A Medley of Potpourri: Oct 4, 2023

Wednesday, October 4, 2023

Water quality

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Water_quality

Water quality refers to the chemical, physical, and biological characteristics of water based on the standards of its usage. It is most frequently used by reference to a set of standards against which compliance, generally achieved through treatment of the water, can be assessed. The most common standards used to monitor and assess water quality convey the health of ecosystems, safety of human contact, extent of water pollution and condition of drinking water. Water quality has a significant impact on water supply and oftentimes determines supply options.

Sampling and measurement

Sample collection

The complexity of water quality as a subject is reflected in the many types of measurements of water quality indicators. Some measurements of water quality are most accurately made on-site, because water exists in equilibrium with its surroundings. Measurements commonly made on-site and in direct contact with the water source in question include temperature, pH, dissolved oxygen, conductivity, oxygen reduction potential (ORP), turbidity, and Secchi disk depth.

Sampling of water for physical or chemical testing can be done by several methods, depending on the accuracy needed and the characteristics of the contaminant. Sampling methods include for example simple random sampling, stratified sampling, systematic and grid sampling, adaptive cluster sampling, grab samples, semi-continuous monitoring and continuous, passive sampling, remote surveillance, remote sensing, and biomonitoring. The use of passive samplers greatly reduces the cost and the need of infrastructure on the sampling location.

Many contamination events are sharply restricted in time, most commonly in association with rain events. For this reason "grab" samples are often inadequate for fully quantifying contaminant levels. Scientists gathering this type of data often employ auto-sampler devices that pump increments of water at either time or discharge intervals.

More complex measurements are often made in a laboratory requiring a water sample to be collected, preserved, transported, and analyzed at another location.

Issues

The process of water sampling introduces two significant problems:

The first problem is the extent to which the sample may be representative of the water source of interest. Water sources vary with time and with location. The measurement of interest may vary seasonally or from day to night or in response to some activity of man or natural populations of aquatic plants and animals. The measurement of interest may vary with distances from the water boundary with overlying atmosphere and underlying or confining soil. The sampler must determine if a single time and location meets the needs of the investigation, or if the water use of interest can be satisfactorily assessed by averaged values of sampling over time and location, or if critical maxima and minima require individual measurements over a range of times, locations or events. The sample collection procedure must assure correct weighting of individual sampling times and locations where averaging is appropriate. Where critical maximum or minimum values exist, statistical methods must be applied to observed variation to determine an adequate number of samples to assess the probability of exceeding those critical values.
The second problem occurs as the sample is removed from the water source and begins to establish chemical equilibrium with its new surroundings – the sample container. Sample containers must be made of materials with minimal reactivity with substances to be measured; pre-cleaning of sample containers is important. The water sample may dissolve part of the sample container and any residue on that container, and chemicals dissolved in the water sample may sorb onto the sample container and remain there when the water is poured out for analysis. Similar physical and chemical interactions may take place with any pumps, piping, or intermediate devices used to transfer the water sample into the sample container. Water collected from depths below the surface will normally be held at the reduced pressure of the atmosphere; so gas dissolved in the water will collect at the top of the container. Atmospheric gas above the water may also dissolve into the water sample. Other chemical reaction equilibria may change if the water sample changes temperature. Finely divided solid particles formerly suspended by water turbulence may settle to the bottom of the sample container, or a solid phase may form from biological growth or chemical precipitation. Microorganisms within the water sample may biochemically alter concentrations of oxygen, carbon dioxide, and organic compounds. Changing carbon dioxide concentrations may alter pH and change solubility of chemicals of interest. These problems are of special concern during measurement of chemicals assumed to be significant at very low concentrations.

Filtering a manually collected water sample (grab sample) for analysis

Sample preservation may partially resolve the second problem. A common procedure is keeping samples cold to slow the rate of chemical reactions and phase change, and analyzing the sample as soon as possible; but this merely minimizes the changes rather than preventing them. A useful procedure for determining influence of sample containers during delay between sample collection and analysis involves preparation for two artificial samples in advance of the sampling event. One sample container is filled with water known from previous analysis to contain no detectable amount of the chemical of interest. This sample, called a "blank", is opened for exposure to the atmosphere when the sample of interest is collected, then resealed and transported to the laboratory with the sample for analysis to determine if sample collection or holding procedures introduced any measurable amount of the chemical of interest. The second artificial sample is collected with the sample of interest, but then "spiked" with a measured additional amount of the chemical of interest at the time of collection. The blank (negative control) and spiked sample (positive control) are carried with the sample of interest and analyzed by the same methods at the same times to determine any changes indicating gains or losses during the elapsed time between collection and analysis.

Testing in response to natural disasters and other emergencies

After events such as earthquakes and tsunamis, there is an immediate response by the aid agencies as relief operations get underway to try and restore basic infrastructure and provide the basic fundamental items that are necessary for survival and subsequent recovery. The threat of disease increases hugely due to the large numbers of people living close together, often in squalid conditions, and without proper sanitation.

After a natural disaster, as far as water quality testing is concerned, there are widespread views on the best course of action to take and a variety of methods can be employed. The key basic water quality parameters that need to be addressed in an emergency are bacteriological indicators of fecal contamination, free chlorine residual, pH, turbidity and possibly conductivity/total dissolved solids. There are many decontamination methods.

After major natural disasters, a considerable length of time might pass before water quality returns to pre-disaster levels. For example, following the 2004 Indian Ocean tsunami the Colombo-based International Water Management Institute (IWMI) monitored the effects of saltwater and concluded that the wells recovered to pre-tsunami drinking water quality one and a half years after the event. IWMI developed protocols for cleaning wells contaminated by saltwater; these were subsequently officially endorsed by the World Health Organization as part of its series of Emergency Guidelines.

Chemical analysis

The simplest methods of chemical analysis are those measuring chemical elements without respect to their form. Elemental analysis for oxygen, as an example, would indicate a concentration of 890 g/L (grams per litre) of water sample because oxygen (O) has 89% mass of the water molecule (H₂O). The method selected to measure dissolved oxygen should differentiate between diatomic oxygen and oxygen combined with other elements. The comparative simplicity of elemental analysis has produced a large amount of sample data and water quality criteria for elements sometimes identified as heavy metals. Water analysis for heavy metals must consider soil particles suspended in the water sample. These suspended soil particles may contain measurable amounts of metal. Although the particles are not dissolved in the water, they may be consumed by people drinking the water. Adding acid to a water sample to prevent loss of dissolved metals onto the sample container may dissolve more metals from suspended soil particles. Filtration of soil particles from the water sample before acid addition, however, may cause loss of dissolved metals onto the filter. The complexities of differentiating similar organic molecules are even more challenging.

Making these complex measurements can be expensive. Because direct measurements of water quality can be expensive, ongoing monitoring programs are typically conducted and results released by government agencies. However, there are local volunteer programs and resources available for some general assessment. Tools available to the general public include on-site test kits, commonly used for home fish tanks, and biological assessment procedures.

Biosensors

Biosensors have the potential for "high sensitivity, selectivity, reliability, simplicity, low-cost and real-time response". For instance, bionanotechnologists reported the development of ROSALIND 2.0, that can detect levels of diverse water pollutants.

Real-time monitoring

Although water quality is usually sampled and analyzed at laboratories, since the late 20th century there has been increasing public interest in the quality of drinking water provided by municipal systems. Many water utilities have developed systems to collect real-time data about source water quality. In the early 21st century, a variety of sensors and remote monitoring systems have been deployed for measuring water pH, turbidity, dissolved oxygen and other parameters. Some remote sensing systems have also been developed for monitoring ambient water quality in riverine, estuarine and coastal water bodies.

The following is a list of indicators often measured by situational category:

Alkalinity
Color of water
pH
Taste and odor (geosmin, 2-Methylisoborneol (MIB), etc.)
Dissolved metals and salts (sodium, chloride, potassium, calcium, manganese, magnesium)
Microorganisms such as fecal coliform bacteria (Escherichia coli), Cryptosporidium, and Giardia lamblia; see Bacteriological water analysis
Dissolved metals and metalloids (lead, mercury, arsenic, etc.)
Dissolved organics: colored dissolved organic matter (CDOM), dissolved organic carbon (DOC)
Radon
Heavy metals
Pharmaceuticals
Hormone analogs

Environmental indicators

Physical indicators

Water temperature
Specific conductance or electrical conductance (EC) or conductivity
Total suspended solids (TSS)
Transparency or turbidity
Water clarity

Total dissolved solids (TDS)
Odour of water
Color of water (such as Forel-Ule scale or Pt/Co scale)
Taste of water

Chemical indicators

Biological indicators

Biological monitoring metrics have been developed in many places, and one widely used family of measurements for freshwater is the presence and abundance of members of the insect orders Ephemeroptera, Plecoptera and Trichoptera (EPT) (of benthic macroinvertebrates whose common names are, respectively, mayfly, stonefly and caddisfly). EPT indexes will naturally vary from region to region, but generally, within a region, the greater the number of taxa from these orders, the better the water quality. Organisations in the United States, such as EPA. offer guidance on developing a monitoring program and identifying members of these and other aquatic insect orders. Many US wastewater dischargers (e.g., factories, power plants, refineries, mines, municipal sewage treatment plants) are required to conduct periodic whole effluent toxicity (WET) tests.

Individuals interested in monitoring water quality who cannot afford or manage lab scale analysis can also use biological indicators to get a general reading of water quality. One example is the IOWATER volunteer water monitoring program of Iowa, which includes an EPT indicator key.

Bivalve molluscs are largely used as bioindicators to monitor the health of aquatic environments in both fresh water and the marine environments. Their population status or structure, physiology, behaviour or the level of contamination with elements or compounds can indicate the state of contamination status of the ecosystem. They are particularly useful since they are sessile so that they are representative of the environment where they are sampled or placed. A typical project is the U.S. Mussel Watch Programme, but today they are used worldwide.

The Southern African Scoring System (SASS) method is a biological water quality monitoring system based on the presence of benthic macroinvertebrates (EPT). The SASS aquatic biomonitoring tool has been refined over the past 30 years and is now on the fifth version (SASS5) which has been specifically modified in accordance with international standards, namely the ISO/IEC 17025 protocol. The SASS5 method is used by the South African Department of Water Affairs as a standard method for River Health Assessment, which feeds the national River Health Programme and the national Rivers Database.

Climate change impacts

Weather and weather-related shocks can affect water quality in several ways. These depend on the local climate and context. Weather-related shocks include water shortages, heavy rain and temperature extremes. They can damage water infrastructure through erosion under heavy rainfall and floods, cause loss of water sources in droughts, and make water quality deteriorate.

Climate change can reduce lower water quality in several ways:

Heavy rainfall can rapidly reduce the water quality in rivers and shallow groundwater. It can affect water quality in reservoirs even if these effects can be slow. Heavy rainfall also impacts groundwater in deeper, unfractured aquifers. But these impacts are less pronounced. Rainfall can increase fecal contamination of water sources.
Floods after heavy rainfalls can mix floodwater with wastewater. Also pollutants can reach water bodies by increased surface runoff.
Groundwater quality may deteriorate due to droughts. The pollution in rivers that feed groundwater becomes less diluted. As groundwater levels drop, rivers may lose direct contact with groundwater.
In coastal regions, more saltwater may mix into freshwater aquifers due to sea level rise and more intense storms. This process is called saltwater intrusion.
Warmer water in lakes, oceans, reservoirs and rivers can cause more eutrophication. This results in more frequent harmful algal blooms. Higher temperatures cause problems for water bodies and aquatic ecosystems because warmer water contains less oxygen.
Permafrost thawing leads to an increased flux of contaminants.
Increased meltwater from glaciers may release contaminants. As glaciers shrink or disappear, the positive effect of seasonal meltwater on downstream water quality through dilution is disappearing.

Standards and reports

In the setting of standards, agencies make political and technical/scientific decisions based on how the water will be used. In the case of natural water bodies, agencies also make some reasonable estimate of pristine conditions. Natural water bodies will vary in response to a region's environmental conditions, whereby water composition is influenced by the surrounding geological features, sediments, and rock types, topography, hydrology, and climate. Environmental scientists and aqueous geochemists work to interpret the parameters and environmental conditions that impact the water quality of a region, which in turn helps to identify the sources and fates of contaminants. Environmental lawyers and policymakers work to define legislation with the intention that water is maintained at an appropriate quality for its identified use.

Another general perception of water quality is that of a simple property that tells whether water is polluted or not. In fact, water quality is a complex subject, in part because water is a complex medium intrinsically tied to the ecology, geology, and anthropogenic activities of a region. Industrial and commercial activities (e.g. manufacturing, mining, construction, transport) are a major cause of water pollution as are runoff from agricultural areas, urban runoff and discharge of treated and untreated sewage.

International

The World Health Organization (WHO) published updated guidelines for drinking-water quality (GDWQ) in 2017.
The International Organization for Standardization (ISO) published regulation of water quality in the section of ICS 13.060, ranging from water sampling, drinking water, industrial class water, sewage, and examination of water for chemical, physical or biological properties. ICS 91.140.60 covers the standards of water supply systems.

National specifications for ambient water and drinking water

European Union

The water policy of the European Union is primarily codified in three directives:

Directive on Urban Waste Water Treatment (91/271/EEC) of 21 May 1991 concerning discharges of municipal and some industrial wastewaters;
The Drinking Water Directive (98/83/EC) of 3 November 1998 concerning potable water quality;
Water Framework Directive (2000/60/EC) of 23 October 2000 concerning water resources management.

India

Indian Council of Medical Research (ICMR) Standards for Drinking Water.

South Africa

Water quality guidelines for South Africa are grouped according to potential user types (e.g. domestic, industrial) in the 1996 Water Quality Guidelines. Drinking water quality is subject to the South African National Standard (SANS) 241 Drinking Water Specification.

United Kingdom

In England and Wales acceptable levels for drinking water supply are listed in the "Water Supply (Water Quality) Regulations 2000."

United States

In the United States, Water Quality Standards are defined by state agencies for various water bodies, guided by the desired uses for the water body (e.g., fish habitat, drinking water supply, recreational use).^[61] The Clean Water Act (CWA) requires each governing jurisdiction (states, territories, and covered tribal entities) to submit a set of biennial reports on the quality of water in their area. These reports are known as the 303(d) and 305(b) reports, named for their respective CWA provisions, and are submitted to, and approved by, EPA. These reports are completed by the governing jurisdiction, typically a state environmental agency. EPA recommends that each state submit a single "Integrated Report" comprising its list of impaired waters and the status of all water bodies in the state. The National Water Quality Inventory Report to Congress is a general report on water quality, providing overall information about the number of miles of streams and rivers and their aggregate condition. The CWA requires states to adopt standards for each of the possible designated uses that they assign to their waters. Should evidence suggest or document that a stream, river or lake has failed to meet the water quality criteria for one or more of its designated uses, it is placed on a list of impaired waters. Once a state has placed a water body on this list, it must develop a management plan establishing Total Maximum Daily Loads (TMDLs) for the pollutant(s) impairing the use of the water. These TMDLs establish the reductions needed to fully support the designated uses.

Drinking water standards, which are applicable to public water systems, are issued by EPA under the Safe Drinking Water Act.

Orbital elements

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Orbital_elements

Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same orbit, but certain schemes, each consisting of a set of six parameters, are commonly used in astronomy and orbital mechanics.

A real orbit and its elements change over time due to gravitational perturbations by other objects and the effects of general relativity. A Kepler orbit is an idealized, mathematical approximation of the orbit at a particular time.

Keplerian elements

In this diagram, the orbital plane (yellow) intersects a reference plane (gray). For Earth-orbiting satellites, the reference plane is usually the Earth's equatorial plane, and for satellites in solar orbits it is the ecliptic plane. The intersection is called the line of nodes, as it connects the center of mass with the ascending and descending nodes. The reference plane, together with the vernal point (♈︎), establishes a reference frame.

The traditional orbital elements are the six Keplerian elements, after Johannes Kepler and his laws of planetary motion.

When viewed from an inertial frame, two orbiting bodies trace out distinct trajectories. Each of these trajectories has its focus at the common center of mass. When viewed from a non-inertial frame centered on one of the bodies, only the trajectory of the opposite body is apparent; Keplerian elements describe these non-inertial trajectories. An orbit has two sets of Keplerian elements depending on which body is used as the point of reference. The reference body (usually the most massive) is called the primary, the other body is called the secondary. The primary does not necessarily possess more mass than the secondary, and even when the bodies are of equal mass, the orbital elements depend on the choice of the primary.

Two elements define the shape and size of the ellipse:

Eccentricity ( $e$ )—shape of the ellipse, describing how much it is elongated compared to a circle (not marked in diagram).
Semimajor axis ( $a$ ) — the sum of the periapsis and apoapsis distances divided by two. For classic two-body orbits, the semimajor axis is the distance between the centers of the bodies, not the distance of the bodies from the center of mass.

Two elements define the orientation of the orbital plane in which the ellipse is embedded:

Inclination ( $i$ ) — vertical tilt of the ellipse with respect to the reference plane, measured at the ascending node (where the orbit passes upward through the reference plane, the green angle $i$ in the diagram). Tilt angle is measured perpendicular to line of intersection between orbital plane and reference plane. Any three distinct points on an ellipse will define the ellipse orbital plane. The plane and the ellipse are both two-dimensional objects defined in three-dimensional space.
Longitude of the ascending node ( $Ω$ ) — horizontally orients the ascending node of the ellipse (where the orbit passes from south to north through the reference plane, symbolized by $☊$ ) with respect to the reference frame's vernal point (symbolized by ♈︎). This is measured in the reference plane, and is shown as the green angle $Ω$ in the diagram.

The remaining two elements are as follows:

Argument of periapsis ( $ω$ ) defines the orientation of the ellipse in the orbital plane, as an angle measured from the ascending node to the periapsis (the closest point the satellite object comes to the primary object around which it orbits), the purple angle $ω$ in the diagram.
True anomaly ( $ν$ , $θ$ , or $f$ ) at epoch ( $t 0$ ) defines the position of the orbiting body along the ellipse at a specific time (the "epoch").

The mean anomaly $M$ is a mathematically convenient fictitious "angle" which varies linearly with time, but which does not correspond to a real geometric angle. It can be converted into the true anomaly $ν$ , which does represent the real geometric angle in the plane of the ellipse, between periapsis (closest approach to the central body) and the position of the orbiting object at any given time. Thus, the true anomaly is shown as the red angle $ν$ in the diagram, and the mean anomaly is not shown.

The angles of inclination, longitude of the ascending node, and argument of periapsis can also be described as the Euler angles defining the orientation of the orbit relative to the reference coordinate system.

Note that non-elliptic trajectories also exist, but are not closed, and are thus not orbits. If the eccentricity is greater than one, the trajectory is a hyperbola. If the eccentricity is equal to one and the angular momentum is zero, the trajectory is radial. If the eccentricity is one and there is angular momentum, the trajectory is a parabola.

Required parameters

Given an inertial frame of reference and an arbitrary epoch (a specified point in time), exactly six parameters are necessary to unambiguously define an arbitrary and unperturbed orbit.

This is because the problem contains six degrees of freedom. These correspond to the three spatial dimensions which define position ( $x$ , $y$ , $z$ in a Cartesian coordinate system), plus the velocity in each of these dimensions. These can be described as orbital state vectors, but this is often an inconvenient way to represent an orbit, which is why Keplerian elements are commonly used instead.

Sometimes the epoch is considered a "seventh" orbital parameter, rather than part of the reference frame.

If the epoch is defined to be at the moment when one of the elements is zero, the number of unspecified elements is reduced to five. (The sixth parameter is still necessary to define the orbit; it is merely numerically set to zero by convention or "moved" into the definition of the epoch with respect to real-world clock time.)

Alternative parametrizations

Keplerian elements can be obtained from orbital state vectors (a three-dimensional vector for the position and another for the velocity) by manual transformations or with computer software.

Other orbital parameters can be computed from the Keplerian elements such as the period, apoapsis, and periapsis. (When orbiting the Earth, the last two terms are known as the apogee and perigee.) It is common to specify the period instead of the semi-major axis in Keplerian element sets, as each can be computed from the other provided the standard gravitational parameter, $GM$ , is given for the central body.

Instead of the mean anomaly at epoch, the mean anomaly $M$ , mean longitude, true anomaly $ν 0$ , or (rarely) the eccentric anomaly might be used.

Using, for example, the "mean anomaly" instead of "mean anomaly at epoch" means that time $t$ must be specified as a seventh orbital element. Sometimes it is assumed that mean anomaly is zero at the epoch (by choosing the appropriate definition of the epoch), leaving only the five other orbital elements to be specified.

Different sets of elements are used for various astronomical bodies. The eccentricity, $e$ , and either the semi-major axis, $a$ , or the distance of periapsis, $q$ , are used to specify the shape and size of an orbit. The longitude of the ascending node, $Ω$ , the inclination, $i$ , and the argument of periapsis, $ω$ , or the longitude of periapsis, $ϖ$ , specify the orientation of the orbit in its plane. Either the longitude at epoch, $L 0$ , the mean anomaly at epoch, $M 0$ , or the time of perihelion passage, $T 0$ , are used to specify a known point in the orbit. The choices made depend whether the vernal equinox or the node are used as the primary reference. The semi-major axis is known if the mean motion and the gravitational mass are known.

It is also quite common to see either the mean anomaly ( $M$ ) or the mean longitude ( $L$ ) expressed directly, without either $M 0$ or $L 0$ as intermediary steps, as a polynomial function with respect to time. This method of expression will consolidate the mean motion ( $n$ ) into the polynomial as one of the coefficients. The appearance will be that $L$ or $M$ are expressed in a more complicated manner, but we will appear to need one fewer orbital element.

Mean motion can also be obscured behind citations of the orbital period $P$ .

Sets of orbital elements
Object	Elements used
Major planet	$e, a, i, Ω, ϖ, L 0$
Comet	$e, q, i, Ω, ω, T 0$
Asteroid	$e, a, i, Ω, ω, M 0$
Two-line elements	$e, i, Ω, ω, n, M 0$

Euler angle transformations

The angles $Ω$ , $i$ , $ω$ are the Euler angles (corresponding to $α$ , $β$ , $γ$ in the notation used in that article) characterizing the orientation of the coordinate system

$x̂$ , $ŷ$ , $ẑ$ from the inertial coordinate frame $Î$ , $Ĵ$ , $K̂$

where:

$Î$ , $Ĵ$ is in the equatorial plane of the central body. $Î$ is in the direction of the vernal equinox. $Ĵ$ is perpendicular to $Î$ and with $Î$ defines the reference plane. $K̂$ is perpendicular to the reference plane. Orbital elements of bodies (planets, comets, asteroids, ...) in the Solar System usually the ecliptic as that plane.
$x̂$ , $ŷ$ are in the orbital plane and with $x̂$ in the direction to the pericenter (periapsis). $ẑ$ is perpendicular to the plane of the orbit. $ŷ$ is mutually perpendicular to $x̂$ and $ẑ$ .

Then, the transformation from the $Î$ , $Ĵ$ , $K̂$ coordinate frame to the $x̂$ , $ŷ$ , $ẑ$ frame with the Euler angles $Ω$ , $i$ , $ω$ is:

\begin{aligned} x_{1} & = \cos Ω \cdot \cos ω - \sin Ω \cdot \cos i \cdot \sin ω; \\ x_{2} & = \sin Ω \cdot \cos ω + \cos Ω \cdot \cos i \cdot \sin ω; \\ x_{3} & = \sin i \cdot \sin ω; \\ y_{1} & = - \cos Ω \cdot \sin ω - \sin Ω \cdot \cos i \cdot \cos ω; \\ y_{2} & = - \sin Ω \cdot \sin ω + \cos Ω \cdot \cos i \cdot \cos ω; \\ y_{3} & = \sin i \cdot \cos ω; \\ z_{1} & = \sin i \cdot \sin Ω; \\ z_{2} & = - \sin i \cdot \cos Ω; \\ z_{3} & = \cos i; \end{aligned}

[\begin{matrix} x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \\ z_{1} & z_{2} & z_{3} \end{matrix}] = [\begin{matrix} \cos ω & \sin ω & 0 \\ - \sin ω & \cos ω & 0 \\ 0 & 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 & 0 \\ 0 & \cos i & \sin i \\ 0 & - \sin i & \cos i \end{matrix}] [\begin{matrix} \cos Ω & \sin Ω & 0 \\ - \sin Ω & \cos Ω & 0 \\ 0 & 0 & 1 \end{matrix}];

where

\begin{aligned} \hat{x} & = x_{1} \hat{I} + x_{2} \hat{J} + x_{3} \hat{K}; \\ \hat{y} & = y_{1} \hat{I} + y_{2} \hat{J} + y_{3} \hat{K}; \\ \hat{z} & = z_{1} \hat{I} + z_{2} \hat{J} + z_{3} \hat{K} . \end{aligned}

The inverse transformation, which computes the 3 coordinates in the I-J-K system given the 3 (or 2) coordinates in the x-y-z system, is represented by the inverse matrix. According to the rules of matrix algebra, the inverse matrix of the product of the 3 rotation matrices is obtained by inverting the order of the three matrices and switching the signs of the three Euler angles.

The transformation from $x̂$ , $ŷ$ , $ẑ$ to Euler angles $Ω$ , $i$ , $ω$ is:

\begin{aligned} Ω & = \arg (- z_{2}, z_{1}) \\ i & = \arg (z_{3}, \sqrt{{z_{1}}^{2} + {z_{2}}^{2}}) \\ ω & = \arg (y_{3}, x_{3}) \end{aligned}

where

arg(x, y)

signifies the polar argument that can be computed with the standard function atan2(y,x) available in many programming languages.

Orbit prediction

Under ideal conditions of a perfectly spherical central body and zero perturbations, all orbital elements except the mean anomaly are constants. The mean anomaly changes linearly with time, scaled by the mean motion,

n = \sqrt{\frac{μ}{a^{3}}} .

where

μ

is the standard gravitational parameter. Hence if at any instant

t 0

the orbital parameters are

(e 0, a 0, i 0, Ω 0, ω 0, M 0)

, then the elements at time

t = t 0 + δt

is given by

(e 0, a 0, i 0, Ω 0, ω 0, M 0 + n δt)

Perturbations and elemental variance

Unperturbed, two-body, Newtonian orbits are always conic sections, so the Keplerian elements define an ellipse, parabola, or hyperbola. Real orbits have perturbations, so a given set of Keplerian elements accurately describes an orbit only at the epoch. Evolution of the orbital elements takes place due to the gravitational pull of bodies other than the primary, the nonsphericity of the primary, atmospheric drag, relativistic effects, radiation pressure, electromagnetic forces, and so on.

Keplerian elements can often be used to produce useful predictions at times near the epoch. Alternatively, real trajectories can be modeled as a sequence of Keplerian orbits that osculate ("kiss" or touch) the real trajectory. They can also be described by the so-called planetary equations, differential equations which come in different forms developed by Lagrange, Gauss, Delaunay, Poincaré, or Hill.

Two-line elements

Keplerian elements parameters can be encoded as text in a number of formats. The most common of them is the NASA / NORAD "two-line elements" (TLE) format, originally designed for use with 80 column punched cards, but still in use because it is the most common format, and can be handled easily by all modern data storages as well.

Depending on the application and object orbit, the data derived from TLEs older than 30 days can become unreliable. Orbital positions can be calculated from TLEs through the SGP / SGP4 / SDP4 / SGP8 / SDP8 algorithms.

Example of a two-line element:

1 27651U 03004A   07083.49636287  .00000119  00000-0  30706-4 0  2692
2 27651 039.9951 132.2059 0025931 073.4582 286.9047 14.81909376225249

Delaunay variables

The Delaunay orbital elements were introduced by Charles-Eugène Delaunay during his study of the motion of the Moon. Commonly called Delaunay variables, they are a set of canonical variables, which are action-angle coordinates. The angles are simple sums of some of the Keplerian angles:

the mean longitude: $ℓ = M + ω + Ω,$
the longitude of periapsis: $g = ω + Ω,$ and
the longitude of the ascending node: $h = Ω$

along with their respective conjugate momenta, $L$ , $G$ , and $H$ . The momenta $L$ , $G$ , and $H$ are the action variables and are more elaborate combinations of the Keplerian elements $a$ , $e$ , and $i$ .

Delaunay variables are used to simplify perturbative calculations in celestial mechanics, for example while investigating the Kozai–Lidov oscillations in hierarchical triple systems. The advantage of the Delaunay variables is that they remain well defined and non-singular (except for $h$ , which can be tolerated) when $e$ and / or $i$ are very small: When the test particle's orbit is very nearly circular ( $e \approx 0$ ), or very nearly “flat” ( $i \approx 0$ ).

A Medley of Potpourri

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Wednesday, October 4, 2023

Water quality

Categories

Human consumption

Industrial and domestic use

Environmental water quality

Sampling and measurement

Sample collection

Issues

Testing in response to natural disasters and other emergencies

Chemical analysis

Biosensors

Real-time monitoring

Environmental indicators

Physical indicators

Chemical indicators

Biological indicators

Climate change impacts

Standards and reports

International

National specifications for ambient water and drinking water

European Union

India

South Africa

United Kingdom

United States

Orbital elements

Keplerian elements

Required parameters

Alternative parametrizations

Euler angle transformations

Orbit prediction

Perturbations and elemental variance

Two-line elements

Delaunay variables

Delayed-choice quantum eraser

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