Stellar Distance and Velocity

In this paper, a method is presented by which it is possible to determine a distance 
between the sun and a star as well as a velocity at which the star moves 
relative to the sun. In order to achieve this, it is sufficient to know three positions of the star and the unit vectors 
determined by the star and three arbitrarily chosen points that do not lie on a 
single line. The method has been tested using the data generated by a computer 
program as well as real data obtained by Gaia mission. In the first case, we 
found the huge differences comparing the results derived by the method to the 
results calculated by the traditional parallax method. In the second case also, 
there are large differences between the obtained and the expected results, but 
primarily because of the form of the input data, that is not fully suited to 
the proposed method. Under certain conditions, one would be able to find a 
velocity at which the sun is moving regarding a stationary coordinate system (K) that will be defined later on.


Introduction
Stellar parallax is defined as an apparent change in position of a star against the background of more distant objects, due to the movement of the earth revolving around the sun. In order to calculate a distance to the star, it is enough to know a distance from the earth to the sun that serves as a base line and the parallax angle that is obtained by two the measurements that have been made six months apart. There are some shortcomings of this method. Firstly, the base line is fixed thus the angles measured are always extremely small. Secondly, the base line is directly affected by the movement of the sun but it is not taken into the consideration. Thirdly, during the time of six months, a star is moving and changing its position which also affects a parallax angle. In some cases, a parallax has a negative value. It is believed that it may arise when the true parallax is smaller than How to cite this paper: Čojanović, M. , , x y z a a a   =   a (10) 1 = a (11) and vice versa if vector , , x y z a a a   =   a is known then, we can easily find its Spherical coordinates.

The Heliocentric-Ecliptic Coordinate System
Let the ( ) , , S x y z represents "The Heliocentric-Ecliptic Coordinate System" (Figure 2). Its origin S is at the center of the sun and the fundamental plane ( ) , S x y coincides with the "ecliptic", plane of the Earth's revolution about the sun. On the first day of Spring a line joining the center of the Earth and the center of the sun points in the direction of positive x-axis. This is called a vernal equinox direction.
The use of The Heliocentric-Ecliptic Coordinate System is obsolete, but in present paper we will use it for a better explanation of the proposed method.
Direction 0 SX represents Vernal equinox, and 1 t the time needed the Earth to move from the point 0 X to the point A. We can say that ( Figure 2) depicts classical explanation how Earth revolves about the sun. Now we will present a different interpretation of the same event. A stationary coordinate system marked as well by (K) is joined to this referential frame. The coordinate system (K) is not moving but rather remains fixed with respect to distant objects while the sun is moving by some velocity v ( Figure 3) regarding the coordinate system (K).
In astronomy, an epoch is a moment in time used as a reference point, so we need to specify a certain time T 0 (TT-Terrestrial Time), which we are using as a reference. In other words an epoch is a moment for when a given position of an astronomical object is valid.    , , , , where 1 T (TT time) is expressed in seconds Thus we have got the coordinates for the center of the Earth (marked by A) regarding the (K). Assuming that the time 1 t is known from Equations (29)-(31) it follows that the coordinates of point A can be expressed as a function of the velocity v . For now v has been considered as a varible.
The astrometric processing uses a coordinate system known as the Barycentric Coordinate Reference System. It has its origin at the solar-system barycentre. Its axes are non-rotating with respect to objects at cosmological distances and coincide with those of the International Celestial Reference Frame.
The positions and proper motions of non-solar system objects derived from Gaia Data Release 1 observations are given in a reference frame that is aligned with the International Celestial Reference Frame (ICRF) [1]. ICRF coordinates are approximately the same as equatorial coordinates.

Coordinate Transformations
The basis vectors in The transformation between the equatorial and galactic systems is given by: The transformation between the galactic and ecliptic systems is given by: Of course, this can not be taken as a fully accurate value.

Determining a Distance to a Star and Its Velocity regarding the Coordinate System (K)
We assume that a star is moving with a uniform, rectilinear space motion u relative to the referential frame (K). At the some momemt T (that will be derived later on) a signal has been emitted from the star ( Figure 4). Postion of the star is marked by A′ . At the instant t (Terrestrial Time) the signal has been recevied at the point A on the Earth. Now we measure right ascension ( α ) and declination ( δ ) of the star regarding the equatorial coordinate system and transform them into the ecliptic longitude and ecliptic latitude ( ) 1 1 , λ β that can be also expressed as a unit vector , , x y y a a a   =   a regarding the coordinate system (K).
In this way the direction in which the star lies has been determined. In order to find a position (in pollar coordinates) of the star at the moment t we need to determine a distance between points A and A′ .
Analogously we have the similiar situation with the pairs of points ( )   , , , , cos cos , , cos cos The points , A B ′ ′ and C′ are collinear therefore we can write a following expression.
where k is for now an unknown coefficient.
, , , , , , , , In order to calculate distances 1 2 3 , , d d d we have to determine a coefficient k. Let 1 t denotes the time when signal is received at the point A. If 1 T denotes the time when the signal was being emitted from the star at the point A′ we will have following equation where c denotes the speed of light in the reference frame K. Analogously for the points B and B′ we will have If 3 T denotes the time when the signal was being emitted from the star at the point C′ and 3 t denoted the time when the signal is received at the point C we will have following equation If u denotes magnitude of the velocity u then from Equation (78) it follows Combining equations (102), (105) and (110) it follows that And finally the Equation (115) can be written in the following form , , The coefficient k is eliminated from the equations, thus the distances ( ) 1 1 , , , , , , We will see later how the change in the value of velocity v affects the distances 1 2 3 , , d d d .
We assume that the coordinates of the points , A B ′ ′ and C′ and corresponding distances 1 2 , d d and 3 d are known thus we are able to determine a velocity u of the star regarding the frame (K). We have got the following equations.

The Case When v = 0
In this section it will be considered a case when v is unknown. Unlike the unit vectors , , a b c that are obtained by measurements and remain unchanged, vector v will be substituted by 0 and u by ∆u . Therefore the formulas given in the [Section 5] get the new forms.
(155) Journal of Applied Mathematics and Physics ...Finally we have got ( ) ...and In that way we are able to determine approximate values for distances 1 2 3 , , d d d and star velocity ∆u regarding the sun.
In the special case, instead of a star, we can observe the barycenter of the Galaxy. Then ∆u denotes the velocity at which the barycenter of the Galaxy "moves" relative to the sun. In other words g = −∆ v u denotes the velocity at which the Sun is moving about the barycenter of the Galaxy.

Determining a Position of a Star
In this section it will be explained how to find a position of the star at some in-  Therefore we get following equations The Equation (187) can be written in the following short form and solved by an unknown There are two roots 1 x and 2 x , but since 2 0 d > it follows that , , Now we will transform the unit vector , , x y z a a a   =   a from ecliptic to equatorial system. Transformed unit vector is marked by _ a eq . (198) [ ] _ _ 1,1 x a eq = a eq (199) [ ] _ _ 1, 2 y a eq = a eq (200) [ ] _ _ 1, 3 z a eq = a eq (201) By transformation from a right-angle coordinate system into the spherical coordinates we obtain the following equations The same procedure will be repeated for unit vector , , (204) [ ] Referring to ( Figure 6) we get

The Second Part
In this section are given descriptions of four programs written in Maxima [4].
The purpose of their writing is testing the results obtained in the previous chapters. Instead of using the real measurements we will use data that are generated by a computer program. These data are presented in spherical coordinates ( ) , λ β or as a corresponding unit vector , , x y z a a a   =   a regarding the coordinate system (K). We assume that velocities v and u as well a distance 1 d between the star and Earth at some instant t are known.
Since the distances between the sun and stars are expressed by large numbers, 64-bits floating-point format which gives from 15 to 17 significant decimal digits is not sufficient to make enough precise arithmetic operations. Therefore, for correct and precise testing, a quad precision (128-bit or 34-digit) is required.

Stellar parallax, proper motion and velocity
If the vectors , , are respectively corresponding unit vectors at points B and C [ Figure 4] it is possible to calculate stellar parallax and its distance from the Earth using well known formulas. The time between two measurements is equal to six months.
AU is as usual defined as astronomical unit. These are field descriptions from the Table 1 and Table 2.

arccos
, ,   T1 with the first row of the table T2, we can conclude that the angular motion of the star in a period of one year is roughly twice as large as its angular motion in six months. Comparing the second row of the Table T1 with the the second row of the Table T2, we can conclude that the angular motion of a star over a period of six months is greater than the one in one year. So in this case it would appear that the star is moving zigzag.
Motion toward or away from the Sun called radial velocity is determined by using the Doppler Effect. Motion perpendicular to the direction to the Sun is called tangential velocity. It is accepted that transverse velocity T V is given by a following formula: where distance is noted by d, proper motion is noted by µ and the factor k comes from the unit conversion.
We are going to test the correctness of this formula.
Let the vector v represents the velocity of the sun and the vector u represents the velocity of the star regarding the (K). Relative velocity , , x y z u u u   ∆ = ∆ ∆ ∆   u of the star regarding the sun is given by following equations.
Let us, regarding the coordinate system (K), define three unit vectors. The first vector noted by _ z radial is directed to the star. Therefore, its spherical coordinates are ( ) , λ β , where λ represents ecliptic longitude and β represents ecliptic latitude. The second one marked by _ x pmlong is determined by star proper motion in longitude direction. Its spherical coordinates are ( ) π 2,0 λ + . And the third one marked by _ y pmlat will be determined by star proper motion in latitude direction. Its spherical coordinates are ( ) cos π 2 cos 0 , cos 0 sin π 2 ,sin 0 λ λ  (237) In this way, a new coordinate system noted by (K') has been defined. Its axes are determined by unit vectors [ ] _ , _ , _ x pmlong y pmlat z radial and its origin is Journal of Applied Mathematics and Physics at the center of the sun. We can find the scalar projections of the vector ∆u onto the unit vectors _ , _ z radial x pmlong and _ y pmlat which is the same as to transform ∆u from the coordinate system (K) to the coordinate system (K').
Let define AZ a transformation matrix from coordinate system (K') to the coordinate system (K) by the following equation.
cos π 2 cos 0 cos 0 sin π 2 sin 0 cos cos π 2 cos π 2 sin sin π 2 cos cos cos sin sin By transforming the velocity ∆u from the coordinate system (K) to the coordinate system (K'), we obtain the following equation where distance between the sun and a star is noted by d and If the formula given by the Equation (230) is valid then we should have that From the two examples shown in Table 3, we can see that the conditions given by (246) and (247) are met.
Matrix AZ is an orthogonal matrix. As a linear transformation, an orthogonal matrix preserves the dot product of vectors. In other words orthogonal transformations preserve lengths of vectors and angles between them. Let . PM LONG Table 3. Transverse velocity of a star determined in two ways. [ ] In this way we have calculated the velocity of the star regarding the sun in two different ways.
The first method is given by the Equations (166)-(168), and the second one by If ′ ∆ ≈ ∆ u u then we can conclude that measurements and proposed methodology are correct. Please refer to the attached program "parallax0.wxmx" for further details, investigation and testing.
Distance and relative velocity of a star if we assume that the sun is stationary regarding the frame (K) In this case we assume that 0 = v and star velocity u is substituted by its relative velocity ∆u regarding the sun.
Let suppose that we observe a star which has a following spherical coordinate regarding the    Comparing the first row of the Table 5 with the second and third one and Comparing the second and the third row with the fourth row of the Table 5 we can conclude that increasing the interval between two measurements increases the accuracy in determining the distance d. This is completely contrary to the situation in the previous section. We can say that the errors in this case are negligible and that they are due to rounding up the numbers into 64-bit format.
Please refer to the attached program "distance_velocity_v.wxmx" for further explanations and data testing.

Discussion
Based on what we have found so far, we can conclude that assuming that 0 = v , the obtained results are far more reliable than the data obtained by the "parallax" method. Problems arise when we increase the interval between two measurements. Increasing this interval increases the error in distance calculation.
Method in which the velocity v is known gives the best results. In addition, by increasing the interval between the two measurements, the angles between the unit vectors , a b and c are increasing which improves the accuracy of the calculations.
Only in cases when the velocity v is known and the measurements have been carried out over a sufficiently long period of time we can obtain reliable data on the distance d and velocity u .

Testing the Proposed Method Using Data Obtained from
Gaia Catalogs DR1 and DR2 The reference system for the source catalogs is the barycentric celestial reference system (BCRS/ICRS). Observations are more naturally expressed in the centre-of-masses reference system (CoMRS) which is defined from the BCRS by special relativistic coordinate transformations. This system moves with the Gaia spacecraft and is defined to be kinematically non-rotating with respect to the BCRS/ICRS [5].
In order to successfully apply the described method, we have to know the unit vectors and the positions of the detector. The exact location of the satellite will be substituted by the center of the earth and the unit vectors will be derived from the data given in the Gaia's catalogs [6] (option "Search"). We should also keep in mind that we need spherical coordinates of a star regarding the coordinate system with its origin at the satellite instead of the sun barycenter.
From the Gaia catalogs we can directly get the following data We will assume that the measurement were taken regarding the Equatorial coordinate system with its origin at the center of the Earth.
In this way, we obtained all the parameters needed to calculate a star relative velocity regarding the sun and the star distance in two different ways.  In order to improve the results we should know the exact coordinates of the the points A, B and C actually the locations of the satellite at the moments when the measurements have been made. The coordinate systems and transformation matrices should be replaced by the appropriate ones. The reference system for the source catalog is the Barycentric Celestial Reference System (BCRS/ICRS), but for the proposed method it is necessary that the Gaia observations are expressed in the Centre-of-Mass Reference System (CoMRS), a system that moves with the Gaia spacecraft.
Therefore, for all these reasons, the results shown in (Table 6) and (Table 7) must be taken with great caution.
Please refer to the attached program "gaia0.wxmx" for further explanations and data testing. In this program is also given a code for extracting data from Gaia's databases.

Determining a Velocity v regarding the (K)
It has already been mentioned why, in the case of very distant objects, it is important that the velocity v is known. One of the methods for determining the velocity v is given in the [7], where the problem of the possible relation between absolute velocity and stellar aberration has been discussed. Journal of Applied Mathematics and Physics If an inequality (268) does not hold then this approach to finding the velocity of the solar system regarding the Galactic barycenter (or a velocity v regarding the stationary frame (K)) did not yield the expected results and therefore it must be rejected.
The same procedure can be repeated, but this time the extra-galactic objects should be observed.

Conclusion
In this paper, we have developed and tested a method by which it is possible to find the distance d between the sun and the arbitrary cosmic object as well as the velocity ∆u at which this object moves relative to the sun. The advantage of this method in comparison to the traditional parallax method is that it can be applied to very distant objects, provided that the measurements are performed over a long period of time and that the velocity at which the solar system moves relative to the coordinate system (K) is known. In addition to distance d and velocity ∆u , it is also possible to derive values for proper motion, parallax and radial velocity and compare them with values obtained by direct measurements.
In this way, we are able to assess to what extent the results obtained by this method are correct.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.