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Optics Effect on the Observed Image about Sgr A* and Orbits of S-Stars

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Yin Zhu  *

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25 July 2024

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26 July 2024

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Abstract
Light is dispersive in gravitational field. It results in that, under the condition of strong gravitational field, the observed numbers, size, distance and spectrogram of a celestial object is different from the real ones. Therefore, the observed jets of the Sgr A* and M87 need be understood further with the orbit of the stars around them. And, the observed orbits of the S-stars around the Sgr A* is different from the real ones. It implies that the observed optical images of the distant celestial body with strong gravitational field need be further understood with other observations and physics laws to accurately and precisely know the real physical configuration of it.
Keywords: 
Subject: Physical Sciences  -   Astronomy and Astrophysics

1. Introduction

The center of the Milky Way is the most unusual place in all of the galaxy. Now, it was observed that there is a black hole, the Sgr A*, at the center of our Milky Way [1,2] and the M87 black hole at the center of the galaxy M87.[3,4] And, the images,[1,2,3,4,5,6] the jet,[5,6,7,8,9,10,11,12,13] the mass[1,2,3] and the size of the two black holes were observed.[4,5,6] And, some of the S-stars orbiting around the SgrA* in the radius of R ≤ 0.5 pc were observed.[14,15,16,17,18,19,20,21]
But, first, new observations, including the fastest stars,[14,15,16,17,18,19,20,21] the circular velocity curve of the Milky Way[22,23,24,25,26,27] and the fast galaxy bar,[28,29,30,31] could show that the traditional galaxy rotation curve, which was originally presented by Babcock, Oort and Rubin from 1939-1980,[32,33] is questioned. And, the traditional galaxy rotation curve was questioned in pure theory.[34,35] We know, the mass of the Sgr A* is related with the traditional galaxy rotation curve. Therefore, the mass of the Sgr A* need be reconsidered with the new observations.[36]
Second, it was well observed that there is a time lag between the light curves with different wavelengths.[37,38] It was presented that the distance of extragalactic system can be measured with “the time lag between variations in the short wavelength and long wavelength light from an active galactic nucleus”.[39] And, it was presented that to have a right measurement of the numbers and spectrogram of the celestial objects and to have an accurate measurement of the size and distance of a celestial object, light dispersion in gravitational field need be considered.[40]
Therefore, in this work, it is presented that, the observation about the Sgr A* is a complex and complicated project. The new observations[14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] are critical to understand Sgr A*. To well understand the Sgr A*, the observed orbits of the S-stars around the Sgr A* and the observed jet of the Sgr A* and M87 need be considered with that there is the time lag between the light curves with different wavelengths[37,38,39] and with that light is dispersive in gravitational field.[40]

2. The Optical Image and the Real Physical Configuration

It is well known that, for a celestial body with a very large distance from us, because a time is needed for a light ray running from the body to us, the body at its present position cannot be observed. We only can observe the body at the retarded time and retarded position. And,
t’ =t−L/c
where L is the distance between the body and the observer, c is the speed of light, t is the present time, t’ is the retarded time. For example, the Sun that we can observe is that at the time almost 8.5 min ago, while it at present time we cannot observe because a time almost 8.5 min is needed for the light running from the Sun to us.
From Eq.(1), the multi retarded time was formulated for the observation that there are time lags between the light curves with different wavelengths:[40]
tλi = t − L/cλi, cλi = c/nλi
where λi refers the ith wavelength of the light, nλi is the refractive index of the light with the wavelength of λi.
It is noted that, correspondent to that there are i retarded times, i images can be produced for one single star. The i images can be connected with or separated from each other as discussed in [40]. Here, the multi retarded time with the multi-images is used to study the observation about the black holes and the orbits around the black holes at the center of a galaxy.

2.1. The Observed Jet of the Sgr A* or M87

The jet of the M87 and Sgr A* was observed.[4,5,6,7,8,9,10,11,12,13] It is important, the time lags between light curves of the Sgr A* was observed: the flux density of the jet is a function of the wavelength of the light.[13] In [40], it was demonstrated that two different kinds of images with different wavelengths can be produced from one single celestial body with super strong gravitational field. If there are several light curves with several wavelengths for a moving body, from Eq.(2), an image as shown in Figure 1 can be produced for the reason as that for the Figure 2B in [40].
The image of the jet of the M87 was observed. It is highly similar to the Figure 1.[4,9] Therefore, a question arise: how to know the observed image of the jet of M87 is a real jet? Or, it is only an optical effect as that shown in the Figure 1?
We know, the jet of the Sgr A* was observed.[5,10,11,12,13] While, the orbits of the S-stars around the Sgr A* was observed.[14,15,16,17,18,19,20,21]If there is a jet extending for 200au and with a big energy emitted from the SgrA*,the orbits of the S-stars should be affected by this jet for that, it is usually thought, the distance between the Sgr A* and many of the S-stars is less than 200au.[14,15,16,17,18,19,20,21] But, till now the effect of the jet of the Sgr A* on the orbits of the S-stars has not been observed. Therefore, there are two different possibilities: first, there should not be a jet of the Sgr A*. The observed image of the jet is only an optical effect. Second, the energy of the jet should be so little that cannot make the orbits varied. Therefore, further observation is needed to right understand the image of the jet.
For the same reason, the jet of the M87 also need be understood with further observation. It is thought that the energy of the jet of M87 is very large and is with a length of 5000 light year.[41] If it is so, then, first, the orbits of the stars around the M87 in the radius of r < 5000 lightyear shall be broken off. Second, as a big mass is limited in almost a cylinder with the radius of R ≈ 500au and the length of l ≈ 5000 lightyear, the orbits of the stars around the mass should be different from that around a sphere. However, no such a kind of orbit has been observed.
Therefore, from the orbits of the S-stars around the Sgr A* and the orbits around the M87, it could be concluded that, in a high probability, the jets of the M87 and Sgr A* is an optical effect, rather than a real one. If there are the real jets, these jets should be with a very little energy that cannot affect the orbits of the stars around the two black holes.

The Superluminal and Accelerating Jet

It was observed that the measured velocity profile of the jet of M87 is not described as a single streamline but rather explained by multiple ones. And, the jet can be accelerated.[41,42,43,44,45,46]. For example, the apparent jet speeds generally increase from ≈0.3c at ≈0.5mas from the core to ≈2.7c at ≈20 mas, and the jet moves at relativistic apparent speeds up to ≈5.8c at distances ≈200–410 mas. It appears that the jet is accelerated from subluminal to superluminal speeds.[43] Now, the observations is difficult to understood. But, as the Figure 1 in this work is considered, the observed jet should be only an image. From [40] we know, the images of one star at different distance from the star can be observed at a same time. It appears that the velocity of the observed image is instantaneous. And, the image of the black hole in the Figure 1 is formed by several analogous images connected with each other.
There is another question about the superluminal jet:[41,42,43,44,45,46] if the jet is with a large energy which is made up of the hadrons particles and is with a very large velocity, this jet could have a very large momentum on the M87. It should make the M87 moved along the converse direction of the jet. But, this effect has not been observed. Therefore, it also could be concluded that the observed jet may be an image, or the jet is with a very little energy.

2.2. The Observed Image of the Orbits of the S-Stars around the Sgr A*

We know, the image of Sgr A*[5,6] with its jet[10,11,12,13] was directly observed. Therefore, the orbits of the S-stars around the Sgr A* can be directly observed. As the Sgr A* is moving along the blue arrow and the S-star is orbiting around the Sgr A*, if the speed of the light emitted from Sgr A* and S-star are csgrA = c/nsgrA and cstar = c/nstar, respectively, the observed image is that as shown in Figure 2. The time for the light running from the Sgr A* to the observer is t − tsgrA = L/csgrA while for that from S-star to the same observer is t − tstar = L/cstar. From the Shapiro time delay we know, the gravitational field is stronger, the speed of light should be slower. Thus, there is csgrA < cstar and tsgrA > tstar. It should result in that, under the condition that the Sgr A* is at the center of the circular orbit for the real orbit, while the observed orbit is an ellipse and the Sgr A is not at the center. And, the observe speed of the orbit is larger than the real one.
The speed of light in a gravitational field is determined mainly with two factors: 1) The strength of the gravitational field. From the Shapiro time delay we know, the field is stronger, the speed of light is slower. 2) The wavelength of the light. It was well observed that there is time lag between the light curves with different wavelengths.[37,38,39,40] The two factors result in that the observed image is radically different from the real physical configuration of the orbit of a star around a center mass. Besides the cases in the Figure 1 and Figure 2, there are these cases: 1) As the velocity of the orbit of the S-star is vertical to the velocity of Sgr A*, the observed Sgr A* is not in the plane of the observed orbit. 2) As the light of one S-star is with several wavelengths, several optical orbits with different speeds of one single S-star can be observed. 3) As the orbits of two stars in one same plane, if the wavelengths of the light of them are different, the observed orbits are not in one plane. The combinations of these cases can make the observed optical image of these orbits very complicated and radically different from the real physical configuration. Therefore, to accurately and precisely know the real physical configuration of the orbits of the S-stars is a very complicated project.

3. Conclusion

The mass of the Sgr A* can be accurately and precisely measured through the orbits of the S-stars around it. But, the multi-image gravitational lensing and the time lag between the light curves with different wavelengths was well observed.[37,38,39] It was known that, to well understand the numbers of the image and the spectrum and to right know the size and the distance of a celestial body, the time lag and the light dispersion in gravitational need be considered.[40] The Sgr A* is with the strongest gravitational field in all of our Milky Way. The speed of light emitted from it or near it is strongly affected by the gravitational field of it. Therefore, to accurately and precisely know the orbits of the S-stars around the Sgr A*, the dispersion of the light in gravitational need be considered.
And, it is important to clarify the effect of the observed image of the jet of the Sgr A* on the orbits of the S-stars. Comparing Figure 1 to the observed image of the jet of M87,[4,9] it is easy to find that, if the light dispersion in gravitational field had not been considered, the observed optical image should not can be completely understood. The optical image of the orbits of the S-stars around the Sgr A* also is affected by light dispersion in gravitational field, and the optical images of the orbits is much more complicated than the real one. Therefore, the observed optical image of the jets of them need be further understood with the orbits of the stars around them. And, the optical image of the orbits of the S-stars around the Sgr A* need be further formulated with Newtonian theory of gravity to know the orbits accurately and precisely.
A fundamental conclusion about the astronomical observation could be presented: For the celestial body with super strong gravitational field, the observed image about the body could be radically different from the real physical configuration produced from the body. Therefore, an observed astronomical image need be further formulated or corrected with other observations and physics laws.

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Preprints 113314 g001
Figure 1. The image of a star observed at the retarded place. The star is moving along the arrow. As the star is arriving at its present place, it cannot be observed by the observer because a time t − tλi = L/cλi is needed for the light running from the star to the observer. Because of the dispersion of light in the gravitational field, the speed of light with different wavelengths is cλi = c/nλi. Correspondent to one of cλi = c/nλi, one retarded position is observed. This image is a combined one by i retarded positions.
Figure 1. The image of a star observed at the retarded place. The star is moving along the arrow. As the star is arriving at its present place, it cannot be observed by the observer because a time t − tλi = L/cλi is needed for the light running from the star to the observer. Because of the dispersion of light in the gravitational field, the speed of light with different wavelengths is cλi = c/nλi. Correspondent to one of cλi = c/nλi, one retarded position is observed. This image is a combined one by i retarded positions.
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Figure 2. The observed optical orbit vs real orbit of the S-star around Sgr A*. The Sgr A* is moving along the blue arrow, the star is along the red one. Under the condition that the real orbit is a circle and the Sgr A* is at the center of the circle, the observed orbit is an ellipse and the Sgr A* is no at the center of the orbit. And, the observed speed of the orbit is larger than the real one.
Figure 2. The observed optical orbit vs real orbit of the S-star around Sgr A*. The Sgr A* is moving along the blue arrow, the star is along the red one. Under the condition that the real orbit is a circle and the Sgr A* is at the center of the circle, the observed orbit is an ellipse and the Sgr A* is no at the center of the orbit. And, the observed speed of the orbit is larger than the real one.
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