with method I ).
Table 17 contains results obtained from equations (3.89), (3.90) (calculated with method II).

86

RESULTS AND CONCLUSIONS

Michelson – Morley’s experiments and the values of interference fringe shifts calculated with
the mathematical model confirm the notion of both the existence of the aether and the
applicability of the Galilean transformation. The speed of light in an inertial system depends
upon the velocity of that system with respect to the aether. By observing shifts of
interference fringes, the absolute speed Vo of the interferometer can be determined. Hence it
is possible to build a speedometer which can measure the absolute speed of an inertial
system (of a spaceship, for example) with no need for the system be linked with any external
frame for reference.

Based on the calculation results, which can be found in the tables, the absolute speed of the
interferometer on the Earth’s surface was determined and expressed with respect to the
speed of light as follows:

104 ≤ V / C  1.5104 (1.124).

oo

Just as J. C. Maxwell had predicted, the speeds of the Earth, the Sun and our Galaxy centers
with respect to the aether were determined by measuring optical phenomena alone.

The values of the interference fringe shifts (see Tables 2-7) can be tested in a very simple
experiment. All that needs to be done is to place the Michelson’s interferometer in a
spaceship traveling at the absolute speed that is specified and linked to the speed of light by
the inequality:

Vo / Co  1.5104 i.e. the satellite of the planet Mercury.
If we consider a changeable mass of a particle (Chapter III), Newton’s second law of motion is
non-invariant with respect to the Galilean transformation, which effectively means that
Newton’s laws of mechanics are different in systems 1 and 2 if the variable mass of a
particle is considered. Hence the absolute speed of an inertial system can be determined with
the help of mechanical experiments performed inside that system (the spaceship).

In this work it was also shown that knowing the difference in times measured by atomic
clocks situated on the Earth’s surface, the absolute speeds of the Earth and the Sun can be
calculated. The elongation of the Earth’s sidereal day with respect to the time measured by
atomic clocks was evidenced as being merely apparent. The clock in system 2 runs slower
when compared to an identical clock in the preferred system 1. The lifetime of unstable
particles in system 2 is longer than the lifetime of identical particles in the preferred
system 1.

87

INDEX OF SYMBOLS

 The velocity of the light in a vacuum with respect to the aether,
C0 the speed of light in a vacuum with respect to the aether,
C0 the speed of light in the semi-transparent plate PP with respect to the
C
aether,
p the absolute velocity of the interferometer and the system 2 (O’EQW),

 the absolute speed of the interferometer and the system 2 (O’EQW),
V
the absolute speed of the interferometer, expressed with respect to
o
the speed of light C0 ,
V the wavelength of light in a vacuum,
the wavelength of light in the semi-transparent plate PP,
o
the refractive index for the semi-transparent plate PP with respect to
V V /C
a vacuum,
w oo angles at which rays of light leave slit S ,

0 0
p
n angles of the light rays refraction in a semi-transparent plate,

2 angle between the OXo and the OX axes and also the angle at which the
interferometer is situated with respect to its absolute velocity V0 ,
,  time intervals in which a ray of light reaches successively points A ,...,A
 ,
15
12
after leaving slit S ,

0
t a1 ,...,t a5
time intervals in which a ray of light reaches successively points B ,...,B
t ,...,t
15
b1 b5
after leaving slit S ,
a ,...,a
0
15
distances between contiguous points So , A ,...,A in the OXY system,
b ,...,b
15
15
distances between contiguous points So , B ,...,B in the OXY system,
a ,...,a
15
1u 5u
distances between contiguous points So , A ,...,A in the O’EQ system,
b ,...,b
15
1u 5u
distances between contiguous points So , B ,...,B in the O’EQ system,
e
15
a5
the coordinate of point A of the screen M reached by a ray of light after
e
5
b5
leaving slit S at angle  ,
Mo
0
e
the coordinate of point B of the screen M reached by a ray of light
o
5
R  l / 
wo after leaving slit S at angle  ,
0
k
R  l /  a point on the screen M (a fixed line in the telescope) in relation to
which the shift of interference fringes is calculated,
rw o the coordinate of the Mo point on the screen M in the O’EQ system.

K the relative difference of distances traveled by the rays of light reaching
r
V one point of screen M in the O’EQ system,
the value of interference fringes shift,
r the relative difference of distances traveled by the rays of light reaching

V distant points A , B of screen M in the O’EQ system,

r 55

the difference of relative differences of distances R ,

rw

the peripheral velocity of a point i.e. a place on the Earth’s surface

where the interferometer is located .
the peripheral speed of a point i.e. a place on the Earth’s surface

where the interferometer is located .

88

 the velocity at which the Earth’s center travels around the Sun,
V the speed at which the Earth’s center travels around the Sun.

zs the velocity at which the Earth’s center travels with respect to the aether,
the speed at which the Earth’s center travels with respect to the aether,
V the velocity at which the Sun’s center travels with respect to the aether,
zs the speed at which the Sun’s center travels with respect to the aether,
Vze
Vze the velocity at which the Sun’s center travels around the center of our Galaxy,
V the speed at which the Sun’s center travels around the center of our Galaxy,

se the velocity at which the center of our Galaxy moves with respect to the

V aether,
se the speed at which the center of our Galaxy moves with respect to the aether,
V
Northern point of the horizon,
sg Southern point of the horizon,
The North Pole,
V
sg The South Pole,
V
the line of intersection between the horizon plane and the meridian plane
ge which run through the point U (, ) ,
the angular speed of the Earth’s rotation,
V inclination of the ecliptic to the celestial equator,
annual precession within ecliptic (in longitude),
ge
true anomaly,
N a radius vector,
S an average Earth-Sun distance,
P a small semi-axis of the Earth’s orbit,
stellar year,
N
tropical year,
P the duration of astronomical winter,

S the absolute velocities of the interferometer in the horizontal system.

N S line  
V  V or V  V ,
 o 01 o 02

p right ascension of the Sun,

r 
a right ascension of the V velocity,
b
T se 
right ascension of the V  V velocity,
rg
se1 se
T right ascension of the V velocity,

rz zs 
Greenwich Hour Angle of velocity V ,
T
 z zs
V ,V
Greenwich Hour Angle of the Sun
01 02
Greenwich Hour Angle of the Aries point,
 Local Hour Angle of velocity V ,
s
zs
 Local Hour Angle of velocity V ,
se
 se 
 Local Hour Angle of velocity V  V ,
se1
 se1 se
 declination of velocity V ,
zs
 zs
GHA declination of velocity V ,

zs se

GHAs
GHAaries

LHA

zs

LHA

se

LHA

se1


zs


se

89


 declination of velocity V  V ,
se1  se1 se

H altitude of velocity V ,
zs
zs altitude of velocity V ,
se 
H altitude of velocity V  V ,
 se1 se
se
altitude of velocity V ,
H 01
altitude of velocity V ,
se1 02
azimuth of velocity V ,
H zs
azimuth of velocity V ,
01 se 
azimuth of velocity V  V ,
H  se1 se
azimuth of velocity V ,
02 01
azimuth of velocity V ,
A 02

zs point U of geographical coordinates ,  in which the interferometer

A is situated,

se rest mass of particle in systems 1 and 2 respectively (Fig.13),

A mass of a particle in motion in systems 1 and 2,

se1 forces acting on a particle in systems 1 and 2,

A particle’s velocity in systems 1 and 2,

01 average life time of unstable particles in system 1 and 2,
frequency of atom vibrations in systems 1 and 2,
A
angular speed of the Earth’s rotation in systems 1 and 2,
02
Earth’s moment of inertia in systems 1 and 2,
U (, )
times measured by atomic clocks in systems 1 and 2,
m ,m
Earth’s sidereal days in systems 1 and 2,
01 02
time measured by the atomic clock in system 2 at   T ,
m ,m 11
1  2 difference of the times T   ,
F ,F 2 2(T1)
1 2
V ,V atomic clocks situated along an Earth’s parallel,

12 an atomic clock situated at the South Pole,

 , the speeds of the ZA , ZA clocks situated on the parallel’s plane,
12
ab
 ,
A1 A2 the absolute speeds of the ZAa , ZAb clocks,
 ,
12 the absolute speed of the ZA clock,
J ,J
12 p

 ,  times measured by the ZAa , ZAb clocks situated on the Earth’s parallel,
12 time measured by the ZA clock situated at the South Pole,
T ,T
12 p

2(T1) the absolute value of the difference in times measured by
R
T the ZAa , ZAp clocks during half-a-sidereal day since the time of their
synchronization.
ZA , ZA the absolute value of the difference in times measured by

ab

ZA

p

V ,V

ra rb

V ,V

0ra 0rb

V

0p

 , 
2ra 2rb

2p
R
pa(T / 2)

R

ba(T / 2)

the ZA , ZA clocks during half-a-sidereal day since the time of their

ab

synchronization.

90

LITERATURE

[1] Wróblewski A.K. , Zakrzewski J.A. : Wstęp do fizyki. T.I . [Introduction to Physics.
Vol.1] PWN Warszawa 1984.

[2] Holliday D. , Resnick R. : Fizyka. T.II . Wydanie VII. [Physics. Vol.2. 7th Ed.] PWN
Warszawa 1984.

[3] Katz R. : Wstęp do szczególnej teorii względności.[Oryg.: An Introduction
to the Special Theory of Relativity]. PWN Zakłady Graficzne w Poznaniu 1967.

[4] Frisz S., Timoriewa A.: Kurs fizyki. T.III. [A Course in Physics. Vol.3] PWN
Warszawa 1959.

[5] Fizyka . Ilustrowana encyklopedia dla wszystkich .[Physics. A Picture Encyclopedia]
WN-T, Warszawa 1991.

[6] Rybka E.: Astronomia Ogólna. Wydanie VII. [General Astronomy. 7th Ed.] PWN
Warszawa 1983.

[7] Kreiner J.M.: Astronomia z Astrofizyką.[Astronomy and Astrophysics] PWN
Warszawa 1988.

[8] Siłka S., Skoczeń S.: Astronawigacja Żeglarska.[Nautical Astronavigation].
Wydawnictwo Sport i Turystyka, Warszawa 1982.

91