The Time System Involved in GPS Measurements

1 Introduction

Time and space are the basic forms of material existence. Time is one of the basic physical quantities, which reflects the sequence and continuity of material movement. People cannot do without time in production, scientific research and daily life.

2. Some basic concepts about the time system

Time is an important physical quantity, which puts forward high requirements on time in GPS measurement. For example, when using the ranging signal transmitted by the GPS satellite to measure the distance between the satellite and the receiver, if the ranging error is required to be less than or equal to $1cm$ , then the error of the measured signal propagation time must be less than or equal to$3\times 10^{-11}\mathrm{s}=0.03\mathrm{ns}$ 。

2.1 Time

Time has two meanings: time interval and moment. The time interval refers to the time process experienced by the movement of things between two (instantaneous) states, which describes the continuous state of the movement of things in time; while the moment refers to the time when a certain phenomenon occurs. The so-called moment is actually a special time interval (between an agreed starting moment), and the time interval refers to the difference between the beginning and end moments of an event. Instantaneous measurement is also known as absolute time measurement, while time interval measurement is known as relative time measurement.

The time system stipulates the standard of time measurement, including the reference base (starting point) of the moment and the scale reference of time interval measurement. The time system is theoretically described by definitions and corresponding regulations, and the time system integration is realized within the same spherical range or within a certain area through timekeeping, time service, and time step rate measurement and comparison techniques and maintain a unified time system. But in actual use, sometimes these two different concepts are not strictly distinguished.

2.2 Time base

Time measurement requires a common standard scale, called time reference or time frequency reference. Generally speaking, any periodic motion that can be observed can be used as a time reference as long as it meets the following conditions:
(1) It can perform continuous periodic motion, and the motion cycle is very stable;
(2) The motion cycle has a good Reproducibility, that is, this periodic movement in different periods and places can be reproduced through observation and experiment.

There are many kinds of motions with the above characteristics in the natural world, such as the early burning incense and hourglass, the later pendulum and the vibrating of quartz crystal, and the modern electromagnetic wave oscillation signal and pulsar pulse signal emitted by atomic transition. So far, there are mainly the following types of more precise time standards in practical application:
(1) The earth's rotation period. It is the time base used to build the world, and its stability is about $10^{-8}$ .${}^{\_}$
(2) The revolution period of the planets around the sun and the revolution period of the moon around the earth. It is the time base used to build the almanac, and its stability is about$10^{-10}$ .${}^{\_}$
(3) The oscillation frequency (period) of the electromagnetic wave signal emitted (or absorbed) by the electrons of the atomic clock when they transition from one energy level to another. It is the time reference used when building atoms, and its stability is about$10^{-14}$ .${}^{\_}$The current best cesium atomic fountain clock has a stability of$10^{-16}$ 级。
（4）脉冲星的自转周期, 最好的毫秒脉冲星的自转周期的稳定度有可能达到 $10^{-19}$ 或更好。目前, 世界各国的科学家们还在为建立具有更高精度(比原子时) 的脉冲星时而努力工 作。

2.3 守时系统(时钟)

守时系统(时钟) 被用来建立和/或维持时间频率基准, 确定任一时刻的时间。守时系 统还可以通过时间频率测量和比对技术来评价该系统内不同时钟的稳定度和准确度, 并据 此给各时钟以不同的权重,以便用多台钟来共同建立和维持时间系统框架。

2.4 授时和时间比对

授时系统可以通过电话、电视、计算机网络系统、专用的长波和短波无线电信号、搬运钟 以及卫星等设备将时间系统所维持的时间信息和频率信息传递给用户。不同用户之间也可 以通过上述设施和方法米实现高精度的时间比对。授时实际上也是一种时间比对,是用户 与标准时间之间进行的时间比对。
不同的时间比对方法具有不同的精度, 其方便程度和所需费用等也不相同, 用户可以根 据需要企择合适的方法。
目前, 国际上有许多单位和机构在建立和维持各种时间系统,并通过各种方式将有关的 时间和频率信息传递给用户, 这些工作统称为时间服务。我国国内的时间服务是由国家授 时中心 (NTSC) 提供的。

2.5 时钟的主要技术指标

时钟是一种重要的守时工具。利用时钟可以连续地向用户提供任一时刻所对应的时间 $t_i$. Since there are errors in any clock, it is necessary to compare it with the standard time regularly or irregularly to obtain the clock difference at the comparison time, and estimate ti at any time after mathematical processing (such as simple linear interpolation) $t_i$Correct the clock difference to obtain a more accurate time. The main technical indicators for evaluating clock performance are frequency accuracy, frequency drift and frequency stability.

3. Sidereal time and solar time

Earth's rotation is a continuous periodic motion. In the early days, due to the limitation of observation accuracy and timing tools, people thought that this rotation was uniform, so it was chosen as the time reference. Both sidereal time and solar time use the earth's rotation as the time reference, and the main difference lies in the reference points selected for measuring the rotation.

3.1. Sidereal time

Sidereal time is based on the vernal equinox as a reference point. Due to the rotation of the earth, the time interval between two consecutive vernal equinoxes passing through the local meridian is one sidereal day. Divide evenly on the sidereal day basis to obtain "hours", "minutes" and "seconds" in the sidereal time system. Sidereal time is numerically equal to the hour angle of the vernal equinox relative to the local meridian. Since sidereal time begins when the vernal equinox passes through the local upper meridian, it is a local time.

Due to the influence of precession and nutation, the direction of the earth's rotation axis in space is constantly changing, so the vernal equinoxes can be divided into true vernal equinoxes and flat vernal equinoxes. The corresponding sidereal time is also divided into true sidereal time and flat sidereal time. Among them, Greenwich Mean Sidereal Time (GAST) and Greenwich Mean Sidereal Time (GMST) often appear in GPS. GAST is the angle between the true vernal equinox and the zero point of longitude (the intersection of the Greenwich origin meridian and the equator). The variation of GAST depends mainly on the rotation of the earth, but is also related to the movement of the true vernal equinox itself due to precession and nutation; GMST is the angle between the vernal equinox and the zero point of longitude, $\mathrm{GAST}-\mathrm{GMST}=Dps\cos \left(e_0+\Delta e\right)$式中,$\Delta \psi$ is the nutation of the Huang Jing;$\Delta \epsilon is$ the angle nutation, which will be introduced in detail later.

3.2. True solar time

The true solar time is based on the center of the sun as a reference point, and the time interval when the center of the sun passes through the meridian of a certain place twice is called a true solar day. Based on it, the "hour", "minute" and "second" in the true solar time system can be obtained after being evenly divided. Therefore, true solar time is a time system established based on the rotation of the earth and the ether as a reference point from the center. True solar time is numerically equal to the hour angle of the center of the sun relative to the local meridian, plus 12 hours. However, since the orbit of the earth around the sun is an ellipse, according to Kepler's three laws of planetary motion, its angular velocity is different. At the perihelion, the angular velocity is the largest; at the aphelion, the angular velocity is the smallest. Since the earth's revolution is located on the ecliptic plane, and the hour angle is measured on the equator plane, the length of true solar time is different. That is to say, true solar time does not have the basic conditions to be a time system .

3.3. Mean solar time

In daily life, people have used the sun to determine the time, arrange work and rest. In order to make up for the unevenness of the true solar time. People imagined to use a fake sun to replace the real sun. This fake sun is also doing the same annual motion as the real sun, but there are two differences: first, its annual motion trajectory is located on the equator plane instead of the ecliptic plane; second, its angular velocity on the equator is constant, equal to the real sun The average angular velocity of the sun. We call this false sun the flat sun. The time system established on the basis of the earth's rotation and the above-mentioned mean solar center as a reference point is called mean solar time. That is to say, the time interval between the hypothetical mean sun passing through the meridian circle of a certain place twice in succession is called a mean solar day. After being evenly divided based on it, the "hour", minute and "second" in the mean solar time system can be obtained. Mean solar time is numerically equal to the hour angle of the mean sun, plus 12 h 12 \mathrm $12\mathrm{h}$ 。

Since the mean sun is a hypothetical invisible celestial body, the mean solar time is actually obtained by observing the stars or the real sun and then calculating according to the mathematical relationship between different time systems.

4. Universal Time and Zone Time

Mean solar time is a local time. At the same moment, the mean solar time of two places located on different longitudes is different. For the convenience of daily life and work, a unified standard time is needed. The International Meridian Conference held in Washington in 1884 decided to divide the world into 24 standard time zones. From the zero meridian of Greenwich, $7.5^{\circ }$ is 0 time zone, then eastward every$15^{\circ }$ 为一个时区, 分别记为 $1,2,\cdots ,23$ 时区。在同一时区, 统一采用该时区中央子午线上的平太阳时, 称为区时。中国幅员辽阁, 从西向东横跨 5 个时区。目前都采用东八区的区时,称为北京时。采用区时后,在一个局部区域内所使用的时间是相对统一的,不同时区间也可以方便地进行换算。
格林尼治起始子午线处的平太阳时称为世界时。世界时是以地球自转周期作为时间基准的,随着科学技术水平的发展及观测精度的提高,人们逐渐发现:
(1) 地球自转的速度是不均匀的, 它不仅有长期减缓的总赲势, 而且也有季节性的变化以及短周期的变化, 情况较为复杂;
(2) 地极在地球上的位置不是固定不变的, 而是在不断移动, 即存在极移现象。 这就意味着世界时已不再严格满足作为一个时间系统的基本条件, 因为它实际上已不是一个完全均匀的时间系统。为了使世界时尽可能均匀, 从 1956 年起,在世界时中引入了极移改正 $\Delta \lambda$ 和地球自转速度的季节性改正 $\Delta T$ 。如果我们把直接根据天文观测测定的世 界时称为UT0, 把经过极移改正后的世界时称为 UT1，把再经过地球自转速度季节性改正后的世界时为 UT2, 则有:
$$\begin{array}{rl}& \mathrm{UT}1=\mathrm{UT}0+\Delta \lambda \\ & \mathrm{UT}2=\mathrm{UT}1+\Delta T=\mathrm{UT}0+\Delta \lambda +\Delta T\end{array}$$
式中, 极移改正 $\Delta \lambda$ 的计算公式为:
$$\Delta \lambda =\frac{1}{15}\left(X_p\sin \lambda -Y_p\cos \lambda \right)\tan If$$
,$X_p、Y_p$are the two components of the pole shift, measured and published by IERS; $\lambda and\phi$ are the longitude and latitude of the station. The seasonal correction of the earth's rotation$Determine\; the\; \Delta T$ value:
$$\Delta T=0.022s\sin 2\pi t-0.012s\cos 2\pi t-0.006\mathrm{s}\sin 4\pi t+0.007s\cos 4\pi t$$tt
in formula$$\text{ (2-8) }$$$t$ is the Bessel year. $\mathrm{t}=($ MJD-51544.03)$/365.2422$ . After the above corrections, the stability of UT2 has improved (about$10^{-8}$ ), but still contains the long-term term, short-period term and some irregular terms in the inhomogeneity of the earth's rotation, so they are not a uniform time system, and cannot be used in high-precision application fields such as GPS measurement.
It should be pointed out that since UT1 reflects the real situation of the earth's rotation and is directly related to the earth's rotation angle, it is an important parameter in the coordinate transformation between GCRS and ITRS (WGS-84) coordinate system.

5. Atomic Time, Coordinated Universal Time and GPS Time

5.1. Atomic time

With the development of productivity and the improvement of science and technology, people have higher and higher requirements for the accuracy and stability of time and frequency. Sidereal time and peace solar time based on the rotation of the earth, and the revolution of planets and moons Almanac has been difficult to meet the requirements. Since the 1950s, people have gradually focused their attention on establishing the atomic time based on the atomic motion inside the material.

When the electrons in an atom jump from one energy level to another, they emit or absorb electromagnetic waves. The frequency of this electromagnetic wave is very stable, and the above phenomenon is easy to reproduce, so it is a good time reference. In 1955, the British National Physical Laboratory NPL cooperated with the US Naval Observatory USNO to accurately measure the oscillation frequency of the electromagnetic wave signal emitted by the ground state of the cesium atom when it transitions between two hyperfine energy levels in a zero magnetic field. In October 1967, the Thirteenth International Conference on Planning and Design adopted the following resolution: cesium $133\left({\mathrm{Cs}}^{133}\right)$The world in which the 9192631770 cycles of transition radiation oscillation between two hyperfine energy levels in the ground state of the atom lasts for 9192631770 cycles is defined as the atomic time is 1s. And the starting point of atomic time is specified as January 1, 1958$0h$ , at this time, the atomic time is aligned with the universal time, but due to technical reasons, it was later found that the atomic time AT and the universal time UT were not aligned accurately at this instant, and there was 0.0039 s 0.0039 \mathrm$0.0039s$ difference, namely:
$$(\mathrm{AT}-\mathrm{UT}{)}_{1958.0}=-0.0039s$$
From this the atomic time can be established. It should be noted that many different types of atomic clocks appeared later, such as rubidium atomic clocks, hydrogen atomic clocks, etc., and their transition signal frequencies were precisely determined to be 6834682605 Hz 6834682605 \mathrm{~Hz$6834682605Hz$ and$1420405757.68Hz$ , so the definition of atomic time is extended to a time system based on the stable frequency of atomic transition.

5.2. International Atomic Time

Atomic time is determined and maintained by atomic clocks, but due to differences in electronic components and external operating environments, the time given by each atomic clock is not strictly the same at the same moment. In order to avoid confusion, it is necessary to establish a more reliable, more accurate, and more authoritative unified time system that can be accepted by all countries in the world——International Atomic Time TAI. TAI was established by the International Time Bureau in 1971 and is now maintained by the time department of the International Bureau of Weights and Measures (BIPM). BIPM is based on the data given by about 240 free-running atomic clocks in about 60 time laboratories around the world, and the international atomic time is given after unified data processing.

5.3. Coordinated Universal Time

Atomic time with good stability and reproducibility can meet the requirements of high-precision time interval measurement, so it is adopted by many departments. However, there are many fields, such as astronomical navigation and terrestrial astronomy, which are closely related to the rotation of the earth and cannot be separated from universal time. Since atomic time is a uniform time system, and the earth's rotation has a long-term trend of slowing down, this means that the second of universal time will become longer and longer, so the difference between atomic time and universal time will become more and more obvious, and it is estimated that by the end of this century, the difference between the two will reach $About\; 2minutes$ . In order to take into account the requirements of the above users at the same time, the International Radio Science Association established the Coordinated Universal Time UTC in the 1960s. The second length of Coordinated Universal Time is strictly equal to the second length of Atomic Time, and the time difference between Coordinated Universal Time and Universal Time UT needs to be kept at$Within\; 0.9s$ , otherwise it will be adjusted by means of leap second. Add$1s$ is called a positive leap second, minus$1s$ is called a negative leap second. Leap seconds generally occur on June 30 and December 31. The specific time of the leap second is notified by the International Bureau of Weights and Measures to the time service agencies of various countries 2 months ago.

For the convenience and timeliness of use, each time laboratory usually uses multiple atomic clocks in the laboratory to establish and maintain a local UTC system for use in the country or region. In order to distinguish, these regional UTC systems should be followed by a bracket indicating which time laboratory was established and maintained. For example, the UTC system established and maintained by the United States Naval Observatory, written as UTC(USNO). The difference between UT1 and UTC (USNO) is given in the GPS navigation message . The globally unified Coordinated Universal Time established by BIPM using the data of various laboratories around the world is directly marked as UTC without brackets.

The second length of atomic time is based on $1900.0$ is defined by the second length of ephemeral time, that is, the length of an atomic time second defined by the 13th International Conference on Weights and Measures is the same as1900.0 1900.0The length of 1 second of ephemeris time is the same at$1900.0\; .$Due to the long-term slowing trend of the earth's rotation, that is to say, the seconds of universal time will become longer and longer. After more than 100 years, there is an obvious difference between the universal time second and the atomic time second, so the jumping seconds become more and more frequent (now about 1 s needs to be adjusted every year) 1 \mathrm{~s}$1s)$ , which brings a lot of inconvenience to use. Some people suggest redefining the second length of atomic time so that it is as consistent as possible with the second length of the current universal time, so as to reduce the number of jumping seconds and keep UTC continuous in a long period of time. But "second" is a very important basic physical quantity. After its definition changes, it will cause a series of parameters such as the speed of light to change. Therefore, there are many opposing opinions, and it needs to be carefully considered and discussed in the long run.
In December 1979, UTC has replaced Universal Time as the standard time in radio communications. At present, many countries have adopted UTC as their own time system, and broadcast time numbers according to UTC time. Users who need to use universal time can obtain UT1 indirectly according to UTC and (UT1-UTC).

5.4. GPS time

GPS time is a time system used by the Global Positioning System, GPS. It is an atomic time established and maintained by the GPS ground monitoring system and the atomic clocks in GPS satellites, and its starting point is 0 h 00 m 00 s 0 \mathrm{~h} 00 \mathrm{~ on January 6, 1980 $0h00m00s$ . At the initial moment, GPS time is aligned with UTC, and the time given by these two time systems is the same. Due to the existence of jumping seconds in UTC, after a period of time, there will be a difference of$n$ whole seconds,$n$ is the accumulated jumping seconds of UTC during this period, which will change with time. Since the starting moment of GPS time is January 6, 1980, the difference between UTC and TAI is$19s$ , so there will always be 19 s between GPS time and international atomic time$19s$ difference, ie TAI-GPST$=19s$ . Theoretically speaking, both TAI and GPST are atomic time and do not jump seconds, so the difference between these two time systems should be strictly$19s$ whole. But TAI (UTC) is a time system jointly maintained by BIPM based on about 240 atomic clocks around the world, and GPST is a localized atomic time maintained by dozens of atomic clocks in the Global Positioning System. In addition to the difference of several whole seconds between systems, there will also be small differences (several or tenths of nanoseconds)$C_0$, $THIS-\mathrm{GPST}=19s+C_0;\mathrm{UTC}-GPST=n$ whole seconds$+$ $C_0$. Thanks to $GPS$ has been widely used in time comparison, and users can obtain high-precision UTC or TAI time through the above relationship. There are special units in the world to measure and publish$C_0$value, its value can generally be kept at $Within10ns$ .

6. A time system based on the framework of the theory of relativity

Before 1984, almanac time was used in calculating the positions of natural and man-made celestial bodies, compiling astronomical ephemeris and satellite ephemeris. Ephemeris time is a time system based on classical Newtonian mechanics. According to Newtonian mechanics, time $t$ is an independent variable in the equation of motion of a celestial body, which has nothing to do with the position of the celestial body in space and the gravitational potential it receives. It can be used not only for the motion of satellites around the earth, but also for the motion of planets around the sun. With the improvement of observation technology and the continuous improvement of the precision of timing tools, the contradiction between this classical theory and observation results began to appear, and became more and more obvious. It is urgent to use a new theory and model to explain and process data.

For this reason, in 1976, the 16th IAU General Assembly made a resolution to officially introduce the relativistic time scale in the field of astronomy, and gave specific definitions of geodynamic time TDT and solar system barycentric dynamic time TDB (however, geodynamic time and solar system barycentric dynamic time These two names were officially determined at the 17th IAU General Assembly in 1979). At the 21st IAU General Assembly held in 1991, it was decided to change the geodynamic time TDT to the earth time TT, and introduced the geocentric coordinate time TCG and the solar system barycentric coordinate time TCB. Concepts such as TDT (TT) and TDB will be involved in the new GPS navigation message.

6.1. Geodynamic Time (TDT)

Geodynamic time is a time system used to solve the equations of motion of celestial bodies (such as artificial satellites) moving around the center of mass of the earth, and to calculate the ephemeris of satellites. TDT is based on the international atomic time TAI, and its second length is equal to the second length of the international atomic time. But there are $A\; difference\; of\; 32.184s$ , ie
$$\mathrm{TDT}=THIS+32.184\mathrm{s}$$
This is because TDT starts at January 1, 1977 $0h$ is the same as the ephemeral time ET (this is done to maintain the continuity of the celestial body's position), and at this time the difference between ET and TAI is$32.184s{}_{\text{。 }}$And TAI and $There\; are\; 19\; s\; between\; GPS$ time$19s$ difference, so there is theoretically$The\; difference\; of\; 51.184s$ :
$$\text{ TDT }=\text{ GPST }+51.184s$$
The above formula does not take into account the slight difference between TAI and GPS time$C_0$, so it is only a theoretical value. It should be noted that the time when a certain time system was established and started to be used is not the same as the starting point of the time system, and the starting point is often earlier than the time when it was started to be used.

In the resolution of the 16th IAU General Assembly, the basic unit of TDT was changed from "second" in atomic time to "day" in astronomy, and 1 day of TDT was defined = 86400 s =86400 \mathrm $=86400s$ (SI). This change has no substantive significance, but only for the convenience of astronomical calculations. In 1991, the 21st IAU General Assembly decided to rename TDT as Earth Time TT in order to avoid using the controversial term Dynamical.

At present, the earth time TT is used in the calculation of the motion equation of the GPS satellite and its ephemeris. Earth time TT can be regarded as an idealized atomic time consistent with SI seconds realized on the geoid.

6.2. The dynamic time TDB of the center of mass of the solar system

The center-of-mass dynamic time of the Paleosystem is simply called the center-of-mass dynamic time. This is a time system used to solve the equations of motion whose origin of coordinates is located at the barycenter of the solar system, and to compile planetary catalogs.

When the IAU introduced TDT (TT) and TDB, in order to prevent a large difference between the two time systems, it was artificially stipulated that no long-term change items are allowed between the two time systems, but only periodic item. That is, there are only slight periodic changes between TT and TDB, but the "average clock speed" of the two time systems is the same within a cycle. Under the constraints specified above, the following relationship exists between TT and TDB:
$$\mathrm{TDB}-\mathrm{TT}=0.001658\mathrm{s}\sin M+0.000014\mathrm{s}\sin 2M+\frac{n_e\cdot \left(\mathbf{x}-{\mathbf{x}}_0\right)}{c^2}$$
where, $M$ is the mean anomaly angle when the earth revolves around the sun;$v_e$, is the revolution velocity vector of the earth's barycenter in the barycentric coordinate system of the solar system; $x_0$is the position quantity of the center of the earth in the barycentric coordinate system of the solar system; $x$ is the position vector of the ground clock in the barycentric coordinate system of the solar system;$\left(x_0-x\right)$is actually the position vector of the ground clock in the geocentric coordinate system;$c$ is the speed of light in vacuum.

There is actually a scale ratio between TT and TDB, that is to say, the length of 1 second in TT is not equal to the length of 1 second in TDB, and the relationship between the two is as follows: Δ TDB Δ TT
$$\frac{\Delta \mathrm{TDB}}{\Delta \mathrm{TT}}=1+L_B$$
In the formula, $L_R=1.55051976772\times 10^{-8}$ 。

This means that in the earth-centered coordinate system and the barycentric coordinate system of the solar system, due to the difference in the moving speed of the coordinate system and the gravitational potential received, under the influence of the theory of relativity, the time units of TT and TDB actually contain a systematic scale ratio $L_B$However, in order not to allow too large a difference between the two time systems, the International Astronomical Union (IAU) artificially defined TDT (TT) and TDB so that no systematic time scale ratio exists between them. , and only allow the existence of periodic variation items (let the average time intervals be equal), in order to keep the speed of light $c$ is constant, so there can only be a scale ratio between the length unit in the earth-centered coordinate system and the length unit in the barycentric coordinate system of the solar system, that is,
$$L_{}=\frac{L_{\mathrm{TT}}}{1-L_B}$$
That is to say, 1 meter in the barycentric coordinate system of Taiyou system is longer than 1 meter in the earth-centered coordinate system.

6.3. TCG in geocentric coordinate time and TCB in solar system barycentric coordinate time

Since introducing $After\; TDT\left(TT\right)$ and TDB, many people raised objections, such as:

• How to interpret the word Dynamical:
• The IAU specifies that only small periodic variations between TDB and TDT(TT) are allowed. However, when the time period is short, it is difficult to strictly distinguish the periodic item from the long-term item, and the periodic item is also equivalent to the long-term item;
• In order to get rid of the long-term term between TDB and TDT, it is necessary to artificially introduce a scale ratio between the geocentric coordinate system and the barycentric coordinate system of the solar system, thus resulting in different astronomical constants in different coordinate systems, and also making Some concepts became vague, so in 1991, at the 21st IAU General Assembly, it was decided to introduce the geocentric coordinate time TCG and the solar system barycentric coordinate time TCB.

Geocentric coordinate time TCG is the fourth dimension coordinate used in the celestial coordinate system whose origin is located at the center of the earth: time coordinate. It is a time-like variable that transforms TDT from the geoid to geocentric time through relativity.
The solar system barycentric time TCB is the fourth dimensional coordinate in the solar system barycentric celestial coordinates. It is the time variable in the equations of motion used to calculate the motion of planets around the sun, and it is also an independent variable when compiling planetary catalogs.
In the time system, we usually refer to the time that can be directly determined by the standard clock as the original time. The original time can be directly measured with precise timing tools, such as atomic time. The time derived under the framework of the theory of relativity is called coordinate time or class time, such as TDB, TCG, TCB, etc. The coordinate time cannot be realized by measurement, but needs to be obtained indirectly through calculation according to the mathematical relation given by the space-time metric, and the space-time law can be obtained through the Einstein field equation. Below we directly give the relationship between TT and TCG without derivation:
$$\text{ TCG }-\mathrm{TT}=L_G(M.J.D-43144.0)\times 86400s$$
where, $L_G$is a constant, its value is equal to $6.969290134\times 10^{-10}$ .${}^{\_}$The starting time of TT and TCG is defined as January 1, 1977 at$0h$ , expressed as 2443144.5 2443144.5in Julian day$2443144.5$ days, expressed as 43144.0 43144.0with simplified Julian day MJD$43144.0$ , it is stipulated that TCG = TTat the starting time$\mathrm{TCG}={\mathrm{TT}}_{\text{。 }}$ 。

And there is the following relationship between TCB and TCG:
$$\begin{array}{rl}\mathrm{TCB}-\mathrm{TCG}=& L_c(M.J.D-43144.0)\times 86400s+0.001658\mathrm{s}\times \sin M\\ & +0.000014\mathrm{s}\times \sin 2M+\frac{v_e}{c^2}\left(\mathbf{x}-{\mathbf{x}}_0\right)\end{array}$$
In the formula, $L_c=1.48082686741\times 10^{-8}$ , the meanings of other symbols are the same as before.
The first term in the above formula is a long-term term, which will increase with the increase of the time interval. On September 4, 2007$At\; 0h$ , this item has reached$14.3335s$ ; The second term is a periodic term related to time, the maximum value can reach$0.001658s$ ; The third term is the periodic term related to the spatial position of the atomic clock, the maximum value is only$2.1\mu \mathrm{s}$ 。

L 2 C , L 5 L_2 C , L_5 in GPS$L_{2}C、L_5$The new navigation messages up-modulated will involve concepts such as TDT (TT), TDB, etc., but due to space limitations, this chapter will not introduce in detail the derivation of various time systems and mutual conversion relations established under the framework of the theory of relativity. Interested students Refer to relevant resolutions of the IAU and reference materials such as space geodesy.

7. Some long time timing methods involved in GPS

在 GPS 导航和 GPS 测量中还会碰到一些计量长时间间隔的计时方法和计时单位, 如年 月日、儒略日和简化儒略日、年积日等, 它们有的涉及历法, 有的则是天文学中的一些术语。 虽然从严格意义上讲,这些内容已超出时间系统的范畴,但由于经常用到,因而也一并作一 介绍。

7.1. 历 法

历法是规定年、月、日的长度以及它们之间的关系、制定时间序列的一套法则。由于地球绕日公转周期和月球绕地球公转的周期均不为整天数, 而历法中规定的年和月的长度则只能为整天数, 所以需要有一套合适的方法来加以编排。目前, 各国使用的历法主要有阳历、阴阳历和阴历三种。
(1) 阳历 ( Solar Calendar)
阳历也称公历, 是以太阳的周年视运动为依据而制定的。太阳中心连续两次通过春分点所经历的时间间隔为一个回归年, 其长度为:
1 回归年 $=365.24218968-0.00000616\times t($ 日 $)$
其中, $t$ 为从 $\mathrm{J}2000.0$ 起算的儒略世纪数, 即
$$t=\frac{\mathrm{JD}-2451545.0}{36525}$$
The length of the regression year corresponding to January 1, 2009 is $365.24218913$ days.

The Julian Calendar
The Julian Calendar is a Gregorian calendar established by the Roman Emperor Julius Caesar in 46 BC. The calendar divides the year into 12 months. Among them, $1,3,5,7,8,10,12$ are big months, 31st of each month;$April,June,SeptemberandNovember$ are small months with 30 days in each month; February has 28 days in normal years and 29 days in leap years. A year that is divisible by 4 is considered a leap year, and a year that is not divisible by 4 is considered an ordinary year. According to the above stipulations, the length of an ordinary year is 365 days, and the length of a leap year is 366 days, and the average length is$365.25$ days. A Julian century is 36525 days. In astronomy and space geodesy, when calculating some parameters that change very slowly, the Julian century is often used as the unit.

The Gregorian Calendar The Gregorian
calendar is the current Gregorian calendar and is widely adopted by countries all over the world. In order to make the average length of each year as consistent as possible with the length of the regression year, in 1582, Pope Gregory modified the regulations for setting leap years in the Julian calendar, stipulating that for century years, only centuries divisible by 400 year is a leap year. In this way, years such as 1700, 1800, and 1900 are all leap years in the Julian calendar, but they all become normal years in the Gregorian calendar, and 2000 is a leap year. Thus, every 400 years in the Gregorian calendar has 3 days less than the 400 years in the Julian calendar. That is, 400 years in the Julian calendar has $365.25\times 400=There\; are\; 146,100$ days, and there are only 146,097 days in the 400 years of the Gregorian calendar. The average length of each year is$365.2425$ days, which is closer to the length of the tropical year.

(2) The lunar calendar (Lunar Calendar)
The lunar calendar is a calendar formulated according to the change cycle of the moon phase (synodic month). The calendar stipulates that a single month has 30 days, a double month has 29 days, and the average of each month is $29.5$ days, and the length of the synodic month$29.53059\dots The\; day$ is approaching. With the beginning of the new moon as the beginning of the month, 12 months make up a year, with a total of 354 days. And the length of 12 synodic months is$354.36708\cdots$ Day, 0.36708 more than the lunar calendar$0.36708...$ day. 30 years is$11.0124$ days. Therefore, every 30 years in the lunar calendar, 11 lunar years should be set, and the$2、5、7、$ 10 、 13 、 16 、 18 、 21 、 24 、 26 、 29 10 、 13 、 16 、 18 、 21 、 24 、 26 、 29 Add one day to the end of December in$10$$,$$13$$,$$16$$,$$18$$,$$21$$,$$24$$,$$26$$,\; and$$29\; ,\; that\; is,\; there\; are\; 355\; days\; in\; a\; leap\; year.$The Hijri calendar used by Islamic countries is a lunar calendar.

(3) Luni-Solar Calendar (Luni-Solar Calendar)
The Luni-Solar Calendar is a calendar that takes into account the characteristics of both the Gregorian and Lunar calendars. The year in the Luni-Solar calendar is based on the return year, while the month is based on the synodic month. The day, the small month is 29 days, the average monthly is $29.5$ days. In order to make the average length of the middle year of the lunar calendar close to the length of the tropical year, the calendar stipulates that 7 of every 19 years are leap years. One month added to a leap year is called a leap month. my country has used the lunar calendar for a long time, and adopted the solar calendar after 1912, but the lunar calendar was not abolished, and it is still widely used among the people, which is called the lunar calendar.

7.2. Julian day and simplified Julian day

(1) Julian Day (Julian Day, JD)
Julian Day is a long-term continuous calendar method that does not involve concepts such as year and month. It is often used in astronomy, space geodesy, and satellite navigation and positioning. This method was proposed by JJScaliger in 1583, named Julian Day in honor of his father Julian. This method is especially convenient when calculating the interval between two moments spanning many years. The starting point of the Julian day is 12h, January 1, 4713 BC, and it is added up day by day. In China's astronomical calendar, there is the Gregorian calendar in this year $\times \times$ month$\times \times$ 日与儒略日的对照表, 供用户查取。此外, 用户也可用下列公式来进行计算。
(1) 根据公历的年 (Y)、月 (M)、日 (D) 来计算对应的儒略日 JD 公式 1:
$$\begin{array}{rl}\mathrm{JD}=& 1721013.5+367\times Y-\mathrm{int}\left\{\frac{7}{4}\left[Y+\mathrm{int}\left(\frac{M+9}{12}\right)\right]\right\}\\ & +d+\frac{h}{24}+\mathrm{int}\left(\frac{275\times \mathrm{M}}{9}\right)\end{array}$$
式中, 常数 $1721013.5$ is January 1, 1 AD0 h 0 \mathrm{~h} Julian day at$0$$h$$Y,M,D$ are the year, month, and day numbers in the Gregorian calendar,$h$ is the hour in universal time, and int is the rounding symbol.

公式 2:
$$\begin{array}{rl}\mathrm{JD}=& \mathrm{int}(365.25\times y)+\mathrm{int}[30.6\\ & +D+\frac{h}{24}+1720981.5\end{array}$$
Current month number $M>2$ , there is$y=Y,m=M$;
$M⩽2$ , there is$y=Y-1,m=M+12$ 。

(2) 根据儒略日反求公历年 、月、日
$$\begin{array}{rl}& a=\mathrm{int}(\mathrm{JD}+0.5)\\ & b=a+1537\\ & c=\mathrm{int}\left[\frac{b-122.1}{365.25}\right]\\ & d=\mathrm{int}(365.25\times c)\\ & e=\mathrm{int}\left(\frac{b-d}{30.600}\right)\\ & D=b-d-\mathrm{int}(30.6001\times e)+\mathrm{FRAC}(\mathrm{JD}+0.5)\\ & M=e-1-12\times \mathrm{int}\left(\frac{e}{14}\right)\\ & Y=c-4715-\mathrm{int}\left(\frac{7+M}{10}\right)\end{array}$$
式中, 符号 $\mathrm{FRAC}(a)$ 表示取数值 $a$ 的小数部分。

IAU 决定从 1984 年起在计算.岁差、章动, 编制天体星表时都采用 J2000.0（即儒峈日 $2451545.0)$ 作为标准历元。任一时刻 $t$ 离标准历元的时间间隔即为 JD $(t)-2451545.0($ day$)$ . (2) Simplified Julian Day (Modified Julian Day, MJD) The starting point of Julian Day has been more than 67 centuries, and the current time expressed by Julian Day has a large value and is inconvenient to use. For this reason, in 1973, the IAU adopted a simpler continuous timekeeping method, the simplified Julian day. The relationship between it and the Julian day is:
$$\text{ MJD }=\mathrm{JD}-2400000.5$$
MJD is the midnight of November 17, 1858 ( JD$=2400000.5$ ) is a continuous timing method used as the starting point of timing. It is convenient to use MJD to indicate the most recent time.

(3) Annual accumulation day
Annual accumulation day is a continuous counting method used only in one year. Every year, January 1 is counted as day 1, February 1 is counted as day 32, and so on. December 31st in an ordinary year is the 365th day, and December 31st in a leap year is the 366th day. It can be used to easily calculate the two moments $t_1$and $t_2$time interval between.

From: Chapter Two of "GPS Measurement and Data Processing".