Precise Distance Measurement for a Near-Earth Asteroid by the Refined Rotational Reflex Velocity Method

Studying near-Earth asteroids (NEAs) is significant to both the formation and evolution of the solar system as well as impact defense. It is believed that NEAs originate from the interactions of celestial bodies in the solar system such as collisions with main-belt asteroids (MBAs; Morbidelli 1999). Later, the Yarkovsky effect is also found to be a contribution to their origin (Morbidelli & Vokrouhlický 2003). Research on the properties of these NEAs also reveal the evolution of the solar system. For example, it is confirmed that encounters with the Earth are the origin of fresh surfaces on NEAs (Binzel et al. 2010). Through spectral class analysis, the secret of the origin of NEA pairs can be found (Moskovitz et al. 2019). In addition to the origin and the evolution of NEAs, scientists are also concerned about the distances between NEAs and the Earth because NEAs might threaten the Earth. Potentially hazardous asteroids (PHAs) are a special kind of NEAs that are considered to possibly have great impact to the Earth. According to the Center for Near-Earth Object Studies, ⁴ PHAs are currently defined by the parameters used to evaluate their impact risks. In particular, all asteroids with an Earth minimum orbit intersection distance of less than 0.05 au (∼19.5 lunar distance) and an absolute magnitude (H) of less than 22.0 are considered PHAs. With the assumption that the albedo is 14%, the diameter of PHAs is larger than 140 m. In history, the Chicxulub impact, which happened ∼65.5 million years ago, is believed to be the cause of the great extinction event (Schulte et al. 2010). Also, a well-known event is the Tunguska event, which occurred in Siberia, Russia in 1908, has flattened an area of ∼2150 square kilometers of trees (Farinella et al. 2001). In 2013, the Chelyabinsk Airburst event occurred. A population of more than 1 million suffered from the impact and atmospheric explosion of a 20 m wide asteroid, which was the largest impact on Earth by an asteroid since 1908 (Popova et al. 2013). Recently, the impact events of asteroids 2008 TC₃ (Jenniskens et al. 2009) and 2022 EB₅ (Geng et al. 2023) are well known. PHAs are large enough to cause significant regional damage, which should be continuously monitored. For the purpose of discovering and monitoring asteroids, several survey programs such as the Panoramic Survey Telescope and Rapid Response System (Chambers & Pan-STARRS Team 2016), the Catalina Sky Survey (Christensen et al. 2018), and Asteroid Terrestrial-impact Last Alert System (Tonry et al. 2018) are carried out. Further, amateurs also join the NEA survey, and up to now over 2200 PHAs have been discovered according to the statistics of the Minor Planet Center. ⁵ However, it is a challenge to evaluate the impact risk of a newly discovered asteroid in a short time. Asteroids within the orbit of the Earth may be observed by a ground-based telescope as it is rushing to the Earth. In this way, accurate and precise distance measurement and impact risk evaluation are of great urgency.

The distance between asteroids and the Earth is the key information for the orbital determination and calculation of the asteroid physical parameters. Radar measurement provides a finer spatial resolution of NEAs than any other type of ground-based observation, which allows us to determine the size, shape, and rotation state of NEAs (Marshall et al. 2021). Radar measurement is a good method to explore the properties of PHAs, and it depends on advanced instruments and analytical technology. By contrast, ground-based telescope measurements are more widely used by astronomers. With ground-based optical-telescope measurements, we can calculate the distances of NEAs after orbital determination. As the topocentric parallax is the key information for near-Earth objects (Zhai et al. 2022), some scholars use alternative parallax methods by observing the same asteroid from different ground-based telescopes. Glukhovsky (2003) reported that their accuracy is better than 1% using three separated telescopes mounted on Connecticut, Netherlands, and California. Adolphson et al. (2015) also used the parallax method by observing the asteroid from two different ground-based telescopes with an accuracy better than 2%. However, sometimes this method is hard to carry out due to the bad weather or lack of observation permissions. Another method for asteroid distance measurement is using the observations from a single telescope over two nights. Alvarez & Buchheim (2012) derived the fitting model of diurnal parallaxes on equatorial coordinates, and their accuracy is better than 5%. Heinze & Metchev (2015; hereafter Paper I) proposed the rotational reflex velocity (RRV) method based on pure kinematics of the Earth with better accuracy than ∼1.6%. Later, Lin et al. (2016; hereafter Paper II) rederived the solution using spherical astronomy, which is easier to understand, and their accuracy has reached ∼0.9%. However, both aimed at measuring the distances of MBAs. In this work, we refined and extended the RRV method to measure the distances of NEAs (PHAs in particular), which has greater significance for impact defense. This measurement approach only requires observations over two nights by a single ground-based telescope. In practice, we reuse the observations of the PHA (99942) Apophis obtained by the 2.4 m telescope at Lijiang Station of Yunnan Observatory for the distance measurements. We also simulate the distance measurements of the four typical NEAs, (1221) Amor, (1862) Apollo, (2062) Aten, and (163693) Atira, at their discovery dates and follow-up dates. Our results show that this method can be used to accurately measure the distances of NEAs (PHAs in particular) in most cases.

The remainder of this paper is arranged as follows. Section 2 provides an in-depth description of the distance measurement method. In Section 3, we elaborate on the procedures of measuring the distance of the PHA Apophis. The results and the discussion are provided in Section 4. The conclusions are drawn in the last section.

The RRV method was originally used to measure the distances of MBAs in Paper I. It only needs two-night observations from a single ground-based telescope to derive an accurate distance. Paper I derived and explained the method based on the pure kinematics of the Earth and the asteroid concerned. In particular, they used the fact that the geocentric angular velocity ω_g when located at the geocenter is different from the topocentric angular velocity ω_o when located at the Earth's surface. Both ω_o and ω_g depend on the geocentric distance of the asteroid. ω_g is the angular velocity due to the Earth's revolution around the Sun, and ω_o is the angular velocity due to both the spin and revolution of the Earth. The difference of the two angular velocities (ω_g –ω_o ), which is also called RRV, is regarded as the angular velocity only due to the spin of the Earth. Later, Paper II rederived the solution according to classical spherical astronomy using calculus. In particular, they used the parallax relation in R.A. (Green 1985) to express the object's positions at topocenter and geocenter, as shown in Equation (1),

$$\begin{eqnarray}&&\alpha -{\alpha }^{{\prime} }=\displaystyle \frac{\rho }{r}\cos {\phi }^{{\prime} }\sin H\sec \delta ,\end{eqnarray} \tag{ 1 }$$

where α and ${\alpha }^{{\prime} }$ are the R.A. of the object before and after the influence of the geocentric parallax, respectively. ρ is the distance between the observer and geocenter. r is the distance between a certain asteroid and the geocenter. ${\phi }^{{\prime} }$ is the latitude of the observer, and H is the observed hour angle. δ is the object's decl. corresponding to α at the time of observation. As shown in Figure 1, the ω_g of the asteroid almost changes linearly with time in  R.A., and (ω_o − ω_g ) changes sinusoidally. Using the RRV method, we can derive the distance of an asteroid using only four frames observed on two nights, including two frames on the first night (observed at t_1a and t_1b ) and two frames on the second night (observed at t_2a and t_2b ). Here, subscripts a and b denote the first and second frames obtained on the same night, respectively. Subscripts 1 and 2 denote the first and second observation nights, respectively. Of course, we can also measure the distances using observations obtained several nights apart. For example, we use two frames obtained on one night (observed at t_1a and t_1b ) and two frames obtained three or four nights later (observed at t_2a and t_2b ). With the assumption that ω_g changes with time linearly, a more precise Equation (2) was derived using calculus. In Equation (2), ${\rho }^{{\prime} }$ is regarded as a constant, which is expressed in Equation (3). $\langle \cos {t}_{x}\rangle$ is a computation related to the time and hour angle. In particular, $\langle \cos {t}_{1}\rangle$ is expressed in Equation (4). Similar to Equation (4), $\langle \cos {t}_{2}\rangle =(\sin {H}_{2b}\,-\sin {H}_{2a})/({t}_{2b}-{t}_{2a})$, $\langle \cos {t}_{a}\rangle =(\sin {H}_{2a}-\sin {H}_{1a})/({t}_{2a}-{t}_{1a})$, and $\langle \cos {t}_{b}\rangle =(\sin {H}_{2b}-\sin {H}_{1b})/({t}_{2b}-{t}_{1b})$. In Equation (2), ω_x is the angular velocity, which can be calculated by the angular distance in R.A. and the time interval Δα/Δt. For example, ω_a = (α_2a − α_1a )/(t_2a − t_1a ) and ω₁ = (α_1b − α_1a )/(t_1b − t_1a ).

$$\begin{eqnarray}&&r={\rho }^{{\prime} }\displaystyle \frac{\left\langle \cos {t}_{1}\right\rangle -\left\langle \cos {t}_{a}\right\rangle +\left\langle \cos {t}_{2}\right\rangle -\left\langle \cos {t}_{b}\right\rangle }{{\omega }_{a}-{\omega }_{1}+{\omega }_{b}-{\omega }_{2}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{\rho }^{{\prime} }=\rho \cos {\phi }^{{\prime} }\sec \delta ,\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\langle \cos {t}_{1}\rangle =\displaystyle \frac{\sin {H}_{1b}-\sin {H}_{1a}}{{t}_{1b}-{t}_{1a}}.\end{eqnarray} \tag{ 4 }$$

Theoretically, the parallax formula in decl. can also be used to deduce an equation like Equation (2) but with larger errors. Therefore, we usually use the parallax relation in R.A. for calculations. Although Paper II obtained accurate distance results of MBAs using the RRV method, some parameters can be improved, and the method applications can be extended compared to Paper II, as we discuss below.

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First, the ${\rho }^{{\prime} }$ value used for calculations in Paper II was not clearly specified. Indeed, Paper II took the geodetic latitude as ${\phi }^{{\prime} }$ and the Earth radius as ρ. Strictly, the geocentric latitude (${\phi }^{{\prime} }$) and the distance between the observer and geocenter (ρ) should be used. According to the Explanatory Supplement to the Astronomical Almanac (Seidelmann & Urban 2010), ${\phi }^{{\prime} }$ and ρ can be accurately calculated using the geodetic longitude, latitude, and altitude of the observatory. For example, as for the location at the Lijiang Station of Yunnan Observatory, ρ and ${\phi }^{{\prime} }$ (used in this work) are 6377.112 km and 26°32${}^{{\prime} }$28'', respectively, after calculation while the geodetic latitude is 26°41${}^{{\prime} }$43''.

Second, both Papers I and II elaborate on the errors of ω₁ and ω₂ (the main errors of ω). This is expressed in Equation (5),

$$\begin{eqnarray}&&{\sigma }_{\omega }=\displaystyle \frac{{\sigma }_{\alpha }\sqrt{2}}{{t}_{b}-{t}_{a}},\end{eqnarray} \tag{ 5 }$$

where σ_α is the standard deviation of the measured R.A. and (t_b − t_a ) is the difference of the two observation time intervals on the same night. However, due to the change in the relative distances of NEAs, the errors derived from the variable distance of an asteroid r should be considered, which was regarded to be negligible in Papers I and II.

We derive the errors by the change in the distance r. Suppose that the distance of an asteroid changes linearly with time during the observation. Thus, we describe the speed along the distance $\dot{r}$. The time interval ΔT_m is the difference between t_2m and t_1m , and r_m is r at the mid-time of t_1m and t_2m . The distances of an asteroid at t_1m and t_2m can be expressed approximately by Equations (6) and (7).

$$\begin{eqnarray}&&{r}_{1}\approx {r}_{m}-\displaystyle \frac{\dot{r}{\rm{\Delta }}{T}_{m}}{2},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{r}_{2}\approx {r}_{m}+\displaystyle \frac{\dot{r}{\rm{\Delta }}{T}_{m}}{2}.\end{eqnarray} \tag{ 7 }$$

Considering the conditions above, Equations (6) and (7) in Paper II should be replaced by Equations (8) and (9).

$$\begin{eqnarray}&&{\omega }_{g}^{1a}+{\omega }_{g}^{1b}\approx 2\left({\omega }_{o1}+\displaystyle \frac{{\rho }^{{\prime} }\langle \cos {t}_{1}\rangle }{r}\left(1+\displaystyle \frac{\dot{r}{\rm{\Delta }}{T}_{m}}{2r}\right)\right),\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{\omega }_{g}^{2a}+{\omega }_{g}^{2b}\approx 2\left({\omega }_{o2}+\displaystyle \frac{{\rho }^{{\prime} }\langle \cos {t}_{2}\rangle }{r}\left(1-\displaystyle \frac{\dot{r}{\rm{\Delta }}{T}_{m}}{2r}\right)\right).\end{eqnarray} \tag{ 9 }$$

Let = $\tfrac{\dot{r}{\rm{\Delta }}{T}_{m}}{2r}$, which is usually very small (for example, = 0.01). The improved equation of the distance measurement should be Equation (10), in which the change in the distance is considered. The relative errors measured using Equation (2) can be estimated using Equation (11).

$$\begin{eqnarray}&&\begin{array}{l}r\,=\,{\rho }^{{\prime} }\displaystyle \frac{\left\langle \cos {t}_{1}\right\rangle (1+\epsilon )-\left\langle \cos {t}_{a}\right\rangle +\left\langle \cos {t}_{2}\right\rangle (1-\epsilon )-\left\langle \cos {t}_{b}\right\rangle }{{\omega }_{a}-{\omega }_{1}+{\omega }_{b}-{\omega }_{2}}.\end{array}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\sigma }_{r}=\displaystyle \frac{\epsilon (\left\langle \cos {t}_{2}\right\rangle -\left\langle \cos {t}_{1}\right\rangle )}{\left\langle \cos {t}_{1}\right\rangle +\left\langle \cos {t}_{2}\right\rangle -\left\langle \cos {t}_{a}\right\rangle -\left\langle \cos {t}_{b}\right\rangle }.\end{eqnarray} \tag{ 11 }$$

In fact, the actual errors will be larger than those estimated using Equation (11) because r and ω_g are not completely linear. To obtain an accurate distance of an asteroid, there are two ways for us to reduce or eliminate the errors from the change in r with the time. The first way is to calculate the value of so that we can measure the distance using the more accurate Equation (10). If we observe an asteroid for more than two nights, for example, for three successive nights, we can measure the distances at the two moments using the observations obtained on the successive two nights according to Equation (2). Using the observations obtained on the first and second nights, we can derive one distance at one moment; using the observations obtained on the second and third nights, we can derive another distance at another moment. Given that the speed along the distance asteroid $\dot{r}$ is linear with time, we can calculate the approximate $\dot{r}$. Once $\dot{r}$ is derived, can be calculated. The second way is to assume $\left\langle \cos {t}_{2}\right\rangle =\left\langle \cos {t}_{1}\right\rangle$ according to Equation (11), i.e.,

$$\begin{eqnarray}&&\displaystyle \frac{\sin {H}_{1b}-\sin {H}_{1a}}{{t}_{1b}-{t}_{1a}}=\displaystyle \frac{\sin {H}_{2b}-\sin {H}_{2a}}{{t}_{2b}-{t}_{2a}}.\end{eqnarray} \tag{ 12 }$$

In all cases during the two-night observation, because both $\tfrac{\sin {H}_{1b}-\sin {H}_{1a}}{{t}_{1b}-{t}_{1a}}$ and $\tfrac{\sin {H}_{2b}-\sin {H}_{2a}}{{t}_{2b}-{t}_{2a}}$ are greater than 0, the condition above is easily satisfied. However, the best condition is to observe an asteroid at two approximate hour angles. In particular, we should observe the asteroid when H_2a ≈ H_1a and H_2b ≈ H_1b . According to the assumptions and error estimation, the following factors are the key elements for obtaining the accurate and precise distances of asteroids.

• 1.  
  The relative distance changes little (i.e., $\tfrac{\dot{r}}{r}$ is small) and linearly.
• 2.  
  The ω_g value of the asteroid changes linearly during observation.
• 3.  
  If we observe an asteroid for more than three nights, we should ensure a longer time interval during one night and use Equation (10) for the distance measurement. If we observe for only two nights, one should follow the hour angle Equation (12) with a time interval as long as possible to reduce the error of ω.

From the derived Equation (10), we know that obtaining the accurate and precise distance does not mean that the asteroid is near its opposition. In fact, we can obtain an even more accurate and precise distance when the object is not near the opposition because ω_g is more significantly linear far from its opposition.

Generally, there are two aspects to discuss regarding the differences between this work and previous works (Papers I and II). The first aspect is the different target and case. Previous works found that the RRV method worked well for MBAs, but application to NEAs has not been tapped. In this work, we find that RRV method works well for both MBAs and NEAs. Further, previous works assumed the method only worked near the opposition of an asteroid, while we find that it also works well even if it is not close to the opposition. The second aspect is the improvement in the equation used for distance measurement. We improved the value of ${\rho }^{{\prime} }$ (using the geocentric latitude and the actual distance between the observer and geocenter). We also improved the errors by the change in the distance r (using a more precise equation with or follow the hour angle equation). In Section 4.1, we use the same observation of Apophis and compare the results in this work with the results obtained using the method in Paper II.

3.1. Observations of Apophis

Apophis is a PHA discovered by Bernardi, Tholen, and Tucker on 2004 June 19 at the Kitt Peak Observatory (Minor Planet Supplement 109613). It had been predicted that there would be an encounter with the Earth on 2029 April 13, with an impact probability of 2.7% , but later it was ruled out (Giorgini et al. 2008; Farnocchia et al. 2013). Some other encounter events were forecast in 2068, 2085, and 2088 (Souchay et al. 2018). Apophis is usually fainter than 20 mag (apparent magnitude) while it can be detected using the stacking method or observing with a large-diameter telescope (Guo et al. 2022). Nevertheless, Wang et al. (2015) and Thuillot et al. (2015) obtained precise astrometric positions when it is much brighter (<17 mag), in the year 2013.

In this work, we reuse the observations of Apophis published by Wang et al. (2015) for distance measurement, which was originally used for position measurement and astrometric calibration. The observations we used in this work were acquired in 2013 February by the 2.4 m telescope at Lijiang Station of Yunnan Observatory (IAU code: O44). The specifications of the telescope and the observations are shown in Tables 1 and 2, respectively. In order to ensure the accuracy of the distance measurement, we use the observations obtained with a large time interval for each night according to Equation (5). In particular, we used the frames observed at the beginning and the end of each night to measure the distance unless the frames were of poor quality. For the observations of each night (see Table 2), a total of 10 frames were used to measure the distance, including 5 frames at the start of observation and 5 frames at the end of observation. The exposure time ranges from 20 to 40 s using the B or I filter in the Johnson–Cousins system. On 2013 February 4, when Apophis was located before the meridian, no frame was observed. On the other three nights, we observed Apophis when it was located before and after the meridian.

Table 1. Specifications of the 2.4 m Telescope at Lijiang Station of Yunnan Observatory and the Attached CCD

Item	Parameter
Geodetic Position	100°1'51''E, 26°42'32''N
Focal Length	1920 cm
Diameter of Primary Mirror	240 cm
F Ratio	8
CCD Field of View	9' × 9'
Size of Pixel	13.5 × 13.5 μm
Size of CCD Array (Effective)	1900 × 1900
Approximate Scale Factor	∼0286 pixel−1

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Table 2. Observations Used to Measure the Distance of Apophis from 2013 February 4 to 7

Set Name	Frame ID	Date (UT)	JD	R.A. (h m s)	Decl. (d m s)	Hour Angle (deg)
A1	A11	2013-2-4	2456328.15250	07 08 21.289	−06 22 05.33	2.6406
	A12	2013-2-4	2456328.15290	07 08 21.215	−06 22 04.22	2.7829
	A13	2013-2-4	2456328.15330	07 08 21.142	−06 22 03.17	2.9253
	A14	2013-2-4	2456328.15369	07 08 21.067	−06 22 02.09	3.0718
	A15	2013-2-4	2456328.15409	07 08 20.994	−06 22 01.01	3.2142
A2	A21	2013-2-4	2456328.18531	07 08 15.242	−06 20 36.71	14.5064
	A22	2013-2-4	2456328.18577	07 08 15.155	−06 20 35.46	14.6739
	A23	2013-2-4	2456328.18623	07 08 15.071	−06 20 34.22	14.8372
	A24	2013-2-4	2456328.18669	07 08 14.987	−06 20 32.96	15.0047
	A25	2013-2-4	2456328.18715	07 08 14.903	−06 20 31.73	15.1721
B1	B11	2013-2-5	2456329.11376	07 05 54.831	−05 39 13.72	−9.7484
	B12	2013-2-5	2456329.11430	07 05 54.735	−05 39 12.28	−9.5558
	B13	2013-2-5	2456329.11482	07 05 54.648	−05 39 10.97	−9.3674
	B14	2013-2-5	2456329.11539	07 05 54.547	−05 39 09.38	−9.1623
	B15	2013-2-5	2456329.11592	07 05 54.452	−05 39 07.99	−8.9697
B2	B21	2013-2-5	2456329.24329	07 05 32.457	−05 33 29.15	37.1016
	B22	2013-2-5	2456329.24383	07 05 32.370	−05 33 27.79	37.2983
	B23	2013-2-5	2456329.24455	07 05 32.248	−05 33 25.85	37.5537
	B24	2013-2-5	2456329.24514	07 05 32.152	−05 33 24.34	37.7714
	B25	2013-2-5	2456329.24573	07 05 32.055	−05 33 22.76	37.9807
C1	C11	2013-2-6	2456330.12909	07 03 28.536	−04 54 42.28	−2.6223
	C12	2013-2-6	2456330.13006	07 03 28.378	−04 54 39.74	−2.2706
	C13	2013-2-6	2456330.13077	07 03 28.260	−04 54 37.88	−2.0153
	C14	2013-2-6	2456330.13139	07 03 28.159	−04 54 36.28	−1.7892
	C15	2013-2-6	2456330.13212	07 03 28.041	−04 54 34.40	−1.5255
C2	C21	2013-2-6	2456330.24602	07 03 09.604	−04 49 36.50	39.6677
	C22	2013-2-6	2456330.24658	07 03 09.514	−04 49 35.07	39.8686
	C23	2013-2-6	2456330.24730	07 03 09.400	−04 49 33.16	40.1281
	C24	2013-2-6	2456330.24784	07 03 09.317	−04 49 31.77	40.3248
	C25	2013-2-6	2456330.24845	07 03 09.221	−04 49 30.13	40.5466
D1	D11	2013-2-7	2456331.07949	07 01 22.240	−04 13 45.19	−19.0184
	D12	2013-2-7	2456331.08034	07 01 22.111	−04 13 43.05	−18.7087
	D13	2013-2-7	2456331.08230	07 01 21.807	−04 13 38.03	−18.0013
	D14	2013-2-7	2456331.08290	07 01 21.718	−04 13 36.56	−17.7837
	D15	2013-2-7	2456331.08350	07 01 21.624	−04 13 34.96	−17.5660
D2	D21	2013-2-7	2456331.13431	07 01 13.792	−04 11 25.18	0.8083
	D22	2013-2-7	2456331.13503	07 01 13.677	−04 11 23.28	1.0678
	D23	2013-2-7	2456331.13572	07 01 13.571	−04 11 21.47	1.3189
	D24	2013-2-7	2456331.13641	07 01 13.462	−04 11 19.67	1.5701
	D25	2013-2-7	2456331.13708	07 01 13.362	−04 11 18.02	1.8087

Note. A total of 40 frames were used, and they are shown in eight frame sets. The first and second columns are the set name and frame ID. The third column is the observed date in UT. The fourth column is the mid-exposure time in JD. The fifth and sixth columns are the measured R.A. and decl. of Apophis, respectively, according to Section 3.2. The last column shows the calculated apparent hour angle in degrees.

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During our observations, Apophis passed through the galactic belt, and there were enough reference Gaia stars (∼600 Gaia stars could be detected at each frame) for astrometric calibration. It was proximate to the Earth with a distance of ∼0.12 au. The changes in ω_g and ω_o in R.A. during the observation time according to JPL ephemeris are shown in Figure 1. The ω_g value of Apophis changes almost linearly in R.A., and (ω_o − ω_g ) changes sinusoidally. The mean relative distance on two successive nights is less than 2% (see the left panel of Figure 2). It was not near its opposition, and the solar phase angle was ∼35°.

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3.2. Astrometric Data Reduction

3.3. Distance Measurement

4.1. Distance Results

4.2. Simulations

According to the requirement of the RRV method, we simulated the distance measurement of NEAs at the time they were discovered. As we know, NEAs are divided into four groups (Amors, Apollos, Atens, and Atiras) according to their perihelion distances, aphelion distances, and semimajor axes of their orbits. We select the typical four NEAs, (1221) Amor, (1862) Apollo, (2062) Aten, and (163693) Atira, for simulation at their discovery dates and follow-up dates. The positions at the successive two nights (the discovery night and the following night) from JPL ephemeris are used after adding Gaussian noise. The standard deviation of the added Gaussian noise is 002 for both R.A. and decl. The time intervals ΔT₁ and ΔT₂ are both 1 hr. We suppose that observations were made by the 2.4 m telescope at Lijiang Station of Yunnan Observatory (details in Table 1). The results simulated for 10,000 iterations are shown in Figure 3 for each NEA. We find that δ is insensitive to the distance results, and the difference of the relative distance is usually less than 0.01% when δ is 1 arc minute different from the mean δ (at t_1a , t_1b , t_2a , and t_2b ). From the simulation results, we can obtain the accurate and precise distance of each NEA. The mean relative distance values of NEAs (1221) Amor, (1862) Apollo, (2062) Aten, and (163693) Atira are −0.0003 (−0.03%), −0.0140 (−1.40%), −0.0002 (−0.02%), and 0.0019 (0.19%), respectively. Their precision (the standard deviation of the mean relative distance) is estimated at 0.0013 (0.13%), 0.0020 (0.20%), 0.0015 (0.15%), and 0.0118 (1.18%).

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4.3. Discussion

In this work, we refined the RRV method and extend it further to the distance measurement of NEAs. This method can accurately and precisely measure the distances of MBAs and NEAs for most cases in an astrometric way without performing orbital determination. This method is not demanding for the positions of asteroids themselves, nor the time coordination or the location of the ground-based telescope. We can measure the distance of an asteroid when it is not near its opposition. In practice, we measure the distance of the PHA Apophis from the acquired CCD frames. Using 40 frames over four nights (10 frames per night), our results show that the total mean relative error for the independent exposure frames obtained on the successive two nights is ∼0.08% (about a factor of 2 improvement in comparison with that in Paper II). We also simulate the distance measurement for the four typical NEAs, Amor, Apollo, Aten, and Atira, at their discovery dates and follow-up dates. The simulation results are accurate and precise for observation time intervals of 1 hr each night. The simulation results show that, for the newly discovered asteroids, we can also derive the accurate and precise distances even if they are observed with a short time. The accurate and precise distances of NEAs computed using this method are important for the study of the dynamics and evolution of the solar system. More importantly, it is promising for Earth security and defense.

We are grateful for the group of the 2.4 m telescope of Lijiang Station, Yunnan Observatory who assisted in obtaining observations. We also deeply thank Prof. Zi Zhu and Dr. Hong Zhang from Nanjing University for valuable discussions with us. Also, we thank Ying Chen, Yaoyao Hong, Jianan Hao, Zhongjie Zheng, Xiao Chen, and Xueqing Fang for their help. This research is supported by the Key Special Project of the Ministry of the National Science and Technology (grant No. 2022YFE0116800), by the National Natural Science Foundation of China (grant Nos.11873026, 11273014, 12203019), by the Joint Research Fund in Astronomy (grant No.U1431227), by the science research grants from the China Manned Space Project with NO. CMS-CSST-2021-B08 and Excellent Postgraduate Recommendation Scientific Research Innovative Cultivation Program of Jinan University. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC; https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.