Urban planning

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https://en.wikipedia.org/wiki/Urban_planning

Partizánske in Slovakia – an example of a typical planned European industrial city founded in 1938 together with a shoemaking factory in which practically all adult inhabitants of the city were employed.

Urban planning, also known as town planning, city planning, regional planning, or rural planning, is a technical and political process that is focused on the development and design of land use and the built environment, including air, water, and the infrastructure passing into and out of urban areas, such as transportation, communications, and distribution networks and their accessibility. Many professional practitioners of urban planning, especially practitioners with the title "urban planner" study urban planning education, while some paraprofessional practitioners are educated in urban studies; others study and work in urban policy - the aspect of public policy used in the public administration subfield of political science that is most aligned with urban planning. Traditionally, urban planning followed a top-down approach in master planning the physical layout of human settlements. The primary concern was the public welfare, which included considerations of efficiency, sanitation, protection and use of the environment, as well as effects of the master plans on the social and economic activities. Over time, urban planning has adopted a focus on the social and environmental bottom-lines that focus on planning as a tool to improve the health and well-being of people while maintaining sustainability standards. Sustainable development was added as one of the main goals of all planning endeavors in the late 20th century when the detrimental economic and the environmental impacts of the previous models of planning had become apparent. Similarly, in the early 21st century, Jane Jacobs's writings on legal and political perspectives to emphasize the interests of residents, businesses and communities effectively influenced urban planners to take into broader consideration of resident experiences and needs while planning.

Urban planning answers questions about how people will live, work and play in a given area and thus, guides orderly development in urban, suburban and rural areas. Although predominantly concerned with the planning of settlements and communities, urban planners are also responsible for planning the efficient transportation of goods, resources, people and waste; the distribution of basic necessities such as water and electricity; a sense of inclusion and opportunity for people of all kinds, culture and needs; economic growth or business development; improving health and conserving areas of natural environmental significance that actively contributes to reduction in CO₂ emissions as well as protecting heritage structures and built environments. Since most urban planning teams consist of highly educated individuals that work for city governments, recent debates focus on how to involve more community members in city planning processes.

Urban planning is an interdisciplinary field that includes aspects of civil engineering, architecture, geography, political science, environmental studies, design sciences, history, economics, sociology, anthropology, business administration, and other fields. Practitioners of urban planning are concerned with research and analysis, strategic thinking, engineering architecture, urban design, public consultation, policy recommendations, implementation and management. It is closely related to the field of urban design and some urban planners provide designs for streets, parks, buildings and other urban areas. Urban planners work with the cognate fields of civil engineering, landscape architecture, architecture, and public administration - especially the urban policy field of public administration - to achieve strategic, policy and sustainability goals. Early urban planners were often members of these cognate fields though today, urban planning is a separate, independent professional discipline. The discipline of urban planning is the broader category that includes different sub-fields such as land-use planning, zoning, economic development, environmental planning, and transportation planning. Creating the plans requires a thorough understanding of penal codes and zonal codes of planning.

Another important aspect of urban planning is that the range of urban planning projects include the large-scale master planning of empty sites or Greenfield projects as well as small-scale interventions and refurbishments of existing structures, buildings and public spaces. Pierre Charles L'Enfant in Washington, D.C., Daniel Burnham in Chicago, Lúcio Costa in Brasília and Georges-Eugene Haussmann in Paris planned cities from scratch, and Robert Moses and Le Corbusier refurbished and transformed cities and neighborhoods to meet their ideas of urban planning.

History

Further information: History of urban planning and Grid Plan

1852 city plan of Pori by G. T. von Chiewitz

Berlin - Siegessäule. August 1963. Spacious and organized city planning in Germany was official government policy dating back to Nazi rule.

There is evidence of urban planning and designed communities dating back to the Mesopotamian, Indus Valley, Minoan, and Egyptian civilizations in the third millennium BCE. Archaeologists studying the ruins of cities in these areas find paved streets that were laid out at right angles in a grid pattern. The idea of a planned out urban area evolved as different civilizations adopted it. Beginning in the 8th century BCE, Greek city states primarily used orthogonal (or grid-like) plans. Hippodamus of Miletus (498–408 BC), the ancient Greek architect and urban planner, is considered to be "the father of European urban planning", and the namesake of the "Hippodamian plan" (grid plan) of city layout.

The ancient Romans, inspired by the Greeks, also used orthogonal plans for their cities. City planning in the Roman world was developed for military defense and public convenience. The spread of the Roman Empire subsequently spread the ideas of urban planning. As the Roman Empire declined, these ideas slowly disappeared. However, many cities in Europe still held onto the planned Roman city center. Cities in Europe from the 9th to 14th centuries, often grew organically and sometimes chaotically. But in the following centuries with the coming of the Renaissance many new cities were enlarged with newly planned extensions. From the 15th century on, much more is recorded of urban design and the people that were involved. In this period, theoretical treatises on architecture and urban planning start to appear in which theoretical questions around planning the main lines, ensuring plans meet the needs of the given population and so forth are addressed and designs of towns and cities are described and depicted. During the Enlightenment period, several European rulers ambitiously attempted to redesign capital cities. During the Second French Empire, Baron Georges-Eugène Haussmann, under the direction of Napoleon III, redesigned the city of Paris into a more modern capital, with long, straight, wide boulevards.

Planning and architecture went through a paradigm shift at the turn of the 20th century. The industrialized cities of the 19th century grew at a tremendous rate. The evils of urban life for the working poor were becoming increasingly evident as a matter of public concern. The laissez-faire style of government management of the economy, in fashion for most of the Victorian era, was starting to give way to a New Liberalism that championed intervention on the part of the poor and disadvantaged. Around 1900, theorists began developing urban planning models to mitigate the consequences of the industrial age, by providing citizens, especially factory workers, with healthier environments. The following century would therefore be globally dominated by a central planning approach to urban planning, not necessarily representing an increment in the overall quality of the urban realm.

At the beginning of the 20th century, urban planning began to be recognized as a separate profession. The Town and Country Planning Association was founded in 1899 and the first academic course in Great Britain on urban planning was offered by the University of Liverpool in 1909. In the 1920s, the ideas of modernism and uniformity began to surface in urban planning, and lasted until the 1970s. In 1933, Le Corbusier presented the Radiant City, a city that grows up in the form of towers, as a solution to the problem of pollution and over-crowding. But many planners started to believe that the ideas of modernism in urban planning led to higher crime rates and social problems.

In the second half of the 20th century, urban planners gradually shifted their focus to individualism and diversity in urban centers.

21st century practices

See also: Mobility transition

Urban planners studying the effects of increasing congestion in urban areas began to address the externalities, the negative impacts caused by induced demand from larger highway systems in western countries such as in the United States. The United Nations Department of Economic and Social Affairs predicted in 2018 that around 2.5 billion more people occupy urban areas by 2050 according to population elements of global migration. New planning theories have adopted non-traditional concepts such as Blue Zones and Innovation Districts to incorporate geographic areas within the city that allow for novel business development and the prioritization of infrastructure that would assist with improving the quality of life of citizens by extending their potential lifespan.

Planning practices have incorporated policy changes to help address anthropocentric global climate change. London began to charge a congestion charge for cars trying to access already crowded places in the city. Cities nowadays stress the importance of public transit and cycling by adopting such policies.

Theories

Further information: Theories of urban planning

See also: Planning cultures

Street Hierarchy and Accessibility

Planning theory is the body of scientific concepts, definitions, behavioral relationships, and assumptions that define the body of knowledge of urban planning. There are eight procedural theories of planning that remain the principal theories of planning procedure today: the rational-comprehensive approach, the incremental approach, the transactive approach, the communicative approach, the advocacy approach, the equity approach, the radical approach, and the humanist or phenomenological approach. Some other conceptual planning theories include Ebenezer Howard's The Three Magnets theory that he envisioned for the future of British settlement, also his Garden Cities, the Concentric Model Zone also called the Burgess Model by sociologist Ernest Burgess, the Radburn Superblock that encourages pedestrian movement, the Sector Model and the Multiple Nuclei Model among others.

Technical aspects

Further information: Technical aspects of urban planning

Technical aspects of urban planning involve the application of scientific, technical processes, considerations and features that are involved in planning for land use, urban design, natural resources, transportation, and infrastructure. Urban planning includes techniques such as: predicting population growth, zoning, geographic mapping and analysis, analyzing park space, surveying the water supply, identifying transportation patterns, recognizing food supply demands, allocating healthcare and social services, and analyzing the impact of land use.

In order to predict how cities will develop and estimate the effects of their interventions, planners use various models. These models can be used to indicate relationships and patterns in demographic, geographic, and economic data. They might deal with short-term issues such as how people move through cities, or long-term issues such as land use and growth. One such model is the Geographic Information System (GIS) that is used to create a model of the existing planning and then to project future impacts on the society, economy and environment.

Building codes and other regulations dovetail with urban planning by governing how cities are constructed and used from the individual level. Enforcement methodologies include governmental zoning, planning permissions, and building codes, as well as private easements and restrictive covenants.

Urban planners

Further information: Urban planner

An urban planner is a professional who works in the field of urban planning for the purpose of optimizing the effectiveness of a community's land use and infrastructure. They formulate plans for the development and management of urban and suburban areas, typically analyzing land use compatibility as well as economic, environmental and social trends. In developing any plan for a community (whether commercial, residential, agricultural, natural or recreational), urban planners must consider a wide array of issues including sustainability, existing and potential pollution, transport including potential congestion, crime, land values, economic development, social equity, zoning codes, and other legislation.

The importance of the urban planner is increasing in the 21st century, as modern society begins to face issues of increased population growth, climate change and unsustainable development. An urban planner could be considered a green collar professional.

Some researchers suggest that urban planners around the world work in different "planning cultures", adapted to their local cities and cultures. However, professionals have identified skills, abilities and basic knowledge sets that are common to urban planners across national and regional boundaries.

Participatory urban planning

Participatory planning in the United States emerged during the 1960s and 1970s. At the same time, participatory planning began to enter the development field, with similar characteristics and agendas. There are many notable urban planners and activists whose work facilitated and shaped participatory planning movements. Jane Jacobs and her work is one of the most significant contributions to participatory planning because of the influence it had across the entire United States. There has also been a recent emergence in engaging youth in urban planning education.

Education

Further information: Urban planning education, Urban studies, and Political science

Many professional practitioners of urban planning, especially practitioners with the title "urban planner" study urban planning education, while some paraprofessional practitioners are educated in urban studies; others study and work in urban policy - the aspect of public policy used in the public administration subfield of political science that is most aligned with urban planning.

Criticisms and debates

The school of neoclassical economics argues that planning is unnecessary, or even harmful, because market efficiency allows for effective land use. A pluralist strain of political thinking argues in a similar vein that the government should not intrude in the political competition between different interest groups which decides how land is used. The traditional justification for urban planning has in response been that the planner does to the city what the engineer or architect does to the home, that is, make it more amenable to the needs and preferences of its inhabitants.

The widely adopted consensus-building model of planning, which seeks to accommodate different preferences within the community has been criticized for being based upon, rather than challenging, the power structures of the community. Instead, agonism has been proposed as a framework for urban planning decision-making.

Another debate within the urban planning field is about who is included and excluded in the urban planning decision-making process. Most urban planning processes use a top-down approach which fails to include the residents of the places where urban planners and city officials are working. Sherry Arnstein's "ladder of citizen participation" is oftentimes used by many urban planners and city governments to determine the degree of inclusivity or exclusivity of their urban planning. One main source of engagement between city officials and residents are city council meetings that are open to the residents and that welcome public comments. Additionally, there are some federal requirements for citizen participation in government-funded infrastructure projects.

Many urban planners and planning agencies rely on community input for their policies and zoning plans. How effective community engagement is can be determined by how member's voices are heard and implemented.

NASA-ESA Mars Sample Return

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/NASA-ESA_Mars_Sample_Return

NASA-ESA MSR Patch

Mars Sample Return Program (Artwork; July 27, 2022)

The NASA-ESA Mars Sample Return is a proposed Mars sample return (MSR) mission to collect Martian rock and soil samples in 43 small, cylindrical, pencil-sized, titanium tubes and return them to Earth around 2033.

The NASA–ESA plan, approved in September 2022, is to return samples using three missions: a sample collection mission (Perseverance), a sample retrieval mission (Sample Retrieval Lander + Mars Ascent Vehicle + Sample Transfer Arm + 2 Ingenuity-class helicopters), and a return mission (Earth Return Orbiter). The mission hopes to resolve the question of whether Mars once harbored life.

Although NASA and ESA's proposal is still in the design stage and facing significant cost overruns as of August 2023, the first leg of gathering samples is currently being executed by the Perseverance rover on Mars and the components of sample retrieval lander (second leg) are in testing phase on earth.

History

See also: Mars sample-return mission § History

2001 to 2004

In the summer of 2001 the Jet Propulsion Laboratory (JPL) requested mission concepts and proposals from industry-led teams (Boeing, Lockheed Martin, and TRW). The science requirements included at least 500 grams (18 oz) of samples, rover mobility to obtain samples at least 1 kilometre (0.62 mi) from the landing spot, and drilling to obtain one sample from a depth of 2 metres (6 ft 7 in). That following winter, JPL made similar requests of certain university aerospace engineering departments (MIT and the University of Michigan).

Also in 2001, a separate set of industry studies was done for the Mars ascent vehicle (MAV) due to the uniqueness and key role of the MAV for MSR. Figure 11 in this reference summarized the need for MAV flight testing at a high altitude over Earth, based on Lockheed Martin's analysis that the risk of mission failure is "extremely high" if launch vehicle components are only tested separately.

In 2003 JPL reported that the mission concepts from 2001 had been deemed too costly, which led to the study of a more affordable plan accepted by two groups of scientists, a new MSR Science Steering Group and the Mars Exploration Program Analysis Group (MEPAG). Instead of a rover and deep drilling, a scoop on the lander would dig 20 centimetres (7.9 in) deep and place multiple samples together into one container. After five years of technology development, the MAV would be flight-tested twice above Earth before the mission PDR (Preliminary Design Review) in 2009.

Based on the simplified mission plan, assuming a launch from Earth in 2013 and two weeks on Mars for a 2016 return, technology development was initiated for ensuring with high reliability that potential Mars microbes would not contaminate Earth, and also that the Mars samples would not be contaminated with Earth-origin biological materials. The sample container would be clean on the outside before departing from Mars, with installation onto the MAV inside an "Earth-clean MAV garage."

In 2004 JPL published an update on the 2003 plan. MSR would use the new large sky crane landing system in development for the Mars Science Laboratory rover (later named Curiosity). A MSR Technology Board was formed, and it was noted that the use of a rover might return to the MSR plan, in light of success with the Spirit and Opportunity rovers that arrived early in 2004. A 285-kilogram (628 lb) ascent rocket would carry 0.5-kilogram (1.1 lb) of samples inside a 5-kilogram (11 lb) payload, the Orbiting Sample (OS). The MAV would transmit enough telemetry to reconstruct events in case of failure on the way up to Mars orbit.

2005 to 2008

As of 2005 a rover had returned to the MSR plan, with a rock core drill in light of results from the Mars Exploration Rover discoveries. Focused technology development would start before the end of 2005 for mission PDR in 2009, followed by launch from Earth in 2013. Related technologies in development included potential advances for Mars arrival (navigation and descent propulsion) and implementing pump-fed liquid launch vehicle technology on a scale small enough for a MAV.

In late 2005 a peer-reviewed analysis showed that ascent trajectories to Mars orbit would differ depending on liquid versus solid propulsion, largely because small solid rocket motors burn faster, requiring a steeper ascent path to avoid excess atmospheric drag, while slower burning liquid propulsion might take advantage of more efficient paths to orbit.

Early in 2006 the Marshall Space Flight Center noted the possibility that a science rover would cache the samples on Mars, then subsequently a mini-rover would be sent along with the MAV on a sample return lander, in which case either the mini-rover or the science rover would deliver the samples to the lander for loading onto the MAV. A two-stage 250-kilogram (550 lb) solid propellant MAV would be gas ejected from a launch tube with its 5-kilogram (11 lb) payload, a 16-centimetre (6.3 in) diameter spherical package containing the samples. The second stage would send telemetry and its steering thrusters would use hydrazine fuel with additives. The authors expected the MAV to need multiple flight tests at a high altitude over Earth.

A peer-reviewed publication in 2007 described testing of autonomous sample capture for Mars orbit rendezvous. Free-floating tests were done on board a NASA aircraft using a parabolic "zero-g" flight path.

In 2007 Alan Stern, then NASA's Associate Administrator for Science, was strongly in favor of completing MSR sooner, and he asked JPL to include sample caching on the Mars Science Laboratory mission (later named Curiosity). A team at the Ames Research Center was designing a hockey puck-sized sample-caching device to be installed as an extra payload on MSL.

A review analysis in 2008 compared Mars ascent to lunar ascent, noting that the MAV would be not only technically daunting, but also a cultural challenge for the planetary community, given that lunar ascent has been done using known technology, and that science missions typically rely on proven propulsion for course corrections and orbit insertion maneuvers, similar to what Earth satellites do routinely.

2009 to 2011

Early in 2009 the In-Space Propulsion Technology project office at the NASA Glenn Research Center (GRC) presented a ranking of six MAV options, concluding that a 285-kilogram (628 lb) two-stage solid rocket with continuous telemetry would be best for delivering a 5-kilogram (11 lb) sample package to Mars orbit. A single-stage pump-fed bipropellant MAV was noted to be less heavy and was ranked second.

Later in 2009 the chief technologist of the Mars Exploration Directorate at JPL referred to a 2008 workshop on MSR technologies at the Lunar and Planetary Institute, and wrote that particularly difficult technology challenges included the MAV, sample acquisition and handling, and back planetary protection, then further commented that "The MAV, in particular, stands out as the system with highest development risk, pointing to the need for an early start" leading to flight testing before preliminary design review (PDR) of the lander that would deliver the MAV.

In October 2009 NASA and ESA established the Mars Exploration Joint Initiative to proceed with the ExoMars program, whose ultimate aim is "the return of samples from Mars in the 2020s". ExoMars's first mission was planned to launch in 2018 with unspecified missions to return samples in the 2020–2022 time frame. As reported to the NASA Advisory Council Science Committee (NAC-SC) early in 2010, MEPAG estimated that MSR "will cost $8-10B, and it is obvious that NASA and ESA can't fund this amount by themselves." The cancellation of the caching rover MAX-C in 2011, and later NASA withdrawal from ExoMars, due to budget limitations, ended the mission. The pull-out was described as "traumatic" for the science community.

In 2010–2011 the NASA In-Space Propulsion Technology (ISPT) program at the Glenn Research Center received proposals and funded industry partners for MAV design studies with contract options to begin technology development, while also considering propulsion needs for Earth return spacecraft. Inserting the spacecraft into Mars orbit, then returning to Earth, was noted to need a high total of velocity changes, leading to a conclusion that solar electric propulsion could reduce mission risk by improving mass margins, compared to the previously assumed use of chemical propulsion along with aerobraking at Mars. The ISPT team also studied scenarios for MAV flight testing over Earth and recommended two flight tests prior to MSR mission PDR, considering the historical low probability of initial success for new launch vehicles.

The NASA–ESA potential mission schedule anticipated launches from Earth in 2018, 2022 and 2024 to send respectively a sample caching rover, a sample return orbiter and a sample retrieval lander for a 2027 Earth arrival, with MAV development starting in 2014 after two years of technology development identified by the MAV design studies. The ISPT program summarized a year of propulsion technology progress for improving Mars arrival, Mars ascent, and Earth return, stating that the first flight test of a MAV engineering model would need to occur in 2018 to meet the 2024 launch date for the sample retrieval lander.

The 2011 MAV industry studies were done by Lockheed-Martin teamed with ATK; Northrop-Grumman; and Firestar Technologies, to deliver a 5-kg (11-lb), 16-cm (6.3-inch) diameter sample sphere to Mars orbit. The Lockheed-Martin-ATK team focused on a solid propellant first stage with either solid or liquid propellant for the upper stage, estimated MAV mass in the range 250 to 300 kg (550 to 660 lb), and identified technologies for development to reduce mass. Northrop-Grumman (the former TRW) similarly estimated a mass below 300 kg using pressure-fed liquid bipropellants for both stages, and had plans for further progress. Firestar Technologies described a single-stage MAV design having liquid fuel and oxidizer blended together in one main propellant tank.

In early 2011 the US National Research Council's Planetary Science Decadal Survey, which laid out mission planning priorities for the period 2013–2022, declared an MSR campaign its highest priority Flagship Mission for that period. In particular, it endorsed the proposed Mars Astrobiology Explorer-Cacher (MAX-C) mission in a "descoped" (less ambitious) form. This mission plan was officially cancelled in April 2011. The plan cancelled in 2011 for budget reasons had been for NASA and ESA to each build a rover to send together in 2018.

2012 to 2013

In 2012 prospects for MSR were slowed further by a 38-percent cut in NASA's Mars program budget for fiscal year 2013, leading to controversy among scientists over whether Mars exploration could thrive on a series of small rover missions. A Mars Program Planning Group (MPPG) was convened as one response to budget cuts.

In mid-2012, eight weeks before Curiosity arrived on Mars, the Lunar and Planetary Institute hosted a NASA-sponsored three-day workshop to gather expertise and ideas from a wide range of professionals and students, as input to help NASA reformulate the Mars Exploration Program, responsive to the latest Planetary Decadal Survey that prioritized MSR. A summary report noted that the workshop was held in response to recent deep budget cuts, 390 submissions were received, 185 people attended and agreed that "credible steps toward MSR" could be done with reduced funding. The MAX-C rover (ultimately implemented as Mars 2020, Perseverance) was considered beyond financial reach at that time, so the report noted that progress toward MSR could include an orbiter mission to test autonomous rendezvous, or a Phoenix-class lander to demonstrate pinpoint landing while delivering a MAV as a technology demonstration. The workshop consisted largely of three breakout group discussions for Technology and Enabling Capabilities, Science and Mission Concepts, and Human Exploration and Precursors.

Wide-ranging discussions were documented by the Technology Panel, which suggested investments for improved drilling and "small is beautiful" rovers with an "emphasis on creative mass-lowering capabilities." The panel stated that MAV "functional technology is not new" but the Mars environment would pose challenges, and referred to MAV technologies as "a risk for most sample return scenarios of any cost range." MAV technology was addressed in numerous written submissions to the workshop, one of which described Mars ascent as "beyond proven technology" (velocity and acceleration in combination for small rockets) and a "huge challenge for the social system," referring to a "Catch-22" dilemma "in which there is no tolerance for new technology if sample return is on the near-term horizon, and no MAV funding if sample return is on the far horizon."

In September 2012 NASA announced its intention to further study MSR strategies as outlined by the MPPG – including a multiple launch scenario, a single-launch scenario, and a multiple-rover scenario – for a mission beginning as early as 2018. A "fetch rover" would retrieve the sample caches and deliver them to a Mars ascent vehicle (MAV). In July 2018, NASA contracted Airbus to produce a "fetch rover" concept. As of late 2012, It was determined that the MAX-C rover concept to collect samples could be implemented for a launch in 2020 (Mars 2020), within available funding using spare parts and mission plans developed for NASA's Curiosity Mars rover

In 2013 the NASA Ames Research Center proposed that a SpaceX Falcon Heavy could deliver two tons of useful payload to the Mars surface, including an Earth return spacecraft that would be launched from Mars by a one-ton single-stage MAV using liquid bipropellants fed by turbopumps. The successful landing of the Curiosity rover directly on its wheels (August 2012) motivated JPL to take a fresh look at carrying the MAV on the back of a rover. A fully guided 300-kg MAV (like Lockheed's 2011 two-stage solid) would avoid the need for a round-trip fetch rover. A smaller 150-kg MAV would permit one rover to also include sample collection while using MSL heritage to reduce mission cost and development time, placing most development risk on the MAV. The 150-kg MAV would be made lightweight by spinning it up before stage separation, although the lack of telemetry data from the spin-stabilized unguided upper stage was noted as a disadvantage.

JPL later presented more details of the 150-kg solid propellant mini-MAV concept of 2012, in a summary of selected past efforts. The absence of telemetry data during the 1999 loss of the Mars Polar Lander had put an emphasis on "critical event communications", subsequently applied to MSR. Then after the MSL landing in 2012, requirements had been revisited with a goal to reduce MAV mass. Single fault tolerance and continuous telemetry data to Mars orbit were questioned. For the 500 grams (1.1 lb) of samples, a 3.6-kg (7.9 lb) payload was deemed possible instead of 5 kg (11 lb). The 2012 mini-MAV concept had single-string avionics, in addition to the spin-stabilized upper stage without telemetry.

2014 to 2017

In 2014–2015 JPL analyzed many options for Mars ascent including solid, hybrid and liquid propellants, for payloads ranging from 6.5 kg to 25 kg. Four MAV concepts using solid propellant had two stages, while one or two stages were considered for hybrid and liquid propellants. Seven options were scored for ten attributes ("figures of merit"). A single stage hybrid received the highest overall score, including the most points for reducing cost and separately for reducing complexity, with the fewest points for technology readiness. Second overall was a single-stage liquid bipropellant MAV using electric pumps. A pressure-fed bipropellant design was third, with the most points for technology readiness. Solid propellant options had lower scores, partly due to receiving very few points for flexibility. JPL and NASA Langley Research Center cautioned that the high thrust and short burn times of solid rocket motors would result in early burnout at a low altitude with substantial atmosphere remaining to coast through at high Mach numbers, raising stability and control concerns. With concurrence from the Mars Program Director, a decision was made in January 2016 to focus limited technology development funds on advancing a hybrid propellant MAV (liquid oxidizer with solid fuel).

Starting in 2015, a new effort for planetary protection moved the backward planetary protection function from the surface of Mars to the sample Return Orbiter, to "break-the-chain" in flight. Concepts for brazing, bagging, and plasma sterilization were studied and tested, with a primary focus on brazing as of 2016.

2018 to 2022

In April 2018 a letter of intent was signed by NASA and ESA that may provide a basis for a Mars sample-return mission. The agreement was dated during the 2nd International Mars Sample Return Conference in Berlin, Germany. The conference program was archived along with 125 technical submissions that covered sample science (anticipated findings, site selection, collection, curation, analysis) and mission implementation (Mars arrival, rovers, rock drills, sample transfer robotics, Mars ascent, autonomous orbit rendezvous, interplanetary propulsion, Earth arrival, planetary protection). In one of many presentations, an international science team noted that collecting sedimentary rock samples would be required to search for ancient life. A joint NASA-ESA presentation described the baseline mission architecture, including sample collection by the Mars 2020 Rover derived from the MAX-C concept, a Sample Retrieval Lander, and an Earth Return Orbiter. An alternative proposal was to use a SpaceX Falcon Heavy to decrease mission cost while delivering more mass to Mars and returning more samples. Another submission to the Berlin conference noted that mission cost could be reduced by advancing MAV technology to enable a significantly smaller MAV for a given sample payload.

In July 2019 a mission architecture was proposed. In 2019, JPL authors summarized sample retrieval, including a sample fetch rover, options for fitting 20 or 30 sample tubes into a 12-kilogram (26 lb) payload on a 400-kilogram (880 lb) single-stage-to-orbit (SSTO) MAV that would use hybrid propellants, a liquid oxidizer with a solid wax fuel, which had been prioritized for propulsion technology development since 2016. Meanwhile, the Marshall Space Flight Center (MSFC) presented a comparison of solid and hybrid propulsion for the MAV. Later in 2019, MSFC and JPL had collaborated on designing a two-stage solid propellant MAV, and noted that an unguided spinning upper stage could reduce mass, but this approach was abandoned at the time due to the potential for orbital variations.

Early in 2020 JPL updated the overall mission plan for an orbiting sample package (the size of a basketball) containing 30 tubes, showing solid and hybrid MAV options in the range 400 to 500 kilograms (880 to 1,100 lb). Adding details, MSFC presented designs for both the solid and hybrid MAV designs, for a target mass of 400 kilograms (880 lb) at Mars liftoff to deliver 20 or 30 sample tubes in a 14-to-16-kilogram (31 to 35 lb) payload package. In April 2020, an updated version of the mission was presented. The decision to adopt a two-stage solid rocket MAV was followed by Design Analysis Cycle 0.0 in the spring of 2020, which refined the MAV to a 525-kilogram (1,157 lb) design having guidance for both stages, leading to reconsideration of an unguided spin-stabilized second stage to save mass.

In October 2020, the MSR Independent Review Board (IRB) released its report recommending overall that the MSR program proceed, then in November NASA responded to detailed IRB recommendations. The IRB noted that MSR would have eight first-time challenges including the first launch from another planet, autonomous orbital rendezvous, and robotic sample handling with sealing to "break-the-chain". The IRB cautioned that the MAV will be unlike any previous launch vehicle, and experience shows that the smaller a launch vehicle, the more likely it is to end up heavier than designed. Referring to the unguided upper stage of the MAV, the IRB stated the importance of telemetry for critical events, "to allow useful reconstruction of a fault during second stage flight". The IRB indicated that the most probable mission cost would be $3.8-$4.4B. As reported to the NAC-SC in April 2021, the Planetary Science Advisory Committee (PAC) was "very concerned about the high cost" of MSR, and wanted to be sure that astrobiology considerations would be included in plans for returned sample laboratories.

Early in 2022 MSFC presented the guided-unguided MAV design for a 125-kilogram (276 lb) mass reduction and documented remaining challenges including aerodynamic complexities during the first stage burn and coast to altitude, a desire to locate hydrazine steering thrusters farther from the center of mass, and stage separation without tip-off rotation. While stage separation and subsequent spin-up would be flight tested, the authors noted that it would be ideal to flight test an entire flight-like MAV, but there would be a large cost.

In April 2022, the United States National Academies released the Planetary Science Decadal Survey report for 2023-2032, a review of plans and priorities for the upcoming ten years, after many committee meetings starting in 2020, with consideration of over 500 independently submitted white papers, more than 100 regarding Mars including comments on science and technology for sample return. The published document noted NASA's 2017 plan for a "focused and rapid" sample return campaign with essential participation from ESA, then recommended, "The highest scientific priority of NASA's robotic exploration efforts this decade should be completion of Mars Sample Return as soon as is practicably possible." Decadal white papers emphasized the importance of MSR for science, included a description of implementing MSR, and noted that the MAV has been underestimated despite needing flight performance beyond the state of the art for small rockets, needs a sustained development effort, and that technology development for a smaller MAV has the potential to reduce MSR mission cost. Decadal Survey committee meetings hosted numerous invited speakers, notably a presentation from the MSR IRB.

Sample collection

Perseverance rover

The Mars 2020 mission landed the Perseverance rover, which is storing samples to be returned to Earth later.

Mars 2020 Perseverance rover

Mapping Perseverance's samples collected to date (The 10 duplicate samples to be left behind at Three Forks Sample Depot are framed in green colour.)

Facsimiles of Perseverance's sample tubes at JPL in Southern California

The Mars 2020 mission landed the Perseverance rover in Jezero crater in February 2021. It collected multiple samples and packed them into cylinders for later return. Jezero appears to be an ancient lakebed, suitable for ground sampling.

At the beginning of August 2021, Perseverance made its first attempt to collect a ground sample by drilling out a finger-size core of Martian rock. This attempt did not succeed. A drill hole was produced, as indicated by instrument readings, and documented by a photograph of the drill hole. However, the sample container turned out to be empty, indicating that the rock sampled was not robust enough to produce a solid core.

Perseverance rover's sampling bits

• The pointed one with two windows on left is Regolith drill
• the two shorter ones on left are Abrasion tools
• the rest in center are Rock drills

A second target rock judged to have a better chance to yield a sufficiently robust sample was sampled at the end of August and the beginning of September 2021. After abrading the rock, cleaning away dust by puffs of pressurized nitrogen, and inspecting the resulting rock surface, a hole was drilled on September 1. A rock sample appeared to be in the tube, but it was not immediately placed in a container. A new procedure of inspecting the tube optically was performed. On September 6, the process was completed and the first sample placed in a container.

In support of the NASA-ESA Mars Sample Return, rock, regolith (Martian soil), and atmosphere samples are being cached by Perseverance. Currently, out of 43 sample tubes, 22 of them have been cached, including 16 rock sample tubes, two regolith sample tubes, an atmosphere sample tube, and three witness tubes. Before launch, 5 of the 43 tubes were designated “witness tubes” and filled with materials that would capture particulates in the ambient environment of Mars. Out of 43 tubes, 3 witness sample tubes will not be returned to Earth and will remain on rover as sample canister will only have 30 tube slots. Alongside, 10 of the 43 tubes are left at backup Three Forks Sample Depot.

From December 21, 2022 Perseverance started a campaign to deposit 10 of its collected samples at the backup depot, Three Forks. This work was completed on January 28, 2023.

Three Forks Sample Depot

After nearly a Martian year of NASA's Perseverance Mars rover's science and sample caching operations for MSR campaign, the rover is currently tasked to deposit ten samples that it has cached from beginning at Three Forks Sample Depot as NASA aims to eventually return them to Earth starting from December 19, 2022. This depot will serve as a backup spot, in case, Perseverance cannot deliver its samples. Perseverance is depositing the samples at a relatively flat terrain known as Three Forks so that NASA and ESA could recover them in its successive missions in the MSR campaign. It is even selected as the backup landing spot for the Sample Retrieval Lander. It is a relatively benign place. It is as flat and smooth as a table top.

Testing a Sample Drop in the Mars Yard with VSTB OPTIMISM Rover

Perseverance's complex Sampling and Caching System takes almost an hour to retrieve the metal tube from inside the rover's belly, view it one last time with its internal Cachecam, and drop the sample ~0.89 m (2 ft 11 in) onto a carefully selected patch of Martian surface.

Mars Perseverance rover – wind lifts a massive dust cloud (June 18, 2021)

The tubes will not be piled up at a single spot. Instead, each tube-drop location will have an "area of operation" ~5.5 m (18 ft) in diameter. To that end, the tubes will be deposited on the surface in an intricate zigzag pattern of 10 spots for 10 tubes, with each sample ~5 m (16 ft) to ~15 m (49 ft) apart from one another near the proposed Sample retrieval lander's landing site. There are various reasons for this plan, biggest for placing them far apart being that is that sample recovery helicopters because they are designed to interact with only one tube at a time. Alongside, they will perform takeoffs and landings, and driving in that spot. To ensure a helicopter could retrieve samples without any problem, the plan is to be executed properly and would span over more than two months.

Perseverance Views Dust Devils Swirling Across Jezero Crater

Before and after Perseverance drops each tube, mission controllers will review a multitude of images from the rover's SHERLOCK Watson camera. Images by the SHERLOC WATSON camera are also used to check for surety that the tube had not rolled into the path of the rover's wheels. They also look to ensure the tube had not landed in such a way that it was standing on its end (each tube has a flat end piece called a "glove" to make it easier to be picked up by future missions). That occurred less than 5% of the time during testing with Perseverance's Earthly twin OPTIMISM in JPL's Mars Yard. In case it does happen on Mars, the mission has written a series of commands for Perseverance to carefully knock the tube over with part of the turret at the end of its robotic arm.

A Map of Perseverance's Sample Depots

These SHERLOCK Watson camera images will also give the Mars Sample Return team the precise data necessary to locate the tubes in the event of the samples becoming covered by dust or sand before they are collected.Mars does get windy, but not like on Earth. But the atmosphere on Mars is 100 times less dense than that of Earth's atmosphere. So winds around here can pick up speed (fastest are Dust devils), but they don't pick up a lot of dust particles. Martian wind can certainly lift fine dust and leave it on surfaces. But even if significant dust is accumulated these images and depositing pattern will help to recover them back. Even a lucky encounter with a dust devil can even remove dust over the samples as in case with the solar panels of Spirit rover and Opportunity rover.

Once this whole task of depositing all the 10 samples is completed, Perseverance will carry on with its mission, traversing to the Crater floor and scaling Delta's summit. The rover be traversing along the edge of the crater and probably, caching more tubes then whilst following the plan of taking single sample at one rock. Till now, several pairs of samples were taken and one samples from pair will be placed at the depot and the other pair will stay on board the rover.

Sample retrieval

The Mars Sample Return mission earlier consisted ESA Sample Fetch Rover and its associated second lander alongside the mars ascent vehicle and its lander that will take the samples to a MAV, from where they will be launched back to Earth. But after consideration and cost overruns, it was decided that given Perseverance's expected longevity, it will be the primary means of transporting samples to Sample Retrieval Lander (SRL).

Sample Retrieval Lander

The sample retrieval mission presently involves launching a 5 solar-array sample return lander in 2028 with the Mars Ascent Vehicle and two sample recovery helicopters as a backup for Perseverance.The SRL lander is about the size of an average two-car garage weighing ~3,375 kg (7,441 lb); tentatively planned to be 7.7 m (25 ft) wide and 2.1 m (6.9 ft) high when fully deployed. The payload mass of the lander is double that of the Perseverance rover, that is ~563 kg (1,241 lb). The lander needs to be close to the Perseverance rover to facilitate the transfer of Mars samples. It must land within 60 m (200 ft) of its target site – much closer than previous Mars rovers and landers. Thus, it will have a secondary battery to power the lander to land on Mars. The lander would take advantage of an enhanced version of NASA's successful Terrain Relative Navigation that helped land Perseverance safely. The new Enhanced Lander Vision System would, among other improvements, add a second camera, an altimeter, and better capabilities to use propulsion for precision landing. It is planned to land near at Three Forks in 2029.

ESA Sample Transfer Arm

The Mars 2020 rover and helicopters will transport the samples to the SRL lander. SRL's ESA-built ~2.40 m (7.9 ft) long, Sample Transfer Arm will be used to extract the samples and load them into the Sample Return Capsule in the Ascent Vehicle.

Mars Sample Recovery Helicopters

Main article: Mars Sample Recovery Helicopter

MSR campaign included Ingenuity-class helicopters, both of which will be collecting the samples with the help of a tiny robotic arm to the SRL, in case Perseverance rover runs into problems.

Mars Ascent Vehicle (MAV)

Mars Ascent Vehicle

Mars Ascent Vehicle mockup on display.
Function	Mars Orbital launch vehicle
Manufacturer	NASA's Marshall Space Flight Center/Lockheed Martin/Northrop Grumman
Country of origin	United States
Size
Height	2.26 m (7.4 ft)
Diameter	0.5 m (1.6 ft)
Mass	450 kg (990 lb)
Stages	2
Capacity
Payload to LAO
Altitude	500 km (310 mi)
Mass	500 g (18 oz)
Associated rockets
Comparable	Mars: Unique Earth: likely a ICBM failing before Max q
Launch history
Status	Under Development
Launch sites	Vector mid-air after release from Sample Retrieval Lander, Three Forks, Jezero Crater
Total launches	1 (planned)
UTC date of spacecraft launch	2030 (planned)
Type of passengers/cargo	Orbiting Sample Container with 30–43 tubes, Radio Beacon (hosted)
First stage
Powered by	1 optimized Star 20 (Altair 3)
Burn time	75 s
Propellant	CTPB
Second stage
Powered by	1 optimized Star 15G
Burn time	20 s
Propellant	HTPB

Mars Ascent Vehicle (MAV) is a two-stage, solid-fueled rocket that will deliver the collected samples from the surface of Mars to the Earth Return Orbiter. Early in 2022, Lockheed Martin was awarded a contract to partner with NASA's Marshall Space Flight Center in developing the MAV and engines from Northrop Grumman. It is planned to be catapulted upward as high as 4.5 m (15 ft) above the lander – or 6.5 m (21 ft) above the Martian surface, into the air just before it ignites, at a rate of 5 m (16 ft) per second, to remove the odds of wrong liftoff like slipping or tilting of SRL under rocket's shear weight and exhaust at liftoff. The front would be tossed a bit harder than the back, causing the rocket to point upward, toward the Martian sky. Thus, the Vertically Ejected Controlled Tip-off Release (VECTOR) system adds a slight rotation during launch, pitching the rocket up and away from the surface. MAV would enter a 380-kilometre (240 mi) orbit. It will remain stowed inside a cylinder on the SRL and will have a thermal protective coating. The rocket's first stage (SRM-1) would be burning for 75 seconds. SRM1 engine can gimbal, but most gimballing solid rocket motor nozzles are designed in a way that can't handle the extreme cold MAV will experience, so the Northrop Grumman team had to come up with something that could: a state-of-the-art trapped ball nozzle featuring a supersonic split line. After SRM1 burnout, the MAV will remain in a coast period for approximately 400s. During this time, the MPA aerodynamic fairing and entire first stage will separate from the vehicle. After stage separation, the second stage will initiate a spin up via side mounted small scale RCS thrusters. The entire second stage will be unguided and spin-stabilized at a rate of approximately 175 RPM. Having achieved the target spin rate, the second stage (SRM-2) will ignite and burn for approximately 18-20s, raising the periapsis and circularizing the orbit. The second stage is planned to be spin-stabilized to save weight in lieu of active guidance, while the Mars samples will result in an unknown payload mass distribution. Spin Stabilization allows the rocket to be lighter, so it wouldn't have to carry active control all the way to orbit. Following SRM2 burnout, the second stage will coast for up to 10 minutes while residual thrust from the SRM2 occurs. Side mounted small scale de-spin motors will then fire, reducing the spin rate to less than 40 RPM. Once the target orbit has been achieved, the MAV will command the MPA to eject the Orbiting Sample Container (OS). The spent second stage of the MAV will remain in orbit, broadcasting a hosted radio beacon signal for up to 25 days. This will aid in the capture of the OS by the ERO.

MAV is scheduled to be launched in 2028 on board the SRL lander.

Components of the Sample Return Landers

 

Concept launch set-up

 

Interior design of MAV, First Extraterrestrial Staging Rocket

 

MAV exterior design

 

MAV flight plan

 

Mars Sample Return 2020–2033 Timeline

Sample return

Earth Return Orbiter (ERO)

ERO is an ESA-developed spacecraft. It includes the NASA-built Capture and Containment and Return System (CCRS) and Electra UHF Communications Package. It will rendezvous with the samples delivered by MAV in low Mars orbit (LMO). ERO orbiter is planned to weigh ~7,000 kg (15,000 lb) (largest Mars Orbiter) and has solar arrays that have a wingspan of more than 38 m (125 ft) (these are some of the largest solar panels ever launched into space).

ERO is scheduled to launch on an Ariane 64 rocket in 2027 and arrive at Mars in 2029, using ion propulsion and a separate chemical propulsion element to gradually reach the proper orbit of 325 km (202 mi) and then rendezvous with the orbiting sample. The MAV's second stage's radio beacon will give controllers the information they need to get the ESA Earth Return Orbiter close enough to the Orbiting Sample to see it through reflective light and capture it for return to earth. To do this, ERO would use high-performance cameras to detect the Orbiting Sample at over 1,000 km (620 mi) distance. Once "locked on" would track it continuously using cameras and LiDARs throughout the rendezvous phase. Once aligned with the sample container, the Capture, Containment, and Return System would power on, open its capture lid, and turn on its capture sensors. ESA's orbiter would then push itself toward the sample container at about 1 to 2 inches (2.5 to 5 centimeters) per second to overtake and "swallow" it. After detecting that the sample container is safely inside, the Capture, Containment, and Return System would quickly close its lid. Thus, the orbiter will retrieve and seal the canisters in orbit and use a NASA-built robotic arm to place the sealed container into an Earth-entry capsule. The 600 kg (1,300 lb) CCRS would be responsible for thoroughly sterilizing the exterior of the Orbiting Sample and double sealing it inside the EES, creating a secondary containment barrier to keep the samples safely isolated and intact for maximum scientific return. It will raise its orbit, jettison the propulsion element (includes ~500 kg (1,100 lb) of CCRS hardware, which is of not use after sterilizing samples), and return to Earth during the 2033 Mars-to-Earth transfer window.

ERO will measure the total radiation dose received throughout the entire flight. Results will help monitor the health of the spacecraft and provide important information on how to protect human explorers in future trips to Mars.

Earth Entry Vehicle (EEV)

The Capture/Containment and Return System (CCRS) would stow the sample in the EEV. EEV would return to Earth and land passively, without a parachute. About a week before arrival at Earth, and only after successfully completing a full system safety check-out, the ERO spacecraft would be configured to perform the Earth return phase. When the orbiter is three days away from Earth, the EES would be released from the main spacecraft and fly a precision entry trajectory to a predetermined landing site. Shortly after separation, the orbiter itself would perform a series of maneuvers to enter orbit around the Sun, never to return to Earth. The desert sand at the Utah Test and Training Range and shock absorbing materials in the vehicle were planned to protect the samples from impact forces. EEV is scheduled to land on Earth in 2033.

Division by zero

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Division_by_zero

The function y = 1/x. As x approaches 0 from the right, y approaches infinity. As x approaches 0 from the left, y approaches negative infinity.

In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as $\frac{a}{0}$, where a is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number that, when multiplied by 0, gives a (assuming $a\ne 0$); thus, division by zero is undefined (a type of singularity). Since any number multiplied by zero is zero, the expression ${\displaystyle \frac{0}{0}}$ is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to $\frac{a}{0}$ is contained in Anglo-Irish philosopher George Berkeley's criticism of infinitesimal calculus in 1734 in The Analyst ("ghosts of departed quantities").

There are mathematical structures in which $\frac{a}{0}$ is defined for some a such as in the Riemann sphere (a model of the extended complex plane) and the projectively extended real line; however, such structures do not satisfy every ordinary rule of arithmetic (the field axioms).

In computing, a program error may result from an attempt to divide by zero. Depending on the programming environment and the type of number (e.g., floating point, integer) being divided by zero, it may generate positive or negative infinity by the IEEE 754 floating-point standard, generate an exception, generate an error message, cause the program to terminate, result in a special not-a-number value, or crash.

Elementary arithmetic

When division is explained at the elementary arithmetic level, it is often considered as splitting a set of objects into equal parts. As an example, consider having ten cookies, and these cookies are to be distributed equally to five people at a table. Each person would receive ${\displaystyle \frac{10}{5}=2}$ cookies. Similarly, if there are ten cookies, and only one person at the table, that person would receive ${\displaystyle \frac{10}{1}=10}$ cookies.

So, for dividing by zero, what is the number of cookies that each person receives when 10 cookies are evenly distributed among 0 people at a table? Certain words can be pinpointed in the question to highlight the problem. The problem with this question is the "when". There is no way to distribute 10 cookies to nobody. Therefore, ${\displaystyle \frac{10}{0}}$—at least in elementary arithmetic—is said to be either meaningless or undefined.

If there are 5 cookies and 2 people, the problem is in "evenly distribute". In any integer partition of 5 things into 2 parts, either one of the parts of the partition will have more elements than the other or there will be a remainder (written as 5/2 = 2 r1). Or, the problem with 5 cookies and 2 people can be solved by cutting one cookie in half, which introduces the idea of fractions (5/2 = 2+1/2) . The problem with 5 cookies and 0 people, on the other hand, cannot be solved in any way that preserves the meaning of "divides".

In elementary algebra, another way of looking at division by zero is that division can always be checked using multiplication. Considering the 10/0 example above, setting x = 10/0, if x equals ten divided by zero, then x times zero equals ten, but there is no x that, when multiplied by zero, gives ten (or any number other than zero). If, instead of x = 10/0, x = 0/0, then every x satisfies the question "what number x, multiplied by zero, gives zero?"

Early attempts

The Brāhmasphuṭasiddhānta of Brahmagupta (c. 598–668) is the earliest text to treat zero as a number in its own right and to define operations involving zero. The author could not explain division by zero in his texts: his definition can be easily proven to lead to algebraic absurdities. According to Brahmagupta,

A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.

In 830, Mahāvīra unsuccessfully tried to correct the mistake Brahmagupta made in his book Ganita Sara Samgraha: "A number remains unchanged when divided by zero."

Algebra

The four basic operations – addition, subtraction, multiplication and division – as applied to whole numbers (positive integers), with some restrictions, in elementary arithmetic are used as a framework to support the extension of the realm of numbers to which they apply. For instance, to make it possible to subtract any whole number from another, the realm of numbers must be expanded to the entire set of integers in order to incorporate the negative integers. Similarly, to support division of any integer by any other, the realm of numbers must expand to the rational numbers. During this gradual expansion of the number system, care is taken to ensure that the "extended operations", when applied to the older numbers, do not produce different results. Loosely speaking, since division by zero has no meaning (is undefined) in the whole number setting, this remains true as the setting expands to the real or even complex numbers.

As the realm of numbers to which these operations can be applied expands there are also changes in how the operations are viewed. For instance, in the realm of integers, subtraction is no longer considered a basic operation since it can be replaced by addition of signed numbers. Similarly, when the realm of numbers expands to include the rational numbers, division is replaced by multiplication by certain rational numbers. In keeping with this change of viewpoint, the question, "Why can't we divide by zero?", becomes "Why can't a rational number have a zero denominator?". Answering this revised question precisely requires close examination of the definition of rational numbers.

In the modern approach to constructing the field of real numbers, the rational numbers appear as an intermediate step in the development that is founded on set theory. First, the natural numbers (including zero) are established on an axiomatic basis such as Peano's axiom system and then this is expanded to the ring of integers. The next step is to define the rational numbers keeping in mind that this must be done using only the sets and operations that have already been established, namely, addition, multiplication and the integers. Starting with the set of ordered pairs of integers, {(a, b)} with b ≠ 0, define a binary relation on this set by (a, b) ≃ (c, d) if and only if ad = bc. This relation is shown to be an equivalence relation and its equivalence classes are then defined to be the rational numbers. It is in the formal proof that this relation is an equivalence relation that the requirement that the second coordinate is not zero is needed (for verifying transitivity).

The above explanation may be too abstract and technical for many purposes, but if one assumes the existence and properties of the rational numbers, as is commonly done in elementary mathematics, the "reason" that division by zero is not allowed is hidden from view. Nevertheless, a (non-rigorous) justification can be given in this setting.

It follows from the properties of the number system we commonly use that if b ≠ 0, then the equation a/b = c is equivalent to a = b × c. If we allowed a zero denominator, we would arrive at either a contradiction, or an equation that was true no matter what value we assigned the "fraction". If a/0 were a number c, then it would follow that a = 0 × c = 0. However, the single number c would then have to be determined by the equation 0 = 0 × c, which is satisfied by every number. We cannot assign a numerical value to 0/0 and instead say that division by zero is not allowed.

Division as the inverse of multiplication

The concept that explains division in algebra is that it is the inverse of multiplication. For example,

$${\displaystyle \frac{6}{3}=2}$$

since 2 is the value for which the unknown quantity in

$${\displaystyle ?\times 3=6}$$

is true. But the expression

$${\displaystyle \frac{6}{0}=x}$$

requires a value to be found for the unknown quantity in

$${\displaystyle x\times 0=6.}$$

But any number multiplied by 0 is 0 and so there is no number that solves the equation.

The expression

$${\displaystyle \frac{0}{0}=x}$$

requires a value to be found for the unknown quantity in

$${\displaystyle x\times 0=0.}$$

Again, any number multiplied by 0 is 0 and so this time every number solves the equation instead of there being a single number that can be taken as the value of 0/0.

In general, a single value can't be assigned to a fraction where the denominator is 0 so the value remains undefined.

Fallacies

Further information: Mathematical fallacy

A compelling reason for not allowing division by zero is that, if it were allowed, many absurd results (i.e., fallacies) would arise. When working with numerical quantities it is easy to determine when an illegal attempt to divide by zero is being made. For example, consider the following computation.

With the assumptions:

$${\displaystyle \begin{array}{rl}0\times 1& =0,\\ 0\times 2& =0,\end{array}}$$

the following is true:

$${\displaystyle 0\times 1=0\times 2.}$$

Dividing both sides by zero gives:

$${\displaystyle \begin{array}{rl}\frac{0\times 1}{0}& =\frac{0\times 2}{0}\\ \frac{0}{0}\times 1& =\frac{0}{0}\times 2.\end{array}}$$

Simplified, this yields:

$${\displaystyle 1=2.}$$

The fallacy here is the assumption that dividing 0 by 0 is a legitimate operation with the same properties as dividing by any other number.

However, it is possible to disguise a division by zero in an algebraic argument, leading to invalid proofs that, for instance, 1 = 2 such as the following:

Let 1 = x.

Multiply by x to get

$${\displaystyle x=x^2.}$$

Subtract 1 from each side to get

$${\displaystyle x-1=x^2-1.}$$

Divide both sides by x − 1

$${\displaystyle \begin{array}{rl}\frac{x-1}{x-1}& =\frac{x^2-1}{x-1}\\ & =\frac{(x+1)(x-1)}{x-1},\end{array}}$$

which simplifies to

$${\displaystyle 1=x+1.}$$

But, since x = 1,

$${\displaystyle 1=1+1=2}$$

and therefore

$${\displaystyle 1=2.}$$

The disguised division by zero occurs since x − 1 = 0 when x = 1.

Analysis

Extended real line

See also: l'Hôpital's rule

At first glance it seems possible to define a/0 by considering the limit of a/b as b approaches 0.

For any positive a, the limit from the right is

$${\displaystyle \underset{b\to 0^{+}}{lim}\frac{a}{b}=+\mathrm{\infty }}$$

however, the limit from the left is

$${\displaystyle \underset{b\to 0^{-}}{lim}\frac{a}{b}=-\mathrm{\infty }}$$

and so the ${\displaystyle \underset{b\to 0}{lim}\frac{a}{b}}$ is undefined (the limit is also undefined for negative a).

Furthermore, there is no obvious definition of 0/0 that can be derived from considering the limit of a ratio. The limit

$${\displaystyle \underset{(a,b)\to (0,0)}{lim}\frac{a}{b}}$$

does not exist. Limits of the form

$${\displaystyle \underset{x\to 0}{lim}\frac{f(x)}{g(x)}}$$

in which both f(x) and g(x) approach 0 as x approaches 0, may equal any real or infinite value, or may not exist at all, depending on the particular functions f and g.

For example, consider:

$${\displaystyle \underset{x\to 1}{lim}\frac{x^2-1}{x-1}}$$

This initially appears to be indeterminate. However:

$${\displaystyle \begin{array}{rl}& =\underset{x\to 1}{lim}\frac{(x-1)(x+1)}{x-1}\\ & =\underset{x\to 1}{lim}(x+1)\\ & =2\end{array}}$$

and so the limit exists, and is equal to ${\displaystyle 2}$.

These and other similar facts show that the expression ${\displaystyle \frac{0}{0}}$ cannot be well-defined as a limit.

Formal operations

A formal calculation is one carried out using rules of arithmetic, without consideration of whether the result of the calculation is well-defined. Thus, it is sometimes useful to think of a/0, where a ≠ 0, as being ${\displaystyle \mathrm{\infty }}$. This infinity can be either positive, negative, or unsigned, depending on context. For example, formally:

$${\displaystyle \underset{x\to 0}{lim}\frac{1}{x}=\frac{\underset{x\to 0}{lim}1}{\underset{x\to 0}{lim}x}=\mathrm{\infty }.}$$

As with any formal calculation, invalid results may be obtained. A logically rigorous (as opposed to formal) computation would assert only that

$${\displaystyle \underset{x\to 0^{+}}{lim}\frac{1}{x}=+\mathrm{\infty }\text{ and }\underset{x\to 0^{-}}{lim}\frac{1}{x}=-\mathrm{\infty }.}$$

Since the one-sided limits are different, the two-sided limit does not exist in the standard framework of the real numbers. Also, the fraction 1/0 is left undefined in the extended real line, therefore it and

$${\displaystyle \frac{\underset{x\to 0}{lim}1}{\underset{x\to 0}{lim}x}}$$

are meaningless expressions.

Projectively extended real line

The set ${\displaystyle \mathbb{R}\cup \{\mathrm{\infty }\}}$ is the projectively extended real line, which is a one-point compactification of the real line. Here ${\displaystyle \mathrm{\infty }}$ means an unsigned infinity or point at infinity, an infinite quantity that is neither positive nor negative. This quantity satisfies ${\displaystyle -\mathrm{\infty }=\mathrm{\infty }}$, which is necessary in this context. In this structure, ${\displaystyle \frac{a}{0}=\mathrm{\infty }}$ can be defined for nonzero a, and ${\displaystyle \frac{a}{\mathrm{\infty }}=0}$ when a is not ${\displaystyle \mathrm{\infty }}$. It is the natural way to view the range of the tangent function and cotangent functions of trigonometry: tan(x) approaches the single point at infinity as x approaches either +π/2 or −π/2 from either direction.

This definition leads to many interesting results. However, the resulting algebraic structure is not a field, and should not be expected to behave like one. For example, ${\displaystyle \mathrm{\infty }+\mathrm{\infty }}$ is undefined in this extension of the real line.

Riemann sphere

The set ${\displaystyle {\mathbb{C}}^{\ast }=\mathbb{C}\cup \{\tilde{\mathrm{\infty }}\}}$ is the Riemann sphere, which is of major importance in complex analysis. Here ${\displaystyle \tilde{\mathrm{\infty }}}$ represents complex infinity, which is also a point at infinity. This set is analogous to the projectively extended real line, except that it is based on the field of complex numbers. In the Riemann sphere, ${\displaystyle \frac{1}{0}=\tilde{\mathrm{\infty }}}$ and ${\displaystyle \frac{1}{\tilde{\mathrm{\infty }}}=0}$, but ${\displaystyle \frac{0}{0}}$, ${\displaystyle \frac{\tilde{\mathrm{\infty }}}{\tilde{\mathrm{\infty }}}}$, and ${\displaystyle 0\times \tilde{\mathrm{\infty }}}$ are undefined.

Higher mathematics

Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures.

Non-standard analysis

In the hyperreal numbers and the surreal numbers, division by zero is still impossible, but division by non-zero infinitesimals is possible.

Distribution theory

In distribution theory one can extend the function $\frac{1}{x}$ to a distribution on the whole space of real numbers (in effect by using Cauchy principal values). It does not, however, make sense to ask for a "value" of this distribution at x = 0; a sophisticated answer refers to the singular support of the distribution.

Linear algebra

In matrix algebra (or linear algebra in general), one can define a pseudo-division, by setting a/b = ab⁺, in which b⁺ represents the pseudoinverse of b. It can be proven that if b⁻¹ exists, then b⁺ = b⁻¹. If b equals 0, then b⁺ = 0.

Abstract algebra

In abstract algebra, the integers, the rational numbers, the real numbers, and the complex numbers can be abstracted to more general algebraic structures, such as a commutative ring, which is a mathematical structure where addition, subtraction, and multiplication behave as they do in the more familiar number systems, but division may not be defined. Adjoining a multiplicative inverses to a commutative ring is called localization. However, the localization of every commutative ring at zero is the trivial ring, where ${\displaystyle 0=1}$, so nontrivial commutative rings do not have inverses at zero, and thus division by zero is undefined for nontrivial commutative rings.

Nevertheless, any number system that forms a commutative ring can be extended to a seldom used structure called a wheel in which division by zero is always possible. However, the resulting mathematical structure is no longer a commutative ring, as multiplication no longer distributes over addition. Furthermore, in a wheel, division of an element by itself no longer results in the multiplicative identity element ${\displaystyle 1}$, and if the original system was an integral domain, the multiplication in the wheel no longer results in a cancellative semigroup.

The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such as rings and fields. In a field, every nonzero element is invertible under multiplication; as above, division poses problems only when attempting to divide by zero. This is likewise true in a skew field (which for this reason is called a division ring). However, in other rings, division by nonzero elements may also pose problems. For example, the ring Z/6Z of integers mod 6. The meaning of the expression $\frac{2}{2}$ should be the solution x of the equation ${\displaystyle 2x=2}$. But in the ring Z/6Z, 2 is a zero divisor. This equation has two distinct solutions, x = 1 and x = 4, so the expression $\frac{2}{2}$ is undefined.

In field theory, the expression $\frac{a}{b}$ is only shorthand for the formal expression ab⁻¹, where b⁻¹ is the multiplicative inverse of b. Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when b is zero. Modern texts, that define fields as a special type of ring, include the axiom 0 ≠ 1 for fields (or its equivalent) so that the zero ring is excluded from being a field. In the zero ring, division by zero is possible, which shows that the other field axioms are not sufficient to exclude division by zero in a field.

Computer arithmetic

Most calculators, such as this Texas Instruments TI-86, will halt execution and display an error message when the user or a running program attempts to divide by zero.

Division by zero on Android 2.2.1's calculator app shows the symbol of infinity.

The IEEE floating-point standard, supported by almost all modern floating-point units, specifies that every floating-point arithmetic operation, including division by zero, has a well-defined result. The standard supports signed zero, as well as infinity and NaN (not a number). There are two zeroes: +0 (positive zero) and −0 (negative zero) and this removes any ambiguity when dividing. In IEEE 754 arithmetic, a ÷ +0 is positive infinity when a is positive, negative infinity when a is negative, and NaN when a = ±0. The infinity signs change when dividing by −0 instead.

Dividing a number by 0 in Windows 11's calculator

The justification for this definition is to preserve the sign of the result in case of arithmetic underflow. For example, in the single-precision computation 1/(x/2), where x = ±2⁻¹⁴⁹, the computation x/2 underflows and produces ±0 with sign matching x, and the result will be ±∞ with sign matching x. The sign will match that of the exact result ±2¹⁵⁰, but the magnitude of the exact result is too large to represent, so infinity is used to indicate overflow.

Terminal Division by 0 in Windows 11

Integer division by zero is usually handled differently from floating point since there is no integer representation for the result. Some processors generate an exception when an attempt is made to divide an integer by zero, although others will simply continue and generate an incorrect result for the division. The result depends on how division is implemented, and can either be zero, or sometimes the largest possible integer.

Because of the improper algebraic results of assigning any value to division by zero, many computer programming languages (including those used by calculators) explicitly forbid the execution of the operation and may prematurely halt a program that attempts it, sometimes reporting a "Divide by zero" error. In these cases, if some special behavior is desired for division by zero, the condition must be explicitly tested (for example, using an if statement). Some programs (especially those that use fixed-point arithmetic where no dedicated floating-point hardware is available) will use behavior similar to the IEEE standard, using large positive and negative numbers to approximate infinities. In some programming languages, an attempt to divide by zero results in undefined behavior. The graphical programming language Scratch 2.0 and 3.0 used in many schools returns Infinity or −Infinity depending on the sign of the dividend.

In two's complement arithmetic, attempts to divide the smallest signed integer by −1 are attended by similar problems, and are handled with the same range of solutions, from explicit error conditions to undefined behavior.

Most calculators will either return an error or state that 1/0 is undefined; however, some TI and HP graphing calculators will evaluate (1/0)² to ∞.

Microsoft Math Solver and Wolfram Mathematica return ComplexInfinity for 1/0. Maple and SageMath return an error message for 1/0, and infinity for 1/0.0 (0.0 tells these systems to use floating-point arithmetic instead of algebraic arithmetic).

Some modern calculators allow division by zero in special cases, where it will be useful to students and, presumably, understood in context by mathematicians. Some calculators, the online Desmos calculator is one example, allow arctangent(1/0). Students are often taught that the inverse cotangent function, arccotangent, should be calculated by taking the arctangent of the reciprocal, and so a calculator may allow arctangent(1/0), giving the output ${\displaystyle \frac{\pi }{2}}$, which is the correct value of arccotangent 0. The mathematical justification is that the limit as x goes to zero of arctangent 1/x is ${\displaystyle \frac{\pi }{2}}$.

Historical accidents

On September 21, 1997, a division by zero error in the "Remote Data Base Manager" aboard USS Yorktown (CG-48) brought down all the machines on the network, causing the ship's propulsion system to fail.