# Version of a glass retroreflector satellite with a submillimeter “target error”

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## Abstract

Specific features of spherical retroreflector arrays for high-precision laser ranging are considered, and errors in distance measurements are analyzed. A version of a glass retroreflector satellite with a submillimeter “target error” is proposed. Its cube-corner reflectors are located in recessions to reduce the effective angular aperture, and their faces have a dielectric interference coating.

## Keywords

Satellite laser ranging Geodetic spherical satellites Cube-corner reflectors Target error## 1 Introduction: retroreflector spherical LEO satellites

The main accuracy characteristics of the spherical satellites include: the error of referencing the results of the satellites laser ranging to the center of the satellites mass (“target error”) and the ballistic coefficient (the ratio of the mass to the area of the cross section of the sphere), which is directly related to the effects of non-gravitational perturbations on the satellite’s precise orbit determination.

The “target error” is caused by the fact that an optical signal, reflected from the satellite, is formed as a result of a superposition of signals from several CCRs, the location of which relative to the center of mass at the moment of the distance measurement is not known exactly. However, while a metal target satellite is in operation, the spin rate about its own center of mass gradually decreases due to the interaction between induced currents and the magnetic field of the Earth. Currently, for example, WESTPAC has stopped to rotate almost completely (the spin period of the satellite became commensurable with the period of its revolution about the Earth).

Moreover, because of the uneven heating of the metal ball caused by the variable location of the satellite in the shadows of the Earth and in the Sun, the far-field diffraction pattern (FFDP) of the reflected radiation is changed and the intensity of the reflected radiation decreases.

Therefore, it is interesting to design glass satellites, the construction of which eliminates some disadvantages inherent to metal spherical satellites. Such satellite with a submillimeter “target error” of distance measurements was implemented in the form of a set of glass spherical layers around the central ball in the BLITS spacecraft. Compared to metal satellites, one major disadvantage of all glass satellites is that their orbits are much more affected by the non-gravitational perturbations caused by their much lower ratio of cross-sectional area to mass.

The goal of this work is to consider the energy and the precision characteristics of an alternative version of a glass satellite: the Geodetic Laser Autonomic Spherical Satellite (GLASS). Such a satellite should has “target error” less than 1 mm and a greater ballistic coefficient than the ballistic coefficient of the existing glass spacecraft BLITS.

## 2 Accuracy characteristics of the retroreflector GLASS satellite with cube-corner reflectors

- 1.
Using only dielectric materials to prevent the influence of the Earth’s magnetic field on the spacecraft spin rate.

- 2.
The reflection of a signal at a given moment mostly occurs from only one CCR to reduce the “target error.”

- 3.
The absence of the so-called dead zones, i.e., the satellite positions relative to the laser ranging station in which reflections do not occur from the CCR.

- 4.
The “target error” does not exceed 1 mm.

*θ*

_{im}is the maximum angle of the ray incidence on the entrance face of the CCR and

*γ*is the central angle between the neighboring CCRs. The angle

*γ*depends on the CCRs number and their arrangement on the sphere.

Let us define *θ*_{im} as an angle of ray incidence upon which the effective area of the CCR aperture decreases by four times. The value of *θ*_{im} is determined by the CCR size, depth of recession and refractive index of the CCR material. *θ*_{im} determines the working aperture of the satellite, i.e., the part of the sphere where the CCR can be conventionally considered as “working.” The CCR position on this part of the sphere is unknown at the moment of laser impulse incidence and determines the ranging error (“target error”) of the spherical satellite. In other words, the distance from the CCR taking part in the act of reflection of an incident photon to the satellite center of mass can be calculated only with a certain error.

To achieve a submillimeter “target error,” we perform an analysis of how this error is affected by the construction and dimensions of the spherical satellites with the CCR if (1) is considered.

Laser ranging involves the concept of the equivalent reflection plane (ERP), a plane that is reached by the wavefront of the incident wave during the time equal to the half time of the impulse flight. In fact, the “target error” is an error in determining the distance from the ERP to the satellite center of mass. Without regard to the influence of the atmosphere, the ERP position is determined by two factors: the position of the reflecting CCR on the sphere and the delay of the laser impulse in the CCR. The delay depends on the refractive index, CCR size and angle of light incidence on its entrance face.

*θ*

_{i}is the angle of incidence of the ray on the CCR entrance face (this angle depends on the CCR position on the sphere) and

*γ*is the central angle between the neighboring CCRs.

*n*is the refractive index of the CCR glass and

*h*is the distance from the vertex to the CCR entrance face.

*θ*

_{i}, i.e., as the CCR moves on the sphere relative to the point

*S*(

*θ*

_{i}= 0), the ERP approaches CCR vertex and the impulse delay in the CCR decreases. However, with an increase in

*θ*

_{i}, the center of the entrance CCR face itself is displaced along the line (in the vertical direction in Fig. 2), parallel to the

*Z*-axis by:

*R*

_{V}is the radius of the sphere tangent to the entrance faces of the CCR.

*S*(point on the sphere nearest to the receiver of the laser rangefinder) to the ERP is determined by the following formula:

*τ*

_{m}

*c*/2 (where

*τ*

_{m}is the measured time of the laser impulse propagation and

*c*is the velocity of light) to the point

*S*:

*O*, the sphere radius should be added to (4):

*Z*equal to \( \Delta Z_{m} = R_{V} (1 - \cos \theta_{im} ) \) and the difference Δ

*V*

_{m}between extreme positions of the ERP at

*θ*

_{i}= 0 and

*θ*

_{i}=

*θ*

_{im}is equal, with allowance for (2):

*S*is determined with allowance for (3) at

*θ*

_{i}= 0 and the obtained value Δ

*V*

_{m}/2:

*S*. For the sphere’s center

*O*, we similarly obtain:

The quantity \( \left\langle {\Delta_{O} } \right\rangle = - \;\left\langle {V_{O} } \right\rangle \) is a “nominal” correction to the distance determination to the satellite, which should be applied by the laser ranging data centers. As the CCR moves on the working region of the sphere (Fig. 2), the corrections \( \left\langle {V_{S} } \right\rangle \) and \( \left\langle {V_{O} } \right\rangle \) are uniformly distributed relative to the average value in the range from − Δ*V*_{m}/2 to + Δ*V*_{m}/2. Thus, one can calculate the root-mean-square deviation of the correction from the average values \( \left\langle {V_{S} } \right\rangle \) or \( \left\langle {V_{O} } \right\rangle \): \( \approx \Delta V_{m} /\sqrt {12} \).

The quantity \( \Delta V_{m} /\sqrt[{}]{12} \) can be considered as the “target error” and determines the accuracy of the given correction of a *single* measurement of the distance.

*θ*

_{im}on the CCR entrance face.

As follows from the analysis of Fig. 3, a decrease in both *θ*_{im} and in *R*_{V} leads to the “target error” reduction. For example, the working angular aperture (the maximum incidence angle on the CCR entrance face) *θ*_{im} = 13° and the radius *R*_{V} = 84 mm provide a “target error” of 0.55 mm (an uncontrolled addition to the average range correction).

## 3 Cube-corner retroreflectors for the GLASS

In laser tracking of the satellites, it is necessary to take into account the velocity aberration of light (Degnan 1993; Arnold 1978, 1979). The effect of the velocity aberration results in the fact that the laser ray is reflected from the CCR not in the location direction but with the angular shift *α* = 2*u*/*c* in the direction of vector **u** (projection of the orbital velocity of the satellite on the plane perpendicular to the line of sight). The value of the velocity aberration is related to the propagation of light in a moving coordinate system of the satellite and amounts to 6′′–10′′ for an orbit height *H* from 1500 to 2000 km of GLASS. It means that a sufficient part of the reflected energy must be contained within 6′′–10′′ of the FFDP.

The FFDP of the reflected radiation, as well as the cross section, depends on the CCR parameters: size, the reflecting properties of faces, an angle between the reflecting faces, etc. One effective way to change and optimize the FFDP is in control of the phase shift of the components of the electric field vector **E** in the process of refraction and reflection of light at the CCR faces, which, in turn, is determined by the kind of the faces coating or its absence (Altamimi et al. 2016; Sadovnikov and Sokolov 2009; Crabtree and Chipman 2010; Sokolov and Murashkin 2011; Sokolov 2013).

**E**. The obtained FFDP consists of six spots without a central spot. Such FFDP corresponds to the perfect dihedral angles of the CCR reflecting faces in case of the normal incidence beam to the CCR entrance face. The FFDP spots of the real CCR depend on the dihedral angular offsets of the CCR reflecting faces and the quality of the interference coating. The spots of the FFDP of the real CCR are displaced to the center or from the center. It depends on the CCR dihedral angular offsets. The energy of radiation reflected from such CCR is distributed at a certain angular distance from the FFDP center.

The CCR orientation on the sphere should provide such a position of the CCR, that is, the line passing through the maximums of FFDP lateral lobes coincides with the vector \( v_{\text{rot}} = \left[ {{\varvec{\upomega}},{\mathbf{r}}} \right] \), where **ω** is the angular satellite velocity vector and **r** is the radius vector drawn from the satellite’s axis of rotation to the center of the CCR entrance face.

The above-described analysis of the diffraction pattern of the light reflected from the CCR is valid only for the case of the ray normal incidence on the CCR entrance face. If the CCR is inclined to the incident ray, the recession depth and diameter play an important part. The larger the height of the hood is and the smaller is its diameter, then the faster the CCR effective area decreases and the incident angle *θ*_{i} increases, and therefore the maximum angle of the ray incidence on the CCR *θ*_{im} (angular aperture) reduces.

Each CCR is fixed with a glass bushing inside the recession with a 25 mm. The recession of the CCRs provides the requirements of the GLASS construction, which does not have any dead zones and reflects the signal by one CCR only.

The slant incidence of light leads to the decrease in the intensity of two of the six FFDP lobes. However, the maximum of the working lateral lobes are shifted to the larger angular distance from the FFDP center; i.e., the FFDP form becomes more favorable for the working angles of the 6′′–10′′ velocity aberration. As follows from Fig. 4, the cross section varies within the range from 1.25 × 10^{6} to 1.5 × 10^{5} m^{2} for angles of 6′′–10′′ (the solid line in Fig. 4).

The polarization state in each point of the FFDP is linear, and the vector **E** rotates with the azimuth φ, making two complete revolutions. Thus, the polarization structure of the reflected radiation is a polarization–symmetrical structure, produced by the second-order Laguerre–Gauss modes. The orientation of this diffraction pattern is determined not by the azimuth of the incident light **E** vector, but by the CCR orientation: If the CCR is rotated by 30°, the spot pattern is rotated by the same angle (Sokolov and Murashkin 2011; Sokolov 2013). The light reflection with circular polarization results in the production of a beam like an optical vortex (Dennis et al. 2009; Rozas et al. 1997; Sokolov 2017). In this case, the state of reflected light polarization is circular in each point of the space, while the phase variation of each component of the vector **E** is proportional to twice the azimuth angle.

## 4 Some features of the GLASS design

Using a dense flint with the density of 4.79 × 10^{3} kg/m^{3} (Bach and Neuroth 1998) as a material of GLASS allows to increase the ballistic coefficient up to the value of 530 kg/m^{2}.

to reduce the heating of the housing by the solar radiation (for this, the absorption coefficient in the visible region of the spectrum should be \( \alpha = 0.1 - 0.2 \) (Oleyink 2018) and the emissivity in the infrared region of the spectrum is \( \varepsilon = 0.8 - 0.9 \));

to ensure a possibility of detecting a satellite after its detachment from the launch vehicle with the help of the ground-based telescopes. (The calculations show that its star magnitude in the estimated satellite orbit is within 11

^{m}–13^{m}.)

The glass ball consists of two halves connected by means of an optical glue. The preliminary estimates confirm the stability of the GLASS design for the distribution of the ball body temperature from 66 to 8 °C due to different positions relative to the Sun: The magnitude of the thermal stress does not exceed 6 MPa. The mechanical strength of the adhesive is at least 9.5 MPa.

Uniform heating or cooling of the CCR does not affect FFDP. On the contrary, temperature gradients inside the CCR, exceeding 3 degrees, are dangerous. In this case, the resulting thermal lens leads to an expansion of the diffraction pattern and the cross section decreases by several times. This problem occurs if the CCR is placed on a metal frame with the structural elements that have inadequate thermal coefficients. For GLASS, which is totally made of glass, according to the ANSYS calculations, the temperature gradient inside the CCRs does not exceed 3 degrees.

The dielectric interference coatings of the reflecting CCR faces provide the cross section increases, and therefore the energy budget in distance measurements increases. The submillimeter “target error” of GLASS is provided due to the decrease in the satellite diameter under the condition of (1) implementing the fundamental principle: The incident laser impulse is reflected by only one CCR at any position of the satellite.

Comparison of retroreflector spherical satellites (RSSs) in low Earth orbits

RSS | LARETS | WESTPAC | LARES | BLITS | BLITS-M | GLASS |
---|---|---|---|---|---|---|

| ||||||

Satellite material | Metal | Metal | Metal | Glass | Glass | Glass |

Sphere diameter, mm | 215 | 215 | 364 | 170 | 220 | 220 |

Hood height (recession depth), mm | 15.0 | 31.5 | 3.3 | – | – | 25.0 |

Mass, kg | 23.90 | 23.40 | 386.80 | 7.53 | 16.6 | 20.3 |

Ballistic coefficient, kg/m | 658 | 644 | 3700 | 330 | 435 | 547 |

Number of CCRs | 60 | 60 | 92 | – | – | 60 |

Orbit height, km | 690 | 835 | 1450 | 832 | 1500 | 1500–2000 |

Target error, mm | 1.50 | 0.60 | 1.70 | 0.10 | 0.10 | 0.55 |

Average value of cross section × 10 | 6.50 | 0.04 | 3.30 | 0.10 | 0.20 | 0.50 |

Initial spin period, s | 0.8 | 3–6 | 11.6 | 5.6 | 6 | 6 |

State | Active | In orbit | Active | Destroyed | Ready to launch | Testing |

It should be noted that GLASS could be launched into an orbit 4000–5000 km high for successful ranging by stations such as “Graz” or the new Russian laser ranging station of the “Tochka” type (the impulse energy 2.5 mJ, the impulse repetition rate 1000 Hz, the receiving aperture 360 mm).

As shown in the ILRS reports (Bloßfeld et al. 2018), up to 11 satellites with the sub-centimeter target errors are required to create the International Terrestrial Reference Frame for the GGOS. The GLASS, like the BLITS-M satellite, can be a worthy representative of such a satellite constellation.

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