# After what time interval do the closest approaches of Mercury to the Earth repeat?

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The sidereal period of Mercury's revolution is 88 days and the synodic period - 116d.

my solution, but in the question featured "the greatest rapprochement." And this is no longer so easy. Because the orbit of Mercury is noticeably elongated and… in turn, it experiences rotation around the Sun. The effect is called "precession of the perihelion of the orbit of Mercury". We must take it into account. The closest approach of Mercury to Earth will be when it is at this point, and it, in turn, is on the Sun-Earth line.

The time of Mercury's revolution around the Sun is 88 days,

Earth around the Sun -365 days,

Let's say we are on Mercury and today there is an Earth opposition, the next one will be when Mercury returns to its original place (having made a circle in its orbit) it is 88 days, but these 88 days the Earth does not stand still, it has shifted in its orbit by 88/365 by about a quarter => Mercury will have to "catch up" with the Earth for 22 days, but during these 22 days the Earth will still shift in orbit by 22/365 approximately one sixteenth => another +5 days. Total 88 days + 22 days + 5 days = 115 days

Answer: oppositions of the Earth for an observer with Mercury are repeated with a frequency of 115 days

I found the answer 7.26 years but how do get it?

The text you quoted (is that from your textbook?) uses a simple approach; it counts every closest approach of Mercury to Earth, not just the ones which occur when Mercury is relatively far from the Sun (and hence closer to Earth). That means you don't need to account for

• the orbit of Mercury is noticeably elongated
• precession of the perihelion of the orbit of Mercury

Especially the latter effect is so small that you don't need to account for it; only very careful measurements at the end of the 19th century revealed there was something wrong with Newtonian mechanics.

The calculation itself is quite simple: Mercury has an angular velocity of $$frac{1}{87.97} ext{day}^{-1}$$ and Earth $$frac{1}{365.24} ext{day}^{-1}$$. If their closest approach is at $$t = 0, ext{day}$$, the next one occurs when Mercury did exactly one revolution more than Earth, so

$$frac{t}{87.97} = 1 + frac{t}{365.24}$$

$$t = 87.97 + 0.24 t$$

$$0.76t = 87.97$$

$$t = 115.75$$

so almost 116 days. I don't really like the quoted 'Achilles and the tortoise paradox'-style approach.

If you do want to take Mercury's rather elliptical orbit into account, you can note that after 3 times 115.75 = 347.25 days, the Earth (and hence Mercury) are at almost the same places in their orbit, so a 'relatively close' closest approach is followed by another one 347 days later.

@Glorfindel's explanation is very clear. I wondered about how big of an effect "everything else" would have, but I couldn't figure out how to add an image to a comment. Here is a graph from 2018 to 2023 of the distance between Mercury and Earth. You can see every 3rd minimum is a bit deeper than the others just like Glorfindel said. The details of each local minimum change in a somewhat irregular way (about 10% of the closest distance variation), but the broad pattern fits the simple explanation well.

Here is python code to generate such a graph yourself for a different time period. It uses skyfield to access JPL's data on the planetary positions and matplotlib to graph it.

from skyfield.api import load import numpy as np import matplotlib.pyplot as plt planets = load("de421.bsp") ts = load.timescale() times = ts.utc(2018,1,np.linspace(0,365*5,10000)) distances = (planets["Mercury"] - planets["Earth"]).at(times).distance() plt.plot(times.utc_datetime(), distances.km); plt.title("Distance from Mercury to Earth in km versus time"); plt.savefig("Mercury-Earth-2018-2023.png">ShareImprove this answeranswered Nov 30 '20 at 1:50Jack SchmidtJack Schmidt26066 bronze badges  Additional Science Textbook Solutions

Physics for Scientists and Engineers An Introduction to Physical Science Horizons: Exploring the Universe (MindTap Course List) Environmental Science (MindTap Course List) Understanding Nutrition (MindTap Course List) Nutrition Through the Life Cycle (MindTap Course List) Introduction to General, Organic and Biochemistry Nutrition: Concepts and Controversies - Standalone book (MindTap Course List)  Venus and Spica, November 2018 Venus seems to be chasing Spica into the sky. These charts show the pair about 30 minutes before sunrise.  November 4 (Figure 4): Only nine days after its conjunction, Venus rises an hour before sunrise this morning it is 4.4° below Spica with the waning crescent moon nearly 27° above Spica notice the contrast of brightness. Venus is about 100 times brighter than Spica.

Update: November 10, 2018  November 6 (Figure 5): Brilliant Venus is 3.6° below Spica with the waning crescent moon 9° to the left of the planet. Watch Venus close the gap on Spica during the next week. November 9: Venus rises at the beginning of twilight, about 100 minutes before sunrise. After today, Venus rises before the beginning of twilight until March 14, 2019.  Figure 6: Venus closes about one degree of Spica in mid-November. There is no conjunction but this is the closest approach — a quasi conjunction  November 14: (Figure 6): The Spica chase ends when brilliant Venus closes to 1.2° of the star. Venus does not pass Spica. Because the separation is less than 5°, this is known as a “quasi-conjunction.”

2 Model for Predicting >100 MeV SEP Events It has been widely accepted that interplanetary coronal mass ejections (CMEs) are the main drivers of interplanetary shocks. It is also widely accepted that the main accelerators of solar energetic particles (SEPs) are interplanetary CME-driven shocks [Tylka et al., 2005 Reames, 2004 ]. The physical role of flares in CME eruptions is still an elusive topic for the solar physics community. Marqué et al. [ 2006 ] and Klein et al. [ 2005 ] proposed a more important role for flares in SEP events. Recently, STEREO results [Richardson et al., 2014 ] on the longitudinal spread of high-energy SEP protons have shown that their relationship with CME shocks cannot be quite as simple as the community thought. Widespread SEP events appear to be more frequent than expected and show longitudinal broadnesses exceeding 300° to 360° [Dresing et al., 2014 ] with multispacecraft observations of 3 He, which are associated with flare acceleration processes, at longitudinal separation angles (of magnetic foot point to parent active region) >60° [Wiedenbeck et al., 2013 Dresing et al., 2014 ].

Although aspects of the physical role of flares in CME eruptions are well known, a quantitative relationship between flare signatures and CME energy is still an open topic of research. One possibility proposed by Chen and Kunkel [ 2010 ] is that the observed duration of soft X-ray emission is comparable to and scales with the duration of the associated CME's poloidal flux injection. This suggests that poloidal flux injection, the main driver of the CME eruption, is physically connected to the flare energy release therefore, the shock strength may be related to some characteristics of the associated flare.

For space weather operational purposes, CME and shock characteristics are less available in real time than quantitative flare observations. Since X-ray flux data may easily be obtained in real time, some empirical models have recently been developed to forecast interplanetary phenomena from flare data only, as a proxy for CME properties. For example, Núñez [ 2011 ] uses flare data and high-energy particle sensors at Earth to detect Sun-Earth magnetic connections along which high-energy particles arrive at the Earth and to predict the subsequent development of >10 MeV SEP events. These recently developed empirical forecasting systems have shown that flares may be used as predictors of interplanetary processes and may be used for operational purposes until physics-based systems are sufficiently developed. Well-connected SEPs follow the Parker spiral and may be detected within several minutes to hours of the solar event in the near-Earth environment. The UMASEP-10 model has shown that at the onset of a well-connected SEP event there is a correlation between the first derivative of the soft X-ray flux and the first derivative of at least one of the GOES differential proton flux channels. This model is also able to predict the subsequent development of the SEP event based on the solar flare signature. The UMASEP's main goal is to infer whether SEPs are abundant on the field lines that connect Sun and Earth. If UMASEP detects this situation, and a large solar flare takes place, then the model predicts an SEP event. UMASEP-10 and UMASEP-100 models correlate the solar activity and the near-Earth proton flux data but use different correlation approaches. While the UMASEP-10 model is based on the measurement of the delayed correlation of two real-value time series (i.e., X-ray and a differential proton channel), the UMASEP-100 model is based on the delayed correlation of two derived time series: the bit-based transformation of the X-ray flux and the bit-based transformation of a differential proton channel. Each of these derived time series are composed by bit values, showing the existence of strong positive derivatives (i.e., a “1”) or not (i.e., a “0”) (Figure 1).

Another difference between UMASEP-10 and UMASEP-100 is the input proton flux channels. UMASEP-10 model analyzes GOES soft X-ray flux and five GOES proton flux channels, from the GOES P3 channel (i.e., 9–15 MeV) to the GOES P7 channel (i.e., 165–500 MeV). UMASEP-100 analyzes the soft X-ray flux and six GOES proton channels, from the GOES P6 channel (i.e., 80–165 MeV) to the GOES P11 channel (i.e., >700 MeV). Another difference is that UMASEP-10 also has a mode for detecting poorly connected events. This mode has been found to be unnecessary for predicting >100 MeV SEP events.

We assume that high-energy particles could arrive at Earth through a magnetic connection when there is a GOES X-ray/proton channel flux correlation. When this correlation is high, and the associated flare is greater than a certain threshold, we predict that the >100 MeV proton flux will surpass the SWPC threshold (1 proton flux unit (pfu)) and predict its evolution for the next 3 h. In other words, the presented algorithm is looking for the onset of high-energy particles, and once it associates them with a strong solar event, it makes a prediction about the subsequent evolution of the event based on an empirical relationship between the GOES X-ray flux and the GOES energetic proton flux channels. The remainder of this section is organized as follows: section 2.1 presents the transformation of the X-ray flux and the differential proton flux into bit-valued time series that indicates the presence (or not) of positive derivatives exceeding critical thresholds. Section 2.2 presents an approach to construct a sequence of cause-consequence pairs between the bit-valued X-ray flux time series and each of the bit-valued differential proton flux channels. Section 2.3 presents an estimate of the cause-consequence correlation between these pairs. Section 2.4 describes the technique for selecting model parameters that optimize the performance of the prediction model.

2.1 Constructing Bit-Valued Time Series to Indicate Strong Positive Derivatives for Solar X-ray and Near-Earth Proton Data In the case of UMASEP-100, in order to detect a causal link between two very fast processes, we look for large positive derivatives. We postulate that certain events at the Sun (i.e., in terms of X-ray fluxes) are associated with a process that will accelerate particles that could arrive at Earth through a magnetic connection producing a rapid increase of differential proton flux values in the near-Earth environment. Every 5 min UMASEP-100 constructs a bit-valued time series for each differential proton channel. The time series show sequences of (yes/no) occurrences of sufficiently large derivatives which are constructed as follows: let us take the X-ray and the differential proton fluxes A and B, respectively. The first derivatives of the most recent sample of size L of these time series, da and db, are calculated and normalized to their maximum values, that is, sDA = <dat-L + 1/maxsDA, dat-L + 2/maxsDA,…, dat/maxsDA> and sDB = <dbt-L + 1/maxsDB, dbt-L + 2/maxsDB,…, dbt/maxsDB>, where maxsDA and maxsDB are the maximum values within the recent sample. The current >10 MeV model calculates the lag correlation between the real-valued sDA and sDB. The new >100 MeV model makes an additional transformation of sDA and sDB to construct two new time series + A and + B, with 1 for the values that surpass a specific percentage p of the maximum value in the present sequence and an absolute threshold d and 0 elsewhere. The threshold d is the minimum derivative value to consider it as strong (i.e., a 1), which is necessary to avoid the false detection of large fluctuations when low solar activity is present. From the two aforementioned intermediate time series, the UMASEP-100 model estimates the lag correlation of + A and + B.

UMASEP-100 constructs the + A and + B time series to identify the positive tails of the distribution of the first derivative time series sDA and sDB with the purpose of measuring the delayed similarity between them. If this real-time tail correlation measure (explained in section 2.3) is high, then we infer that there is evidence that there is causal linkage.

2.2 Identifying Cause-Consequence Pairs of X-ray and Proton Channel Fluctuations We assume that the solar-event-associated flux A influences the near-Earth associated flux B when the highest fluctuations of A are lag correlated with the highest fluctuations in B, and this correlation is expected within some lag thresholds Lmin and Lmax.  A condition regarding Lmin: we assume that particles might take, at least, 8.5 min to traverse the Sun-Earth distance at nearly the speed of light. In fact, light-speed protons could arrive nearly at the same time at 1 AU as the associated solar signature in electromagnetic phenomena (if at 0 pitch angle and neglecting the curvature of the Parker spiral). In other words, a first condition to accept a pair composed of the fluctuations i and j is that time(i) + Lmin, ≤time(j), where Lmin is 8.5 min. A condition regarding Lmax: we have empirically found that there is a very low probability that i may cause the fluctuation j after 5 h that is, the second condition to accept a pair composed of the fluctuations i and j is that time(j) ≤ time(i) + Lmax, where Lmax is 5 h. A condition regarding the accepted sequence order of fluctuations: we assume that a sequence of fluctuations in + A is lag correlated to a sequence of fluctuations in + B in other words, if there is a sequence of pairs x and y, in which the X-ray fluctuation of x occurs before the X-ray fluctuation of y, then the proton channel fluctuation of the pair x must also occur before the proton channel fluctuation of the pair y. If there is not yet a previously constructed pair, this condition is always met.

Each identified pair has a lag time, called here pairSeparation, which is time(j) − time(i), that is, the time from the large X-ray fluctuation to the time of the large proton channel fluctuation. The identification of a sequence of pairs does not imply any correlation, because the pairs could have very different lags, which reduces the degree of lag correlation as explained in section 2.3. Those fluctuations in + A and + B that do not fulfill the aforementioned conditions are unpaired fluctuations, also called odds in this section. Figure 2 illustrates the identification of pairs from the time series of large fluctuations in + A and + B. Note that i, the first fluctuation in + A (in red), is paired to j (also in red), because the j takes place after time(i) + Lmin (i.e., the first condition is met) and before time(i) + Lmax (i.e., the second condition is met). The third condition is also met because there is not yet a previously constructed pair. Note that the last fluctuation in + A cannot be paired because there is nothing left in + B to pair with, and the first two fluctuations in + B are unpaired fluctuations because there are no possible causing fluctuations in + A that meet the first aforementioned condition.

2.3 Empirical Estimation of the Solar X-ray and Near-Earth Proton Flux Correlations This section introduces a novel approach to estimate the cause-consequence correlation measure between large positive X-ray and large positive proton fluctuations. A high correlation measure should be obtained if a sequence of X-ray fluctuations is similar to a sequence of strong positive proton channel fluctuations and has the same lag time. Note that pairs are solely identified if both the causing fluctuation and the consequence fluctuation are equal to 1, and the three conditions explained in section 2.2 are met. The goal of the correlation measure FluctuationCorrelation will be higher as a fewer number of odds is encountered and more homogeneity in lag times of the discovered pairs (i.e., less standard deviation of the pair separation of the identified pairs) is estimated. In order to meet the aforementioned empirical design strategy, we define the FluctuationCorrelation measure as the product of a factor ∈ [0, 1] that assesses the proportion of identified pairs and odds and a factor ∈ [0, 1] related to the standard deviation of the discovered lags of the identified pairs.

(1) 2.4 Real-Time Inferences

After calculating the fluctuation correlations of X-ray and all differential proton channel fluxes, the maximum value is selected as the generalFluctuationCorrelation at time t. If the generalFluctuationCorrelation is greater than a threshold r, we assume that there is statistical evidence that the correlation is due to the occurrence of a magnetic connectivity of strength m given by the generalFluctuationCorrelation value. We also assume that PairSeparationMean plus light travel time from the Sun to Earth is the approximated transit time of protons along the magnetic field line that presumably connects the Sun with the Earth. If this X-ray flux peak is greater than a threshold f, then a >100 MeV SEP event forecast is issued. This X-ray flux peak is found by seeking the highest flux in A. Our model calibration is an optimization process with the goal of maximizing the performance metrics. The most common metrics for measuring the performance of SEP event predictors are the probability of detection (POD), false alarm ratio (FAR), and warning time. These metrics have been widely used in papers and presentations about SEP forecasters [Núñez, 2011 Balch, 2008 Laurenza et al., 2009 Posner, 2007 ]. In this paper POD = A/(A + C) and FAR = B/(A + B), where A is the number of correct forecasts, B is the number of false alarms, C is the number of missed events, and D is the number of correct nulls.

An optimization goal of the prediction model is that the POD is high another optimization goal is that the FAR is low (i.e., the Recall is high). The multigoal optimization strategy is to find an optimal configuration of the model thresholds L, p, d, r, and f that maximizes 0.5 · Precision + 0.5 · Recall (which we call a general forecasting performance), where Recall is the probability of detection, Precision is 1—false alarm ratio [Davis and Goadrich, 2006 ], L is the length of the analyzed time interval, p is the percentage for recognizing positive strong fluctuations (see section 2.1), d is the minimum derivative value to detect large fluctuations, r is the threshold of the minimum generalFluctuationCorrelation (see section 2.3), and f is the minimum X-ray flux of the associated flare for issuing a SEP prediction. In order to find the optimal solution, we ran candidate models, each one with a different threshold configuration, measuring the general forecasting performance for the period from January 1994 to September 2013. We found that the best candidate model was that one with L = 5 h, p = 81%, d = 0.6 × 10 −5 , r = 0.5, and f = M3.5.

It is important to say that the f threshold is directly related to the miss rate. The UMASEP-100 approach is not able to predict >100 MeV SEP events associated with flares whose peak is lower than M3.5, which includes those SEP events associated with faint solar activity behind the limb. Regarding the prediction of the intensity of the expected solar radiation storm, we found that the intensity is proportional to m (that is, to the highest correlation between X-ray and differential proton activity) we also found that the first hours of the >100 MeV integrated flux (up to 3 h of the SEP start time), I3h, is proportional to the time integrated soft X-ray flux s. We empirically concluded that a linear relation (i.e., I3h = a m + b s + c) did not work well. Alternatively, we manually explored several possible nonlinear combinations between m and s and we empirically concluded that the multiplication m.s is an important factor in predicting the intensity of the prompt component. Therefore, the formula we use for predicting the intensity I at 3 h after the SEP start time is I3h = a.(m.s) + b. Finally, we adjusted the coefficients a and b by minimizing the obtained mean absolute error (i.e., the mean of the absolute values of the differences between the forecasted intensity and the observed intensity).

After issuing an SEP prediction, UMASEP-100 retrieves the NOAA Event list file (ftp://ftp.swpc.noaa.gov/pub/indices/events/events.txt) with the details of the associated flare, that is, its active region and heliolongitude. This file collects solar events of different kinds, including the flares with its corresponding active regions and heliolocalization. The performance results of the current version of the >100 MeV model for the period from January 1994 to September 2013 are presented in the next section.

Regarding the necessary quality of the GOES data, note that the UMASEP-100 approach is principally based on the correct identification of the largest X-ray/proton fluctuations (see section 2.1). For this reason, the UMASEP-100 forecasting approach presented in section 2 is less sensitive (i.e., more robust) to GOES flux calibration issues [Smart and Shea, 1999 ].  The mass, gravity field, and ephemeris of Mercury This paper represents a final report on the gravity analysis of radio Doppler and range data generated by the Deep Space Network (DSN) with Mariner 10 during two of its encounters with Mercury in March 1974 and March 1975. A combined least-squares fit to Doppler data from both encounters has resulted in a determination of two second degree gravity harmonics, J2 = (6.0 ± 2.0) × 10 −5 and C22 = (1.0 ± 0.5) × 10 −5 , referred to an equatorial radius of 2439 km, plus an indication of a gravity anomaly in the region of closest approach of Mariner 10 to Mercury in March 1975 amounting to a mass deficiency of about GM = −0.1 km 3 sec −2 . An analysis is included that defends the integrity of previously published values for the mass of Mercury (H. T. Howard et al. 1974, Science 185, 179–180 P. B. Esposito, J. D. Anderson, and A. T. Y. Ng 1978, COSPAR: Space Res. 17, 639–644). This is in response to a published suggestion by R. A. Lyttleton (1980, Q. J. R. Astron. Soc. 21, 400–413 1981, Q. J. R. Astron. Soc. 22, 322–323) that the accepted values may be in error by more than 30%. We conclude that there is no basis for being suspicious of the earlier determinations and obtain a mass GM = 22,032.09 ± 0.91 km 3 sec −2 or a Sun to Mercury mass ratio of 6,023,600 ± 250. The corresponding mean density of Mercury is 5.43 ± 0.01 g cm −3 . The one-sigma error limits on the gravity results include an assessment of systematic error, including the possibility that harmonics other than J2and C22 are significantly different from zero. A discussion of the utility of the DSN radio range data obtained with Mariner 10 is included. These data are most applicable to the improvement of the ephemeris of Mercury, in particular the determination of the precession of the perihelion.

Professor Colombo contributed to this work while a Visiting Distinguished Scientist at JPL. He did not see the final version of the manuscript before his death in February 1984. He did review early drafts on the gravity field and the range fixes to Mercury. He had little or no interest in reanalyzing the data for the mass determination which he considered settled. The other authors are responsible for that.  Abstract

The paper contains the results of orbital evolution construction for Near-Earth asteroid (NEA) 137924 2000 BD19 with a small perihelion distance on the interval (−7500, 5000) years. Probabilistic orbital evolution of the asteroid was studied and some motion features, such as close and multiple approaches to Mercury and the Earth and the 3:4 mean motion resonance (MMR) with Venus, were revealed. Also, chaoticity of the NEA was estimated by indicator OMEGNO (Orthogonal Mean Exponential Growth factor of Nearby Orbits). The perturbations from planets, the Moon, relativistic effects of the Sun, the Sun oblateness, and the Yarkovsky effect were taken into account during the study of the asteroid dynamics. Since the Yarkovsky effect can have a significant influence on the asteroid motion due to the small perihelion distance, the paper presents the results of a study of its dynamics, both with and without this effect. The behavior of confidence regions has been analyzed and reasons for their changes have been revealed in both cases.

4 Empirical Model of the Index σb This section constructs a linear empirical model for σb on the sphere shown in Figure 10a. We first design a geodesic grid on the sphere comprising a cloud of nearly evenly separated points, see Figure 11a. Then the footprints of the spacecraft are binned according to their nearest geodesic point. In each bin, we estimate the coefficients , and of the following linear function through a least squares regression:  σb distribution over the dayside polar cap under (a) antisunward and (b) sunward IMF, with (c) their difference. (d–f, g–i, and j–l) Same as Figures 11a–11c but for downward and duskward IMF (±IMFy), southward and northern IMF(±IMFz), and at a minimum and maximum Mercury's heliocentric distance ( rmin and rmax ), respectively. These maps are estimated from the empirical model constructed in section 4, with IMF inputs specified on the top right corner in each panel. The “plus” and “minus” signs at the very top represent that the first two panels in a given row show the results of a pair of opposite condition, and the “delta” symbol represents the difference between the previous two columns. In Figure 11a, the crosses display the grid on which the regression is carried out separately. The size of the cross measures the number of samples used for the each regression, with a median of 1787. The blue dashed line shows a referring level σb = 1 nT. (2) Here t is the time when the data are collected, TM = 88.0 Earth days is the Mercury year, and i = is the imaginary unit. The harmonic term describes Mercury's orbit phase and thus the solar wind particle density and ram pressure as it changes with heliospheric distance [Zhong et al., 2015b ]. The regression implements a robust algorithm given by Huber and Ronchett [ 2009 ], and coefficients characterized by a confidence level below 0.99 are set to zero. The resultant coefficients are used to construct a linear model according to equation 2.

Note that, as explained in section 2.3, the IMF estimation captures only a part of the power in the IMF fluctuations. Accordingly, the IMF-dependent coefficients βIMF are also expected being underestimated, under the linear assumption of equation 2. An essential solution relies on simultaneous observations both in near Mercury orbit and in the solar wind, as will be conducted by BepiColombo [Glassmeier et al., 2010 ].

1. Introduction Systems of interacting agents arise in a wide variety of disciplines, including Physics, Biology, Ecology, Neurobiology, Social Sciences, and Economics (e.g., refs. 1 ⇓ ⇓ –4 and references therein). Agents may represent particles, atoms, cells, animals, neurons, people, rational agents, opinions, etc. The understanding of agent interactions at the appropriate scale in these systems is as fundamental a problem as the understanding of interaction laws of particles in Physics. How can laws of interaction between agents be discovered? In Physics, vast knowledge and intuition exist to formulate hypotheses about the form of interactions, inspiring careful experiments and accurate measurements, that together lead to the inference of interaction laws. This is a classical area of research, dating back to at least Gauss, Lagrange, and Laplace (5), that plays a fundamental role in many disciplines. In the context of interacting agents at the scale of complex organisms, there are fewer controlled experiments possible and few “canonical” choices for modeling the interactions. Different types and models of interactions have been proposed in different scientific fields and fit to experimental data, which in turn may suggest new modeling approaches, in a model–data validation loop. Often, the form of governing interaction laws is chosen a priori, within perhaps a small parametric family, and the aim is often to reproduce only qualitatively, and not quantitatively, some of the macroscopic features of the observed dynamics, such as the formation of certain patterns.

Our work fits at the boundary between statistical/machine learning and dynamical systems, where equations are estimated from observed trajectory data, and inference takes into account assumptions about the form of the equations governing the dynamics. Since the past decade, the rapidly increasing acquisition of data, due to decreasing costs of sensors and measurements, has made the learning of large and complex systems possible, and there has been an increasing interest in inference techniques that are model-agnostic and scalable to high-dimensional systems and large datasets.

We establish statistically sound, dynamically accurate, computationally efficient techniques* for inferring these interaction laws from trajectory data. We propose a nonparametric approach for learning interaction laws in particle and agent systems, based on observations of trajectories of the states (e.g., position, opinion, etc.) of the systems, on the assumption that the interaction kernel depends on pairwise distances only, unlike recent efforts that either require feature libraries or parametric forms for such interactions (6 ⇓ ⇓ ⇓ –10), or aim at identifying only the type of interaction from a small set of possible types (11 ⇓ –13). We consider a least-squares (LS) estimator, classical in the area of inverse problems (dating back to Legendre and Gauss), suitably regularized and tuned to the learning of the interaction kernel in agent-based systems.

The unknown is the interaction kernel, a function of pairwise distances between agents of the systems. While the values of this function are not observed, in contrast to the standard regression problems, we are able to show that our estimator converges at an optimal rate as if we were in the 1D regression setting. In particular, the learning rate has no dependency on the dimension of the state space of the system, therefore avoiding any curse of dimensionality, and making these estimators well-suited for the modern high-dimensional data regime. It may be easily extended to a variety of complex systems here, we consider first- and second-order models, with single and multiple types of agents, and with interactions with simple environments. We demonstrate with examples that the theoretical guarantees on the performance of the estimator make it suitable for testing hypotheses on underlying models of interactions, assisting an investigator in choosing among different possible (nonparametric) models.

Finally, our estimator is constructed with algorithms that are computationally efficient (with complexity O ( L N 2 M ) when the interaction kernel is Lipschitz SI Appendix, section 2F) and may be implemented in a streaming fashion: It is, therefore, well-suited for large datasets.  Mars closest to Earth October 6 Photo above: View at EarthSky Community Photos. | Paulette Haws captured the planet Mars on September 21, 2020, reflecting in Little Tupper Lake in New York state. Thanks, Paulette! Remember the historically close approach of Mars in 2003? At that time, Mars was closer than it had been in some 60,000 years. Mars was only slightly farther, but still very close, in 2018. On October 6, 2020, at about 14 UTC, Mars is closest for this two-year period, only a bit farther away than in 2003 or 2018. October 6 of this year presents Earth and Mars closer together than they will be again for another 15 years, or until September 2035. For the continental U.S. and Canada, Mars’ closest approach comes on October 6, 2020, at at 10 a.m. EDT, 9 a.m. CDT, 8 a.m. MDT, 7 a.m. PDT, 6 a.m. Alaskan Time and at 4 a.m. Hawaiian Time. At that precise time, Mars is about 38.57 million miles (62.07 million km) from us. Of course, these moments of closest approach are fleeting as both Earth and Mars move in their orbits around the sun.

At its 2003 close approach – on August 27, 2003 – Mars was 34.65 million miles (55.76 million km) away. At the 2018 close approach – on July 31, 2018 – Mars was 35.78 million miles (57.59 million km) away.

Mars won’t beat its 2003 performance until until August 28, 2287, when the red planet will be 34.60 miles (55.69 million km) away. But, as we said above, Earth and Mars are closer on October 6, 2020 than they will be again for another 15 years, or until September 2035. Have you seen Mars yet? You can see it easily with the eye alone as the resplendent red “star” in the east every evening, and in the west before dawn. In fact, dazzling Mars is easily the brightest starlike object to light up the evening sky. Only the planet Venus – the third-brightest celestial object, after the sun and moon – beams brighter than Mars. Yet Mars lords over the nighttime from evening until dawn, whereas Venus is relegated to the eastern morning sky.   View at EarthSky Community Photos. | Veteran meteor observer Eliot Herman in Tucson used an automatic all-sky camera to capture this cool image of a bright meteor and Mars over Tucson, Arizona, on September 22, 2020. He wrote: “Looks like it was shot from Mars – not really, of course – but it does look like Mars shot it toward Earth. First time I have caught such a conjunction.” View this image full-sized. Thank you, Eliot!

Mars is closest in spite of the fact that Earth will swing between Mars and the sun at its opposition on October 13, 2020. Why aren’t we closest to Mars on the day we pass between it and the sun? If both the Earth and Mars circled the sun in perfect circles, and on the same exact plane, the distance between Earth and Mars would always be least on the day of Mars’ opposition. But we don’t live in such a symmetrical universe. All planets have elliptical orbits and a perihelion (closest point) and aphelion (farthest point) from the sun. Mars’ orbit around the sun takes 687 days in contrast to 365 days for Earth. It has a year nearly twice as long as ours. Earth’s farthest point from the sun comes yearly, in early July. Mars was at its closest to the sun on August 3, 2020. Ever since July 4, 2020, Earth has been moving closer to the sun and ever since August 3, 2020, Mars has been edging away from the sun.

At its opposition on October 13 – when Earth will be directly between Mars and the sun – Mars will be farther from than sun than on October 6, 2020. On the other hand, Earth will be closer to the sun (and therefore farther from Mars) on October 13 than on October 6. That all adds up to Earth being slightly closer to Mars on October 6 than October 13. The time interval between a Mars opposition and its least distance from Earth can be as long as 8.5 days (1969), or as little as 10 minutes (2208 and 2232). Generally speaking, Mars is at its brightest in 2020 throughout the month of October 2020. It is now shining more brilliantly than the planet Jupiter, and it’s not very often that Mars outshines the king planet!

Artist’s concept of the orbits of Earth and Mars, via NASA. Is Mars brightest when it’s closest? Not necessarily. You might think Mars should be brighter when closest to Earth on October 6 than at opposition on October 13. But it’s not (although it’s still plenty bright). Mars is a tiny bit fainter now than it will be at its October 13 opposition. That’s because of something known as opposition surge. Mars reflects sunlight most directly back to Earth at opposition. This directness accentuates Mars’ brilliance. Before and after opposition, sunlight is reflected at a slightly slanted angle relative to Earth, thereby reducing Mars’ brightness. Earth swings between Mars and the sun every other year, at progressively later dates. Earth will next lap Mars on December 8, 2022. Its closest approach to Earth that year will be December 1, 2022. After that, Earth will next lap Mars on January 16, 2025, but its closest approach will come on January 12, 2025. At both of those oppositions of Mars – and at every opposition for some years to come – Mars will appear fainter, and fainter, in our sky. That’s because those oppositions will happen closer and closer to Mars’ aphelion date.

In the year 2027, Mars’ opposition comes on February 19, 2027, and Mars sweeps closest to Earth on February 20, 2929. At a distance of 63.02 million miles (101.42 million km), this will present Mars’s most distant opposition in the 21st century (2001 to 2100). Mars reaches aphelion – it farthest distance from the sun – on March 2, 2027. So enjoy Mars in October 2020! You won’t see it this bright again until September 2035.

Mars is out almost all night long now. It looks like a bright reddish “star,” shining with a steadier light than the true stars. In mid-October 2020, look for Mars in the east at nightfall – highest in the sky near midnight – and in the west as morning dawn starts to light the sky. Clouded out tonight? Look tomorrow or the next night! Mars will remain dazzlingly bright in our sky for all of October.   View at EarthSky Community Photos. | The telescopic view of Mars is at its best now! Marcelo Barbosa in Texas captured this telescopic image of Mars on September 27, 2020. Thank you, Marcelo! Bottom line: The Mars opposition – when Earth flies between the sun and Mars – comes on October 13, 2020. But the distance between Mars and Earth is least on October 6, 2020. You can see Mars easily with the eye alone. It looks like a bright red “star” in the east every evening, in the west before dawn.

Mystery reddish orange object Could a satellite look twice as bright as Venus and have a very reddish color to it? I am puzzled to say the least. Yes. You may have seen a very rare and, in my opinion, beautiful satellite known as Trumpet 3. I've written about it in various posts here but seeing that this is mostly a "faint fuzzy" kind of audience it gets looked over. Trumpet 3/USA 136 has a 500 foot-wide mesh antenna. Due to the coating on the antenna it gives off a reddish color. I've only spotted it once. If for some reason what you saw was not Trumpet 3 it could have been its two sisters. The only other reddish-colored large satellites on a polar orbit I know of color are USA 133 and USA 152. They will only appear to be red if the sun hits the radar antenna at just the right angle. The reason you didn't see any of these satellites in Stellarium is because they get their TLEs from a source which doesn't have the intelligence gathering satellites in the catalogue. You need to use different software to get that list.

Congrats on spotting one of these birds. Quite a sight isn't it? #3 BrooksObs Your sighting, particularly since you indicate great brilliance, multiple surfaces in close proximity visible with optical aid, an abrupt disappeance, and its speed of motion all sound very much like what one expects when seeing reflections off a high flying jet a while past sunset. Just how long after sunset would be key here to a better understanding of the circumstances surrounding the event. Certainly, I've witnessed almost precisely the same sort of event many times with aircraft viewed in twilight. #4 Qwickdraw #5 Qwickdraw Your sighting, particularly since you indicate great brilliance, multiple surfaces in close proximity visible with optical aid, an abrupt disappeance, and its speed of motion all sound very much like what one expects when seeing reflections off a high flying jet a while past sunset. Just how long after sunset would be key here to a better understanding of the circumstances surrounding the event. Certainly, I've witnessed almost precisely the same sort of event many times with aircraft viewed in twilight.

The sun was set for about 15 minutes. No contrail whatsoever. I see plenty of jets and coincidently I just observed one in my scope a few minutes before this, found its contrail in my finder and followed it to the jet. The other thing is almost all jets in my Ann Arbor,mi area fly east/west. I would say this was precisely from N to S. I truly believe I could have distinguished a jet in my scope.  #6 BrooksObs That your sighting came just 15 minutes after sunset almost assures this was nothing more than a very high-flying jet aircraft. No very large flat surface on a satellite could maintain an extremely bright reflection angle for a given ground location over the time interval and distance travelled that you indicate. Likewise, it need not have been a commercial aircraft and contrails only form at altitudes where moisture is sufficient. Dry air often found at higher altitudes will not support them.

#7 Qwickdraw That your sighting came just 15 minutes after sunset almost assures this was nothing more than a very high-flying jet aircraft. No very large flat surface on a satellite could maintain an extremely bright reflection angle for a given ground location over the time interval and distance travelled that you indicate. Likewise, it need not have been a commercial aircraft and contrails only form at altitudes where moisture is sufficient. Dry air often found at higher altitudes will not support them. #8 GlennLeDrew A jet at any altitude, when illuminated by the Sun, is obviously so with any optical aid. This observation does seem to be of a satellite.

What magnification was used? Compared to a familiar object like Jupiter or Saturn at that magnification, what was its apparent size? What do you estimate its maximum elevation above the horizon to have been? I take it that so soon after sunset no stars were yet visible?  #9 rdandrea #10 BrooksObs A jet at any altitude, when illuminated by the Sun, is obviously so with any optical aid. This observation does seem to be of a satellite. May I point out the fact that the OP implies an object of considerable negative magnitude (-3 at a minimum since he states it was the brightest thing in the sky and Venus is more than -3.5 currently) for it to be noticed accidentally, yet quickly apparent to the eye, just 15 minutes after sunset. He also indicates the object maintained this level of brightness while traversing a significant span of sky.

The only satellite capable of maintaining such an obviously brilliant luster over the course of two or more minutes might be the ISS in a quite favorable pass. Beyond that object, a satellite as the source of the sighting can be ruled out. Even an Iridium satellite will not continue at a considerable negative magnitude for more than than 10-20 seconds. Likewise, as I've already pointed out, changing orientation relative to the observer of any moving satellite's large flat surface will only create a short duration flaring. Only the ISS might serve as an explanation and that should be easily checked for. Likewise, since the object showed some obvious structure telescopically, again only the ISS would be a reasonable possibility. So if not the ISS then this was no satellite and more likely a very high flying jet whose appearance seemed distorted by it giving off multiple reflections of the setting Sun from its body, wings and/or engines. #11 GlennLeDrew How about a sounding balloon? (Although the angular rate of motion might have been on the high side.) Again, an aircraft passing some 45 degrees above the horizon will present a pretty unambiguous planform at binocular magnifications, let alone that provided by an 8" aperture. I once tracked a jet at cruising altitude (with contrail) with a 60mm refractor at 60X, and clearly saw a brief cloud of spray coming out the rear underside, suggesting a toilet dump (?). Windows were resolved, too. It's a neat sight, seeing such details on a plane up in the lower stratosphere!  #12 Qwickdraw A jet at any altitude, when illuminated by the Sun, is obviously so with any optical aid. This observation does seem to be of a satellite. What magnification was used? Compared to a familiar object like Jupiter or Saturn at that magnification, what was its apparent size? What do you estimate its maximum elevation above the horizon to have been? I take it that so soon after sunset no stars were yet visible? Glenn,I was using an 8" F6 newt with a 26mm EP so I figure a magnification of about X47. Apparent size filled about a 10th of the FOV so I believe that would be at least twice as large as jupiter. Looking at Stellarium it was at about 50 deg wwhen it was due west and then I lost it. My first thoughts were it was an Iridium flare and I did not know they could be red so once I lost it I didnt look to recover it again. It may have still been in view but not bright.

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