Stellar distance and velocity (III)

The use of parallax angles is one of the standard methods for determining stellar distance. The problem that arises in using this method is how to measure that angle. In order for the measurement to be correct, it is necessary for the object we are observing to be stationary in relation to the sun. This is generally not true. One way to overcome this problem is to observe the object from two different places at the same time. This would be technically possible but will probably never be realized. Another way to determine the distance is given in [1]. With certain assumptions, this is a mathematically completely correct method. After the publication of the third Gaia’s catalog [2], we are now able to test the proposed method using real data. Unfortunately, for the majority of stars it is not possible to obtain the distance directly, but with the help of some additional measurements we would be able to indirectly determine the distance of such stars.


Determining the stellar distance and velocity
Suppose that the observed star Z is moving with a uniform, rectilinear space motion regarding the sun Fig. 1. Let us denote by τ 1 the time when the signal was sent from the point noted by Z 1 and by t 1 the time when the signal is registered at point noted by A. We assume that τ 1 and t 1 are expressed in the same units of time. The unit vector of the direction AZ 1 is denoted byâ. In an analogous way, we will define triples (τ 2 , t 2 ,b) and (τ 3 , t 3 ,ĉ) for pairs of points (B, Z 2 ) and (C, Z 3 ), respectively. • C(t 3 ) abĉ Figure 1: Star Z moves uniformly regarding the sun. There is one nontrivial solution.
Coordinate system (K) is heliocentric ecliptic coordinate system. Coordinate axes are determined in accordance with the ICRS standard. This is a list of constants that will be used in the calculations: We will now define the times when the measurements were made, and the time t 0 that will be considered as the initial time. The spherical coordinates (lon2015, lat2015), (lon2015.5, lat2015.5) and (lon2016, lat2016) are given in the Gaia's catalogs: lon2015 -Ecliptic longitude of the source in ICRS at the reference epoch J2015.0 lat2015 -Ecliptic latitude of the source in ICRS at the reference epoch J2015.0 lon2015.5 -Ecliptic longitude of the source in ICRS at the reference epoch J2015.5 lat2015.5 -Ecliptic latitude of the source in ICRS at the reference epoch J2015.5 lon2016 -Ecliptic longitude of the source in ICRS at the reference epoch J2016.0 lat2016 -Ecliptic latitude of the source in ICRS at the reference epoch J2016.0 We will now transform the spherical coordinates into Cartesian coordinates.
c y = cos(lat2016) * sin(lon2016) (16) The origin of the (K) coordinate system is at the barycenter of the solar system, therefore the velocity v at which the solar system moves relative to the (K) is equal to zero [1].
Let the angles α, β and γ be defined as follows: We will now determine the coordinates of points A(t 1 ), B(t 2 ), and C(t 3 ) , which indicate the positions of the observer at the time the measurements were made Fig. 1.
Let define the time intervals ∆t 1 and ∆t 2 as follows: One can define the matrices M and N : With τ 1 , τ 2 and τ 3 we denote the times when the signals have been sent Fig. (1). Of course, these times mean nothing to us, but we can define the times ∆τ 1 , ∆τ 2 as it follows: The time on the star Z may be slower or faster than the time on Earth, but we assume that the stellar time ∆τ i is converted to terrestrial time.
Coefficient k is defined in the following way.
It has been proved in [1] that : Let us define matrices D, D 1 , D 2 , D 3 in following way: Assuming that det(D) = 0 , we found [1] that the distances d 1 , d 2 , d 3 are given by the following equations : The triple (d 1 , d 2 , d 3 ) represents a unique solution. Therefore, the collinear points Z 1 , Z 2 and Z 3 are uniquely determined. If there were some other three collinear points Z 1 , Z 2 and Z 3 which would correspond to the three positions of the observed object, then we would have two different solutions, which is contrary to the fact that there is only one triple as a solution. In this way we proved that: d 1 -denotes distance between the Earth (satellite) and a star at the time (t 1 ) d 2 -denotes distance between the Earth (satellite) and a star at the time (t 2 ) d 3 -denotes distance between the Earth (satellite) and a star at the time (t 3 ) For brevity in writing this method will be denoted by (3P ).
Let us define the arithmetic mean noted by d , standard deviation noted by σ and the coefficient of variation (CV ) noted by c v as follows: The distances d 1 , d 2 and d 3 are defined in different ways but we will assume that the differences between them are relatively very "small" compared to d.
In fact Equation (56 ) should be written in the following form: Where total proper motion is noted by P M while ∆t 2 and ∆τ 2 are defined by equations (39) and (43). Time interval ∆τ 2 can be measured using Doppler effect [4]. The difference between the times ∆τ i and ∆t i is due for two reasons, first the speed of light is finite and second the distance d is constantly changing. For example if the observed object moves away then ∆t 2 > ∆τ 2 .

Analysis of the obtained results
Depending on whether it is possible to determine the distance d and with what precision, all observed cosmic objects can be divided into three groups: where ε represents some small number and In this case we will say that the star Z moves uniformly at a distance d from the Earth Fig. (1). The accuracy in determining the distance d is significantly greater when points A, B and C form a triangle instead of lying on one line. This means that the intervals between the two measurements should be four (sixteen,twenty-eight..) instead of six months. This case is presented in Table 1.
where ε represents some small number and Figure 2: The star moves along a curve that is close to a straight line. All distances d i are positive, but we are still unable to determine the distance d In this case we will say the measurements are not accurate enough or that the star does not move uniformly with respect to the sun Fig. (2). The time difference between the two measurements is six months. If we were to reduce this time to 2-4 months, then in some cases the trajectory along which the observing star moves would be closer to a straight line. This means that the accuracy in determining the distance d would eventually increase. This case is presented in Table 2.   If at least one of d i is less than zero it means that d does not represent the distance between the earth and the star Z. The points S 1 , S 2 and S 3 lie on one line and since there is one solution it means that the points Z 1 , Z 2 and Z 3 are not collinear but lie on some curve Fig. (3). As indicated in Table 3, the distance between the observer and the star is not possible to determine.  Table 3: Some of distances d i are less than zero. The star moves along the curve and it is not possible to determine distant d Therefore it could be concluded, in order to obtain optimal results in determining the distance d, the time interval between two measurements should not be fixed, but chosen according to which of the three groups the observed object belongs to.

Comparison between the two methods
After testing 320, 000 randomly selected stars, assuming that the c v was equal to 0.001, for only 321 stars we were able to determine the distance d. In percentage it is about 0.1%. We have considered only those cases where ra error <  Table 4: Distances obtained by the (3P) method and the standard parallax method. Table 4 shows only a few examples but also in all other cases there are significant differences between the two methods in determining the distance d.

Determining the distance d for stars with negative parallax
In the (3P) method, the parallax angle does not play any role, so it is completely irrelevant whether the parallax is positive or not.  Using the proposed method, as shown in Table 5, it is easy to find the distance d for those stars for which this would not be possible if we used the standard parallax method.

The distance for stars whose parallax is greater than 10 [mas]
Just for comparison between the two methods, the distances of several stars, whose parallax is greater than 10, are shown in Table 6 Table 6: Distances for stars whose parallax is greater than 10 [mas] The distance of the star marked with #8 is far greater than the distance of the star marked with #7, although the parallax angle of the star #8 is greater than the parallax angle of the star marked with #7. After we selected all the objects so that abs(l − 180) < 1 and among them we chose the three that have the greatest distances we got the final result shown in Table 7  The same procedure was repeated assuming that ((l < 0.5) ∨ (360 − l) < 0.5). The final result is shown in Table 8.

How big is the Milky Way galaxy
If we assume that GA = GB then it follows that: The galaxy does not have a perfectly symmetrical shape, so this result result should be taken with caution.
Tables 9 and 10 were defined in a similar way, taking into account those objects so that (90 − b < 1) and (b + 90 < 1), respectively.   Obviously, for objects #2 in Table 9 and #1 and #2 in Table 10, the distances are extremely large, which is one of the reasons why the transverse velocities are extremely high. We can consider two possibilities. First, if they were Galactic objects then it is most likely a measurement error. If they were extragalactic objects then again there are two possibilities. Firstly, it could be a random measurement error. Another possibility is that due to the distortion of the space around the Galaxy, the line connecting the object and the observer is not a straight line but a curve. In this case the proposed algorithm cannot be applied.

Star constellations distance
For some stars from a stellar constellation it is not possible to determine the distance. Therefore, it is necessary to find those stars for which it is possible to find the distance. In this way, we can indirectly determine the distance of other stars if we are able to prove that these stars move in the orbit of some of the stars whose distance has already been determined. Table 11 shows one such simple example. This is just an example and it has not been proved that stars marked with #2, #3 and #4 move in the orbit of the star marked with #1.  Things get much more complicated because generally each constellation is made of several hundred (thousand) stars. But the principle should remain the same. It is necessary to find those stars for which it is possible to determine the distance and then the others that move in their orbits. The star from the first group could be called a "mother star" while the star from the second group could be called a "daughter star". We can say that it is easy to determine whether a star belongs to the mother-star group, but the question is how to find its mother-star for a daughter-star. We will not deal with this problem in this paper, because it requires much more observation.