title: Harold Ford Jr.

text: When Harold Sr. decided not to seek a 12th term in Congress in 1996, Harold Jr. entered the race and became the favorite in the Democratic primary, which was widely regarded as the real contest in the heavily Democratic, black-majority 9th district. Ford arranged his schedule for his last semester of law school so he would not have Monday or Friday classes and would be able to fly home to Memphis for an extended weekend each week to continue his campaign. As was expected, he easily won the Democratic primary, followed by his election in November. Taking office at the age of 26, he was one of the youngest members of Congress in US history and the youngest in the 105th and 106th Congresses. He was reelected four times without substantive Republican opposition, by an average of 80 percent of the vote. In 2000, Ford was the keynote speaker for the 2000 Democratic National Convention supporting then Vice President Al Gore for the Democratic nomination for President.


title: Harold Ford Jr.

text: On 4 November 1999, Ford voted in favor of the Gramm–Leach–Bliley Act. This act repealed much of the Glass–Steagall Act of 1933, which had been enacted to prevent any one organization from acting as any combination of an investment bank, a commercial bank, and an insurance company. The resulting repeal allowed many banks and insurance companies to gamble with money raised from savings and checking bank accounts or insurance policies. Several economists, notably Nobel laureate Joseph Stiglitz, point to the repeal of Glass–Steagall as helping to create the conditions of the 2007 financial crisis.


title: Harold Ford Jr.

text: On October 10, 2002, he was among the 81 House Democrats who voted in favor of authorizing the invasion of Iraq.


title: Harold Ford Jr.

text: After the Democrats lost seven Congressional seats in the 2002 elections, Ford announced his candidacy for House Democratic Leader, challenging then-House Minority Whip Nancy Pelosi, arguing that current leadership was ineffective. Ford was defeated but exceeded initial expectations in the amount of support he received. Although his name was mentioned as a possible Democratic vice presidential candidate in 2004, he was ineligible for the office due to his age (four months shy of 35 on Inauguration Day 2005).


title: Harold Ford Jr.

text: A June 7, 2005, article in "The Washington Times" reported that from 1998 to 2003, Ford took 61 privately funded trips but did not file travel disclosure forms with the House clerk for the trips, as required by the chamber's ethics rules, until August 2003. Ford's office called the late filings a "mere oversight", since Ford had filed the required financial disclosure statements for the trips at the time they occurred.


title: Harold Ford Jr.

text: In November 2005, when Ohio Republican Congresswoman Jean Schmidt implied that Pennsylvania Democrat John Murtha was a "coward" in response to Murtha's proposal for a withdrawal of American forces from Iraq, Ford charged across the House floor to the Republican side during the resulting uproar in the chamber, shouting "Say it to Murtha!" (or "Say Murtha's name!" depending on the source) while waving his finger at Schmidt. He had to be restrained by fellow Democrat Dave Obey of Wisconsin. Like many Democrats, Ford believed Schmidt's remarks (which she later withdrew) were an unwarranted "cheap shot" against Murtha, a veteran of the Marine Corps.


title: Harold Ford Jr.

text: In Congress, Ford opposed benefits for same-sex couples and supported the Federal Marriage Amendment, which would have defined marriage as one man and one woman. He told Democrats they should be more supportive of the Iraq War, and criticized Senate Democrats who attempted to filibuster the nomination of Samuel Alito. He was one of the few Democrats who voted for the bankruptcy bill, and he defined himself as an anti-abortion candidate, supporting some restrictions on abortion, including a ban on intact dilation and extraction, called "partial-birth abortion" by opponents. However, the National Right to Life Committee says that while in Congress he voted against the anti-abortion position 87% of the time. He supported the Republican effort to intercede in the Terri Schiavo case. He opposed President George W. Bush's energy proposals (including oil drilling in Arctic National Wildlife Refuge), demonstrated support for adoption rights of same-sex couples, is in favor of federal funding of embryonic stem cell research, supported universal healthcare coverage, opposed the death penalty and indicated a willingness to reform drug policy.


title: Harold Ford Jr.

text: In addition, Ford sat on the House Budget Committee and the House Committee on Financial Services. He also served on the Transformation Advisory Group, a group of political, military and academic leaders who worked with the Department of Defense to assess the needs of the armed forces. Ford was a member of the New Democrat Coalition, the Congressional Black Caucus and the Blue Dog Coalition.


title: Harold Ford Jr.

text: In 2002, Ford was mentioned as a possible Democratic candidate for the Senate seat being vacated by Fred Thompson, but he declined to run. Instead, he supported fellow Congressman Bob Clement who would lose to former Republican Governor Lamar Alexander in the general election.


title: Harold Ford Jr.

text: In 2006, Ford did not run for re-election to the House of Representatives, due to his campaign for the United States Senate in Tennessee, which he lost. His younger brother, Jake Ford, ran for the 9th district seat as an Independent, but lost to Democrat Steve Cohen.


title: Harold Ford Jr.

text: On April 6, 2005, during an interview on C-SPAN's call-in show "Washington Journal", Ford confirmed that he would be running for the Senate. He filed the papers necessary to officially begin his Senate campaign on May 25, 2005.


title: Harold Ford Jr.

text: Democratic State Senator Rosalind Kurita briefly challenged Ford for the nomination but dropped out of the primary because of inadequate fundraising, effectively handing Ford the nomination. On August 3, 2006, Ford overwhelmingly won the Democratic primary. After the primary, Ford's supporters held a large victory celebration at Nashville's LP Field. Among the speakers was former U.S. President Bill Clinton.


title: Harold Ford Jr.

text: Ford faced Republican Bob Corker in the November 2006 election. Not long after Corker's primary victory was assured, Ford challenged Corker to seven televised debates across the state. In response, Corker said he would debate Ford, though he did not agree to seven debates.


title: Harold Ford Jr.

text: the Republican Party ran radio and television ads characterized by some as racist. A radio spot, referred to by critics as the "jungle drums" ad, had drums playing when Ford's name was mentioned and patriotic music when Corker's name was spoken. This ad was criticized as attacking Ford's race by evoking images of primitive, chanting African tribes. A television ad that received more attention featured satirical "man‑on‑the‑street" interviews purporting to support Ford, including one in which a blond white woman (implied to be nude) talks about meeting Ford at "the Playboy party"; she returns at the end of the ad to wink and whisper in a seductive tone, "Harold, call me." The ad was denounced by many people, including Republican former Senator William Cohen, who called it "a very serious appeal to a racist sentiment", and Corker asked the Republican leadership to pull the ad. The ad was retired one day after Republican National Committee Chairman Ken Mehlman said he had no authority to discontinue the ad and disagreed with the negative characterizations of it.


title: Harold Ford Jr.

text: Corker and Ford participated in a televised debate in Memphis on October 7, in Corker's hometown of Chattanooga on October 10, and in Nashville on October 28. In January 2006, NBC's "Meet the Press" extended an open invitation for the candidates to debate on the nationally televised show.


title: Harold Ford Jr.

text: On November 8, Ford conceded the election to Corker, who defeated Ford by less than three percentage points.


title: Harold Ford Jr.

text: In December 2006, the "Los Angeles Times" reported that Ford told students at an L.A.-area school that he might run again in 2008 for the Senate seat held by Republican Lamar Alexander, but in January 2007 Ford said that he had no plans to challenge the incumbent. Instead, Ford has said that he "hopes to spend a lot of time at home, perhaps do some teaching and work with Governor Bredesen on some issues in Tennessee."


title: Harold Ford Jr.

text: In March 2007, Ford joined the financial services firm Merrill Lynch as a vice chairman and senior policy adviser. In the same month he was hired by Fox News Channel as a political contributor. In March 2008, he moved from Fox to MSNBC as a news analyst, appearing as a panelist on David Gregory's "Race for the White House", "Hardball", and "Morning Joe".


title: Harold Ford Jr.

text: Ford was appointed visiting professor of public policy at Vanderbilt University in 2007 and taught a class on American political leadership. In October 2007, Ford was appointed as the inaugural Barbara Jordan Visiting Professor at the Lyndon B. Johnson School of Public Affairs at the University of Texas at Austin. As of the spring 2010 semester, he is a visiting professor at New York University's Wagner School of Public Service where he teaches Policy Formation: U.S. Domestic Policy. In the fall of 2015 he was a visiting faculty member at the University of Michigan Ford School of Public Policy.


title: Harold Ford Jr.

text: Ford worked at the Wall Street firm Morgan Stanley as a managing director. In late 2017, erroneous media reports said that he was fired for sexual misconduct and Morgan Stanley issued a statement the following month saying that no sexual misconduct took place. In January 2018, "The New York Times" reported that Ford reached a legal settlement with Morgan Stanley.


title: Harold Ford Jr.

text: Ford is on the Board of Selectors of Jefferson Awards for Public Service. He is also a member of the ReFormers Caucus of Issue One.


title: Harold Ford Jr.

text: Ford considered a primary challenge to Senator Kirsten Gillibrand in New York in 2010. Sources close to Mayor Michael Bloomberg of New York have said that he would consider supporting someone "of Mr. Ford's stature". However, it was also reported that Mayor Bloomberg "reassured Mr. Reid that he was not personally involved in the effort to promote a Ford candidacy." Chuck Schumer sought to dissuade Ford from running. Ford's spokesperson on January 11, 2010, stated Ford had become a supporter of same-sex marriage after "listening to the debate in state legislatures across the country".


title: Harold Ford Jr.

text: On February 5, 2010, Gillibrand held a press conference at which she raised questions as to whether Ford, as an executive of Merrill Lynch, received taxpayer-backed bonuses from Bank of America, stemming from the federal bailout. She was joined by New York City Public Advocate Bill de Blasio, who endorsed her candidacy.


title: Harold Ford Jr.

text: On February 12, it was reported that Ford's NBC contract was suspended due to his potential campaign, and that he had taken an unpaid leave of absence from Merrill Lynch for the same reason. On March 1, Ford stated in an op-ed article published by "The New York Times" that he would not run against Gillibrand.


title: Harold Ford Jr.

text: Ford married Emily Threlkeld, who works in public relations for Carolina Herrera in New York, on April 26, 2008. They have a daughter, Georgia Walker Ford, who was born in December 2013 in New York City. She is named for Ford's great-grandmother Georgia, and Threlkeld's mother's maiden name, Walker. Their second child, a son, Harold Eugene Ford III, was born in May 2015.


title: Harold Ford Jr.

text: <nowiki>*</nowiki>Write-in and minor candidate notes: In 1996, Mary D. Taylor received 498 votes; Anthony Burton received 424 votes; Greg Voehringer received 327 votes; Tom Jeanette received 222 votes; Del Gill received 199 votes; Bill Taylor received 179 votes; Johnny E. Kelly received 156 votes; Don Fox received 146 votes; and write-ins received 10 votes. In 1998, Johnny Kelly received 775 votes; Greg Voehringer received 567 votes; and write-ins received 2 votes. In 2000, write-ins received 36 votes. In 2002, write-ins received 148 votes. In 2004, Jim Maynard received 166 votes.


title: Harold Ford Jr.

text: <nowiki>*</nowiki>Write-in and minor candidate notes: In 2006, David "None of the Above" Gatchell received 3,746 votes, Emory "Bo" Heyward received 3,580 votes, H. Gary Keplinger received 3,033 votes and Chris Lugo (Green) received 2,589 votes.


title: Rocket engine

text: A rocket engine uses stored rocket propellants as the reaction mass for forming a high-speed propulsive jet of fluid, usually high-temperature gas. Rocket engines are reaction engines, producing thrust by ejecting mass rearward, in accordance with Newton's third law. Most rocket engines use the combustion of reactive chemicals to supply the necessary energy, but non-combusting forms such as cold gas thrusters and nuclear thermal rockets also exist. Vehicles propelled by rocket engines are commonly called rockets. Rocket vehicles carry their own oxidiser, unlike most combustion engines, so rocket engines can be used in a vacuum to propel spacecraft and ballistic missiles.


title: Rocket engine

text: Compared to other types of jet engine, rocket engines are the lightest and have the highest thrust, but are the least propellant-efficient (they have the lowest specific impulse). The ideal exhaust is hydrogen, the lightest of all elements, but chemical rockets produce a mix of heavier species, reducing the exhaust velocity.


title: Rocket engine

text: Thermal rockets use an inert propellant, heated by electricity (electrothermal propulsion) or a nuclear reactor (nuclear thermal rocket).


title: Rocket engine

text: Rocket engines produce thrust by the expulsion of an exhaust fluid that has been accelerated to high speed through a propelling nozzle. The fluid is usually a gas created by high pressure () combustion of solid or liquid propellants, consisting of fuel and oxidiser components, within a combustion chamber. As the gases expand through the nozzle, they are accelerated to very high (supersonic) speed, and the reaction to this pushes the engine in the opposite direction. Combustion is most frequently used for practical rockets, as the laws of thermodynamics (specifically Carnot's theorem) dictate that high temperatures and pressures are desirable for the best thermal efficiency. Nuclear thermal rockets are capable of higher efficiencies, but currently have environmental problems which preclude their routine use in the Earth's atmosphere and cislunar space.


title: Rocket engine

text: For model rocketry, an available alternative to combustion is the water rocket pressurized by compressed air, carbon dioxide, nitrogen, or any other readily available, inert gas.


title: Rocket engine

text: Rocket propellant is mass that is stored, usually in some form of tank, or within the combustion chamber itself, prior to being ejected from a rocket engine in the form of a fluid jet to produce thrust.


title: Rocket engine

text: Chemical rocket propellants are the most commonly used. These undergo exothermic chemical reactions producing a hot gas jet for propulsion. Alternatively, a chemically inert reaction mass can be heated by a high-energy power source through a heat exchanger in lieu of a combustion chamber.


title: Rocket engine

text: Solid rocket propellants are prepared in a mixture of fuel and oxidising components called "grain", and the propellant storage casing effectively becomes the combustion chamber.


title: Rocket engine

text: Liquid-fuelled rockets force separate fuel and oxidiser components into the combustion chamber, where they mix and burn. Hybrid rocket engines use a combination of solid and liquid or gaseous propellants. Both liquid and hybrid rockets use "injectors" to introduce the propellant into the chamber. These are often an array of simple jets – holes through which the propellant escapes under pressure; but sometimes may be more complex spray nozzles. When two or more propellants are injected, the jets usually deliberately cause the propellants to collide as this breaks up the flow into smaller droplets that burn more easily.


title: Rocket engine

text: For chemical rockets the combustion chamber is typically cylindrical, and flame holders, used to hold a part of the combustion in a slower-flowing portion of the combustion chamber, are not needed. The dimensions of the cylinder are such that the propellant is able to combust thoroughly; different rocket propellants require different combustion chamber sizes for this to occur.


title: Rocket engine

text: The temperatures and pressures typically reached in a rocket combustion chamber in order to achieve practical thermal efficiency are extreme compared to a non-afterburning airbreathing jet engine. No atmospheric nitrogen is present to dilute and cool the combustion, so the propellant mixture can reach true stoichiometric ratios. This, in combination with the high pressures, means that the rate of heat conduction through the walls is very high.


title: Rocket engine

text: In order for fuel and oxidiser to flow into the chamber, the pressure of the propellants entering the combustion chamber must exceed the pressure inside the combustion chamber itself. This may be accomplished by a variety of design approaches including turbopumps or, in simpler engines, via sufficient tank pressure to advance fluid flow. Tank pressure may be maintained by several means, including a high-pressure helium pressurization system common to many large rocket engines or, in some newer rocket systems, by a bleed-off of high-pressure gas from the engine cycle to autogenously pressurize the propellant tanks For example, the self-pressurization gas system of the SpaceX Starship is a critical part of SpaceX strategy to reduce launch vehicle fluids from five in their legacy Falcon 9 vehicle family to just two in Starship, eliminating not only the helium tank pressurant but all hypergolic propellants as well as nitrogen for cold-gas reaction-control thrusters.


title: Rocket engine

text: The hot gas produced in the combustion chamber is permitted to escape through an opening (the "throat"), and then through a diverging expansion section. When sufficient pressure is provided to the nozzle (about 2.5–3 times ambient pressure), the nozzle "chokes" and a supersonic jet is formed, dramatically accelerating the gas, converting most of the thermal energy into kinetic energy. Exhaust speeds vary, depending on the expansion ratio the nozzle is designed for, but exhaust speeds as high as ten times the speed of sound in air at sea level are not uncommon. About half of the rocket engine's thrust comes from the unbalanced pressures inside the combustion chamber, and the rest comes from the pressures acting against the inside of the nozzle (see diagram). As the gas expands (adiabatically) the pressure against the nozzle's walls forces the rocket engine in one direction while accelerating the gas in the other.


title: Rocket engine

text: The most commonly used nozzle is the de Laval nozzle, a fixed geometry nozzle with a high expansion-ratio. The large bell- or cone-shaped nozzle extension beyond the throat gives the rocket engine its characteristic shape.


title: Rocket engine

text: The exit static pressure of the exhaust jet depends on the chamber pressure and the ratio of exit to throat area of the nozzle. As exit pressure varies from the ambient (atmospheric) pressure, a choked nozzle is said to be


title: Rocket engine

text: In practice, perfect expansion is only achievable with a variable-exit area nozzle (since ambient pressure decreases as altitude increases), and is not possible above a certain altitude as ambient pressure approaches zero. If the nozzle is not perfectly expanded, then loss of efficiency occurs. Grossly over-expanded nozzles lose less efficiency, but can cause mechanical problems with the nozzle. Fixed-area nozzles become progressively more under-expanded as they gain altitude. Almost all de Laval nozzles will be momentarily grossly over-expanded during startup in an atmosphere.


title: Rocket engine

text: Nozzle efficiency is affected by operation in the atmosphere because atmospheric pressure changes with altitude; but due to the supersonic speeds of the gas exiting from a rocket engine, the pressure of the jet may be either below or above ambient, and equilibrium between the two is not reached at all altitudes (see diagram).


title: Rocket engine

text: For optimal performance, the pressure of the gas at the end of the nozzle should just equal the ambient pressure: if the exhaust's pressure is lower than the ambient pressure, then the vehicle will be slowed by the difference in pressure between the top of the engine and the exit; on the other hand, if the exhaust's pressure is higher, then exhaust pressure that could have been converted into thrust is not converted, and energy is wasted.


title: Rocket engine

text: To maintain this ideal of equality between the exhaust's exit pressure and the ambient pressure, the diameter of the nozzle would need to increase with altitude, giving the pressure a longer nozzle to act on (and reducing the exit pressure and temperature). This increase is difficult to arrange in a lightweight fashion, although is routinely done with other forms of jet engines. In rocketry a lightweight compromise nozzle is generally used and some reduction in atmospheric performance occurs when used at other than the 'design altitude' or when throttled. To improve on this, various exotic nozzle designs such as the plug nozzle, stepped nozzles, the expanding nozzle and the aerospike have been proposed, each providing some way to adapt to changing ambient air pressure and each allowing the gas to expand further against the nozzle, giving extra thrust at higher altitudes.


title: Rocket engine

text: When exhausting into a sufficiently low ambient pressure (vacuum) several issues arise. One is the sheer weight of the nozzle—beyond a certain point, for a particular vehicle, the extra weight of the nozzle outweighs any performance gained. Secondly, as the exhaust gases adiabatically expand within the nozzle they cool, and eventually some of the chemicals can freeze, producing 'snow' within the jet. This causes instabilities in the jet and must be avoided.


title: Rocket engine

text: On a de Laval nozzle, exhaust gas flow detachment will occur in a grossly over-expanded nozzle. As the detachment point will not be uniform around the axis of the engine, a side force may be imparted to the engine. This side force may change over time and result in control problems with the launch vehicle.


title: Rocket engine

text: Advanced altitude-compensating designs, such as the aerospike or plug nozzle, attempt to minimize performance losses by adjusting to varying expansion ratio caused by changing altitude.


title: Rocket engine

text: For a rocket engine to be propellant efficient, it is important that the maximum pressures possible be created on the walls of the chamber and nozzle by a specific amount of propellant; as this is the source of the thrust. This can be achieved by all of:


title: Rocket engine

text: Since all of these things minimise the mass of the propellant used, and since pressure is proportional to the mass of propellant present to be accelerated as it pushes on the engine, and since from Newton's third law the pressure that acts on the engine also reciprocally acts on the propellant, it turns out that for any given engine, the speed that the propellant leaves the chamber is unaffected by the chamber pressure (although the thrust is proportional). However, speed is significantly affected by all three of the above factors and the exhaust speed is an excellent measure of the engine propellant efficiency. This is termed "exhaust velocity", and after allowance is made for factors that can reduce it, the effective exhaust velocity is one of the most important parameters of a rocket engine (although weight, cost, ease of manufacture etc. are usually also very important).


title: Rocket engine

text: For aerodynamic reasons the flow goes sonic ("chokes") at the narrowest part of the nozzle, the 'throat'. Since the speed of sound in gases increases with the square root of temperature, the use of hot exhaust gas greatly improves performance. By comparison, at room temperature the speed of sound in air is about 340 m/s while the speed of sound in the hot gas of a rocket engine can be over 1700 m/s; much of this performance is due to the higher temperature, but additionally rocket propellants are chosen to be of low molecular mass, and this also gives a higher velocity compared to air.


title: Rocket engine

text: Expansion in the rocket nozzle then further multiplies the speed, typically between 1.5 and 2 times, giving a highly collimated hypersonic exhaust jet. The speed increase of a rocket nozzle is mostly determined by its area expansion ratio—the ratio of the area of the exit to the area of the throat, but detailed properties of the gas are also important. Larger ratio nozzles are more massive but are able to extract more heat from the combustion gases, increasing the exhaust velocity.


title: Rocket engine

text: Vehicles typically require the overall thrust to change direction over the length of the burn. A number of different ways to achieve this have been flown:


title: Rocket engine

text: Rocket technology can combine very high thrust (meganewtons), very high exhaust speeds (around 10 times the speed of sound in air at sea level) and very high thrust/weight ratios (>100) "simultaneously" as well as being able to operate outside the atmosphere, and while permitting the use of low pressure and hence lightweight tanks and structure.


title: Rocket engine

text: Rockets can be further optimised to even more extreme performance along one or more of these axes at the expense of the others.


title: Rocket engine

text: The most important metric for the efficiency of a rocket engine is impulse per unit of propellant, this is called specific impulse (usually written formula_5). This is either measured as a speed (the "effective exhaust velocity" formula_6 in metres/second or ft/s) or as a time (seconds). For example, if an engine producing 100 pounds of thrust runs for 320 seconds and burns 100 pounds of propellant, then the specific impulse is 320 seconds. The higher the specific impulse, the less propellant is required to provide the desired impulse.


title: Rocket engine

text: The specific impulse that can be achieved is primarily a function of the propellant mix (and ultimately would limit the specific impulse), but practical limits on chamber pressures and the nozzle expansion ratios reduce the performance that can be achieved.


title: Rocket engine

text: Since, unlike a jet engine, a conventional rocket motor lacks an air intake, there is no 'ram drag' to deduct from the gross thrust. Consequently, the net thrust of a rocket motor is equal to the gross thrust (apart from static back pressure).


title: Rocket engine

text: The formula_7 term represents the momentum thrust, which remains constant at a given throttle setting, whereas the formula_8 term represents the pressure thrust term. At full throttle, the net thrust of a rocket motor improves slightly with increasing altitude, because as atmospheric pressure decreases with altitude, the pressure thrust term increases. At the surface of the Earth the pressure thrust may be reduced by up to 30%, depending on the engine design. This reduction drops roughly exponentially to zero with increasing altitude.


title: Rocket engine

text: Maximum efficiency for a rocket engine is achieved by maximising the momentum contribution of the equation without incurring penalties from over expanding the exhaust. This occurs when formula_9. Since ambient pressure changes with altitude, most rocket engines spend very little time operating at peak efficiency.


title: Rocket engine

text: Since specific impulse is force divided by the rate of mass flow, this equation means that the specific impulse varies with altitude.


title: Rocket engine

text: Due to the specific impulse varying with pressure, a quantity that is easy to compare and calculate with is useful. Because rockets choke at the throat, and because the supersonic exhaust prevents external pressure influences travelling upstream, it turns out that the pressure at the exit is ideally exactly proportional to the propellant flow formula_10, provided the mixture ratios and combustion efficiencies are maintained. It is thus quite usual to rearrange the above equation slightly:


title: Rocket engine

text: Rockets can be throttled by controlling the propellant combustion rate formula_10 (usually measured in kg/s or lb/s). In liquid and hybrid rockets, the propellant flow entering the chamber is controlled using valves, in solid rockets it is controlled by changing the area of propellant that is burning and this can be designed into the propellant grain (and hence cannot be controlled in real-time).


title: Rocket engine

text: Rockets can usually be throttled down to an exit pressure of about one-third of ambient pressure (often limited by flow separation in nozzles) and up to a maximum limit determined only by the mechanical strength of the engine.


title: Rocket engine

text: In practice, the degree to which rockets can be throttled varies greatly, but most rockets can be throttled by a factor of 2 without great difficulty; the typical limitation is combustion stability, as for example, injectors need a minimum pressure to avoid triggering damaging oscillations (chugging or combustion instabilities); but injectors can be optimised and tested for wider ranges.


title: Rocket engine

text: For example, some more recent liquid-propellant engine designs that have been optimised for greater throttling capability (BE-3, Raptor) can be throttled to as low as 18–20 percent of rated thrust.


title: Rocket engine

text: Solid rockets can be throttled by using shaped grains that will vary their surface area over the course of the burn.


title: Rocket engine

text: Rocket engine nozzles are surprisingly efficient heat engines for generating a high speed jet, as a consequence of the high combustion temperature and high compression ratio. Rocket nozzles give an excellent approximation to adiabatic expansion which is a reversible process, and hence they give efficiencies which are very close to that of the Carnot cycle. Given the temperatures reached, over 60% efficiency can be achieved with chemical rockets.


title: Rocket engine

text: For a "vehicle" employing a rocket engine the energetic efficiency is very good if the vehicle speed approaches or somewhat exceeds the exhaust velocity (relative to launch); but at low speeds the energy efficiency goes to 0% at zero speed (as with all jet propulsion). See Rocket energy efficiency for more details.


title: Rocket engine

text: Rockets, of all the jet engines, indeed of essentially all engines, have the highest thrust to weight ratio. This is especially true for liquid-fuelled rocket engines.


title: Rocket engine

text: This high performance is due to the small volume of pressure vessels that make up the engine—the pumps, pipes and combustion chambers involved. The lack of inlet duct and the use of dense liquid propellant allows the pressurisation system to be small and lightweight, whereas duct engines have to deal with air which has around three orders of magnitude lower density.


title: Rocket engine

text: Of the liquid propellants used, density is lowest for liquid hydrogen. Although this propellant has the highest specific impulse, its very low density (about one fourteenth that of water) requires larger and heavier turbopumps and pipework, which decreases the engine's thrust-to-weight ratio (for example the RS-25) compared to those that do not (NK-33).


title: Rocket engine

text: For efficiency reasons, higher temperatures are desirable, but materials lose their strength if the temperature becomes too high. Rockets run with combustion temperatures that can reach .


title: Rocket engine

text: Most other jet engines have gas turbines in the hot exhaust. Due to their larger surface area, they are harder to cool and hence there is a need to run the combustion processes at much lower temperatures, losing efficiency. In addition, duct engines use air as an oxidant, which contains 78% largely unreactive nitrogen, which dilutes the reaction and lowers the temperatures. Rockets have none of these inherent combustion temperature limiters.


title: Rocket engine

text: The temperatures reached by combustion in rocket engines often substantially exceed the melting points of the nozzle and combustion chamber materials (about 1,200 K for copper). Most construction materials will also combust if exposed to high temperature oxidiser, which leads to a number of design challenges. The nozzle and combustion chamber walls must not be allowed to combust, melt, or vaporize (sometimes facetiously termed an "engine-rich exhaust").


title: Rocket engine

text: Rockets that use the common construction materials such as aluminium, steel, nickel or copper alloys must employ cooling systems to limit the temperatures that engine structures experience. Regenerative cooling, where the propellant is passed through tubes around the combustion chamber or nozzle, and other techniques, such as film cooling, are employed to give longer nozzle and chamber life. These techniques ensure that a gaseous thermal boundary layer touching the material is kept below the temperature which would cause the material to catastrophically fail.


title: Rocket engine

text: Material exceptions that can sustain rocket combustion temperatures to a certain degree are carbon–carbon materials and rhenium, although both are subject to oxidation under certain conditions.


title: Rocket engine

text: Other refractory alloys, such as alumina, molybdenum, tantalum or tungsten have been tried, but were given up on due to various issues.


title: Rocket engine

text: In rockets, the heat fluxes that can pass through the wall are among the highest in engineering; fluxes are generally in the range of 0.8–80 MW/m (0.5-50 BTU/in-sec). The strongest heat fluxes are found at the throat, which often sees twice that found in the associated chamber and nozzle. This is due to the combination of high speeds (which gives a very thin boundary layer), and although lower than the chamber, the high temperatures seen there. (See above for temperatures in nozzle).


title: Rocket engine

text: In all cases, another effect that aids in cooling the rocket engine chamber wall is a thin layer of combustion gases (a boundary layer) that is notably cooler than the combustion temperature. Disruption of the boundary layer may occur during cooling failures or combustion instabilities, and wall failure typically occurs soon after.


title: Rocket engine

text: With regenerative cooling a second boundary layer is found in the coolant channels around the chamber. This boundary layer thickness needs to be as small as possible, since the boundary layer acts as an insulator between the wall and the coolant. This may be achieved by making the coolant velocity in the channels as high as possible.


title: Rocket engine

text: Liquid-fuelled engines are often run fuel-rich, which lowers combustion temperatures. This reduces heat loads on the engine and allows lower cost materials and a simplified cooling system. This can also "increase" performance by lowering the average molecular weight of the exhaust and increasing the efficiency with which combustion heat is converted to kinetic exhaust energy.


title: Rocket engine

text: Rocket combustion chambers are normally operated at fairly high pressure, typically 10–200bar (1–20MPa, 150–3,000psi). When operated within significant atmospheric pressure, higher combustion chamber pressures give better performance by permitting a larger and more efficient nozzle to be fitted without it being grossly overexpanded.


title: Rocket engine

text: However, these high pressures cause the outermost part of the chamber to be under very large hoop stresses – rocket engines are pressure vessels.


title: Rocket engine

text: Worse, due to the high temperatures created in rocket engines the materials used tend to have a significantly lowered working tensile strength.


title: Rocket engine

text: In addition, significant temperature gradients are set up in the walls of the chamber and nozzle, these cause differential expansion of the inner liner that create internal stresses.


title: Rocket engine

text: The extreme vibration and acoustic environment inside a rocket motor commonly result in peak stresses well above mean values, especially in the presence of organ pipe-like resonances and gas turbulence.


title: Rocket engine

text: The combustion may display undesired instabilities, of sudden or periodic nature. The pressure in the injection chamber may increase until the propellant flow through the injector plate decreases; a moment later the pressure drops and the flow increases, injecting more propellant in the combustion chamber which burns a moment later, and again increases the chamber pressure, repeating the cycle. This may lead to high-amplitude pressure oscillations, often in ultrasonic range, which may damage the motor. Oscillations of ±200 psi at 25 kHz were the cause of failures of early versions of the Titan II missile second stage engines. The other failure mode is a deflagration to detonation transition; the supersonic pressure wave formed in the combustion chamber may destroy the engine.


title: Rocket engine

text: Combustion instability was also a problem during Atlas development. The Rocketdyne engines used in the Atlas family were found to suffer from this effect in several static firing tests, and three missile launches exploded on the pad due to rough combustion in the booster engines. In most cases, it occurred while attempting to start the engines with a "dry start" method whereby the igniter mechanism would be activated prior to propellant injection. During the process of man-rating Atlas for Project Mercury, solving combustion instability was a high priority, and the final two Mercury flights sported an upgraded propulsion system with baffled injectors and a hypergolic igniter.


title: Rocket engine

text: The problem affecting Atlas vehicles was mainly the so-called "racetrack" phenomenon, where burning propellant would swirl around in a circle at faster and faster speeds, eventually producing vibration strong enough to rupture the engine, leading to complete destruction of the rocket. It was eventually solved by adding several baffles around the injector face to break up swirling propellant.


title: Rocket engine

text: More significantly, combustion instability was a problem with the Saturn F-1 engines. Some of the early units tested exploded during static firing, which led to the addition of injector baffles.


title: Rocket engine

text: In the Soviet space program, combustion instability also proved a problem on some rocket engines, including the RD-107 engine used in the R-7 family and the RD-216 used in the R-14 family, and several failures of these vehicles occurred before the problem was solved. Soviet engineering and manufacturing processes never satisfactorily resolved combustion instability in larger RP-1/LOX engines, so the RD-171 engine used to power the Zenit family still used four smaller thrust chambers fed by a common engine mechanism.


title: Rocket engine

text: The combustion instabilities can be provoked by remains of cleaning solvents in the engine (e.g. the first attempted launch of a Titan II in 1962), reflected shock wave, initial instability after ignition, explosion near the nozzle that reflects into the combustion chamber, and many more factors. In stable engine designs the oscillations are quickly suppressed; in unstable designs they persist for prolonged periods. Oscillation suppressors are commonly used.


title: Rocket engine

text: A low frequency oscillation in chamber pressure below 200 Hertz. Usually it is caused by pressure variations in feed lines due to variations in acceleration of the vehicle, when rocket engines are building up thrust, are shut down or are being throttled.


title: Rocket engine

text: Chugging can cause a worsening feedback loop, as cyclic variation in thrust causes longitudinal vibrations to travel up the rocket, causing the fuel lines to vibrate, which in turn do not deliver propellant smoothly into the engines. This phenomenon is known as "pogo oscillations" or "pogo", named after the pogo stick.


title: Rocket engine

text: In the worst case, this may result in damage to the payload or vehicle. Chugging can be minimised by using several methods, such as installing energy-absorbing devices on feed lines. Chugging may cause Screaming.


title: Rocket engine

text: An intermediate frequency oscillation in chamber pressure between 200 and 1000 Hertz. Usually caused due to insufficient pressure drop across the injectors. It generally is mostly annoying, rather than being damaging.


title: Rocket engine

text: Buzzing is known to have adverse effects on engine performance and reliability, primarily as it causes material fatigue. In extreme cases combustion can end up being forced backwards through the injectors – this can cause explosions with monopropellants. Buzzing may cause Screaming.


title: Rocket engine

text: A high frequency oscillation in chamber pressure above 1000 Hertz, sometimes called screaming or squealing. The most immediately damaging, and the hardest to control. It is due to acoustics within the combustion chamber that often couples to the chemical combustion processes that are the primary drivers of the energy release, and can lead to unstable resonant "screeching" that commonly leads to catastrophic failure due to thinning of the insulating thermal boundary layer. Acoustic oscillations can be excited by thermal processes, such as the flow of hot air through a pipe or combustion in a chamber. Specifically, standing acoustic waves inside a chamber can be intensified if combustion occurs more intensely in regions where the pressure of the acoustic wave is maximal.


title: Rocket engine

text: Such effects are very difficult to predict analytically during the design process, and have usually been addressed by expensive, time-consuming and extensive testing, combined with trial and error remedial correction measures.


title: Rocket engine

text: Screeching is often dealt with by detailed changes to injectors, changes in the propellant chemistry, vaporising the propellant before injection or use of Helmholtz dampers within the combustion chambers to change the resonant modes of the chamber.


title: Rocket engine

text: Testing for the possibility of screeching is sometimes done by exploding small explosive charges outside the combustion chamber with a tube set tangentially to the combustion chamber near the injectors to determine the engine's impulse response and then evaluating the time response of the chamber pressure- a fast recovery indicates a stable system.


title: Rocket engine

text: For all but the very smallest sizes, rocket exhaust compared to other engines is generally very noisy. As the hypersonic exhaust mixes with the ambient air, shock waves are formed. The Space Shuttle generated over 200 dB(A) of noise around its base. To reduce this, and the risk of payload damage or injury to the crew atop the stack, the mobile launcher platform was fitted with a Sound Suppression System that sprayed of water around the base of the rocket in 41 seconds at launch time. Using this system kept sound levels within the payload bay to 142 dB.


title: Rocket engine

text: The sound intensity from the shock waves generated depends on the size of the rocket and on the exhaust velocity. Such shock waves seem to account for the characteristic crackling and popping sounds produced by large rocket engines when heard live. These noise peaks typically overload microphones and audio electronics, and so are generally weakened or entirely absent in recorded or broadcast audio reproductions. For large rockets at close range, the acoustic effects could actually kill.


title: Rocket engine

text: More worryingly for space agencies, such sound levels can also damage the launch structure, or worse, be reflected back at the comparatively delicate rocket above. This is why so much water is typically used at launches. The water spray changes the acoustic qualities of the air and reduces or deflects the sound energy away from the rocket.


title: Rocket engine

text: Generally speaking, noise is most intense when a rocket is close to the ground, since the noise from the engines radiates up away from the jet, as well as reflecting off the ground. Also, when the vehicle is moving slowly, little of the chemical energy input to the engine can go into increasing the kinetic energy of the rocket (since useful power "P" transmitted to the vehicle is formula_12 for thrust "F" and speed "V"). Then the largest portion of the energy is dissipated in the exhaust's interaction with the ambient air, producing noise. This noise can be reduced somewhat by flame trenches with roofs, by water injection around the jet and by deflecting the jet at an angle.


title: Rocket engine

text: Rocket engines are usually statically tested at a test facility before being put into production. For high altitude engines, either a shorter nozzle must be used, or the rocket must be tested in a large vacuum chamber.


title: Rocket engine

text: Rocket vehicles have a reputation for unreliability and danger; especially catastrophic failures. Contrary to this reputation, carefully designed rockets can be made arbitrarily reliable. In military use, rockets are not unreliable. However, one of the main non-military uses of rockets is for orbital launch. In this application, the premium has typically been placed on minimum weight, and it is difficult to achieve high reliability and low weight simultaneously. In addition, if the number of flights launched is low, there is a very high chance of a design, operations or manufacturing error causing destruction of the vehicle.


title: Rocket engine

text: The Rocketdyne H-1 engine, used in a cluster of eight in the first stage of the Saturn I and Saturn IB launch vehicles, had no catastrophic failures in 152 engine-flights. The Pratt and Whitney RL10 engine, used in a cluster of six in the Saturn I second stage, had no catastrophic failures in 36 engine-flights. The Rocketdyne F-1 engine, used in a cluster of five in the first stage of the Saturn V, had no failures in 65 engine-flights. The Rocketdyne J-2 engine, used in a cluster of five in the Saturn V second stage, and singly in the Saturn IB second stage and Saturn V third stage, had no catastrophic failures in 86 engine-flights.


title: Rocket engine

text: The Space Shuttle Solid Rocket Booster, used in pairs, caused one notable catastrophic failure in 270 engine-flights.


title: Rocket engine

text: The RS-25, used in a cluster of three, flew in 46 refurbished engine units. These made a total of 405 engine-flights with no catastrophic in-flight failures. A single in-flight RS-25 engine failure occurred during 's STS-51-F mission. This failure had no effect on mission objectives or duration.


title: Rocket engine

text: Rocket propellants require a high energy per unit mass (specific energy), which must be balanced against the tendency of highly energetic propellants to spontaneously explode. Assuming that the chemical potential energy of the propellants can be safely stored, the combustion process results in a great deal of heat being released. A significant fraction of this heat is transferred to kinetic energy in the engine nozzle, propelling the rocket forward in combination with the mass of combustion products released.


title: Rocket engine

text: Ideally all the reaction energy appears as kinetic energy of the exhaust gases, as exhaust velocity is the single most important performance parameter of an engine. However, real exhaust species are molecules, which typically have translation, vibrational, and rotational modes with which to dissipate energy. Of these, only translation can do useful work to the vehicle, and while energy does transfer between modes this process occurs on a timescale far in excess of the time required for the exhaust to leave the nozzle.


title: Rocket engine

text: The more chemical bonds an exhaust molecule has, the more rotational and vibrational modes it will have. Consequently, it is generally desirable for the exhaust species to be as simple as possible, with a diatomic molecule composed of light, abundant atoms such as H being ideal in practical terms. However, in the case of a chemical rocket, hydrogen is a reactant and reducing agent, not a product. An oxidizing agent, most typically oxygen or an oxygen-rich species, must be introduced into the combustion process, adding mass and chemical bonds to the exhaust species.


title: Rocket engine

text: An additional advantage of light molecules is that they may be accelerated to high velocity at temperatures that can be contained by currently available materials - the high gas temperatures in rocket engines pose serious problems for the engineering of survivable motors.


title: Rocket engine

text: Liquid hydrogen (LH2) and oxygen (LOX, or LO2), are the most effective propellants in terms of exhaust velocity that have been widely used to date, though a few exotic combinations involving boron or liquid ozone are potentially somewhat better in theory if various practical problems could be solved.


title: Rocket engine

text: It is important to note that, when computing the specific reaction energy of a given propellant combination, the entire mass of the propellants (both fuel and oxidiser) must be included. The exception is in the case of air-breathing engines, which use atmospheric oxygen and consequently have to carry less mass for a given energy output. Fuels for car or turbojet engines have a much better effective energy output per unit mass of propellant that must be carried, but are similar per unit mass of fuel.


title: Rocket engine

text: With liquid and hybrid rockets, immediate ignition of the propellants as they first enter the combustion chamber is essential.


title: Rocket engine

text: With liquid propellants (but not gaseous), failure to ignite within milliseconds usually causes too much liquid propellant to be inside the chamber, and if/when ignition occurs the amount of hot gas created can exceed the maximum design pressure of the chamber, causing a catastrophic failure of the pressure vessel.


title: Rocket engine

text: Ignition can be achieved by a number of different methods; a pyrotechnic charge can be used, a plasma torch can be used, or electric spark ignition may be employed. Some fuel/oxidiser combinations ignite on contact (hypergolic), and non-hypergolic fuels can be "chemically ignited" by priming the fuel lines with hypergolic propellants (popular in Russian engines).


title: Rocket engine

text: Gaseous propellants generally will not cause hard starts, with rockets the total injector area is less than the throat thus the chamber pressure tends to ambient prior to ignition and high pressures cannot form even if the entire chamber is full of flammable gas at ignition.


title: Rocket engine

text: Solid propellants are usually ignited with one-shot pyrotechnic devices and combustion usually proceeds through total consumption of the propellants.


title: Rocket engine

text: Once ignited, rocket chambers are self-sustaining and igniters are not needed and combustion usually proceeds through total consumption of the propellants. Indeed, chambers often spontaneously reignite if they are restarted after being shut down for a few seconds. Unless designed for re-ignition, when cooled, many rockets cannot be restarted without at least minor maintenance, such as replacement of the pyrotechnic igniter or even refueling of the propellants.


title: Rocket engine

text: Rocket jets vary depending on the rocket engine, design altitude, altitude, thrust and other factors.


title: Rocket engine

text: Carbon-rich exhausts from kerosene-based fuels such as RP-1 are often orange in colour due to the black-body radiation of the unburnt particles, in addition to the blue Swan bands. Peroxide oxidiser-based rockets and hydrogen rocket jets contain largely steam and are nearly invisible to the naked eye but shine brightly in the ultraviolet and infrared ranges. Jets from solid-propellant rockets can be highly visible, as the propellant frequently contains metals such as elemental aluminium which burns with an orange-white flame and adds energy to the combustion process. Rocket engines which burn liquid hydrogen and oxygen will exhibit a nearly transparent exhaust, due to it being mostly superheated steam (water vapour), plus some unburned hydrogen.


title: Rocket engine

text: The nozzle is usually over-expanded at sea level, and the exhaust can exhibit visible shock diamonds through a schlieren effect caused by the incandescence of the exhaust gas.


title: Rocket engine

text: The shape of the jet varies for a fixed-area nozzle as the expansion ratio varies with altitude: at high altitude all rockets are grossly under-expanded, and a quite small percentage of exhaust gases actually end up expanding forwards.


title: Rocket engine

text: The solar thermal rocket would make use of solar power to directly heat reaction mass, and therefore does not require an electrical generator as most other forms of solar-powered propulsion do. A solar thermal rocket only has to carry the means of capturing solar energy, such as concentrators and mirrors. The heated propellant is fed through a conventional rocket nozzle to produce thrust. The engine thrust is directly related to the surface area of the solar collector and to the local intensity of the solar radiation and inversely proportional to the "I".


title: Rocket engine

text: Nuclear propulsion includes a wide variety of propulsion methods that use some form of nuclear reaction as their primary power source. Various types of nuclear propulsion have been proposed, and some of them tested, for spacecraft applications:


title: Rocket engine

text: According to the writings of the Roman Aulus Gellius, the earliest known example of jet propulsion was in c. 400 BC, when a Greek Pythagorean named Archytas, propelled a wooden bird along wires using steam. However, it was not powerful enough to take off under its own thrust.


title: Rocket engine

text: The "aeolipile" described in the first century BC, often known as "Hero's engine", consisted of a pair of steam rocket nozzles mounted on a bearing. It was created almost two millennia before the Industrial Revolution but the principles behind it were not well understood, and it was not developed into a practical power source.


title: Rocket engine

text: The availability of black powder to propel projectiles was a precursor to the development of the first solid rocket. Ninth Century Chinese Taoist alchemists discovered black powder in a search for the elixir of life; this accidental discovery led to fire arrows which were the first rocket engines to leave the ground.


title: Rocket engine

text: It is stated that "the reactive forces of incendiaries were probably not applied to the propulsion of projectiles prior to the 13th century". A turning point in rocket technology emerged with a short manuscript entitled "Liber Ignium ad Comburendos Hostes" (abbreviated as "The Book of Fires"). The manuscript is composed of recipes for creating incendiary weapons from the mid-eighth to the end of the thirteenth centuries—two of which are rockets. The first recipe calls for one part of colophonium and sulfur added to six parts of saltpeter (potassium nitrate) dissolved in laurel oil, then inserted into hollow wood and lit to "fly away suddenly to whatever place you wish and burn up everything". The second recipe combines one pound of sulfur, two pounds of charcoal, and six pounds of saltpeter—all finely powdered on a marble slab. This powder mixture is packed firmly into a long and narrow case. The introduction of saltpeter into pyrotechnic mixtures connected the shift from hurled Greek fire into self-propelled rocketry. .


title: Rocket engine

text: Articles and books on the subject of rocketry appeared increasingly from the fifteenth through seventeenth centuries. In the sixteenth century, German military engineer Conrad Haas (1509–1576) wrote a manuscript which introduced the construction of multi-staged rockets.


title: Rocket engine

text: Rocket engines were also put in use by Tippu Sultan, the king of Mysore. These usually consisted of a tube of soft hammered iron about long and diameter, closed at one end, packed with black powder propellant and strapped to a shaft of bamboo about long. A rocket carrying about one pound of powder could travel almost . These 'rockets', fitted with swords, would travel several meters in the air before coming down with sword edges facing the enemy. These were used very effectively against the British empire.


title: Rocket engine

text: Slow development of this technology continued up to the later 19th century, when Russian Konstantin Tsiolkovsky first wrote about liquid-fuelled rocket engines. He was the first to develop the Tsiolkovsky rocket equation, though it was not published widely for some years.


title: Rocket engine

text: The modern solid- and liquid-fuelled engines became realities early in the 20th century, thanks to the American physicist Robert Goddard. Goddard was the first to use a De Laval nozzle on a solid-propellant (gunpowder) rocket engine, doubling the thrust and increasing the efficiency by a factor of about twenty-five. This was the birth of the modern rocket engine. He calculated from his independently derived rocket equation that a reasonably sized rocket, using solid fuel, could place a one-pound payload on the Moon.


title: Rocket engine

text: Fritz von Opel was instrumental in popularizing rockets as means of propulsion. In the 1920s, he initiated together with Max Valier, co-founder of the "Verein für Raumschiffahrt", the world's first rocket program, Opel-RAK, leading to speed records for automobiles, rail vehicles and the first manned rocket-powered flight in September of 1929. Months earlier in 1928, one of his rocket-powered prototypes, the Opel RAK2, reached piloted by von Opel himself at the AVUS speedway in Berlin a record speed of 238 km/h, watched by 3000 spectators and world media. A world record for rail vehicles was reached with RAK3 and a top speed of 256 km/h. After these successes, von Opel piloted the world's first public rocket-powered flight using Opel RAK.1, a rocket plane designed by Julius Hatry. World media reported on these efforts, including UNIVERSAL Newsreel of the US, causing as "Raketen-Rummel" or "Rocket Rumble" immense global public excitement, and in particular in Germany, where inter alia Wernher von Braun was highly influenced. The Great Depression led to an end of the Opel-RAK program, but Max Valier continued the efforts. After switching from solid-fuel to liquid-fuel rockets, he died while testing and is considered the first fatality of the dawning space age.


title: Rocket engine

text: Goddard began to use liquid propellants in 1921, and in 1926 became the first to launch a liquid-fuelled rocket. Goddard pioneered the use of the De Laval nozzle, lightweight propellant tanks, small light turbopumps, thrust vectoring, the smoothly-throttled liquid fuel engine, regenerative cooling, and curtain cooling.


title: Rocket engine

text: During the late 1930s, German scientists, such as Wernher von Braun and Hellmuth Walter, investigated installing liquid-fuelled rockets in military aircraft (Heinkel He 112, He 111, He 176 and Messerschmitt Me 163).


title: Rocket engine

text: The turbopump was employed by German scientists in World War II. Until then cooling the nozzle had been problematic, and the A4 ballistic missile used dilute alcohol for the fuel, which reduced the combustion temperature sufficiently.


title: Rocket engine

text: Staged combustion ("Замкнутая схема") was first proposed by Alexey Isaev in 1949. The first staged combustion engine was the S1.5400 used in the Soviet planetary rocket, designed by Melnikov, a former assistant to Isaev. About the same time (1959), Nikolai Kuznetsov began work on the closed cycle engine NK-9 for Korolev's orbital ICBM, GR-1. Kuznetsov later evolved that design into the NK-15 and NK-33 engines for the unsuccessful Lunar N1 rocket.


title: Rocket engine

text: In the West, the first laboratory staged-combustion test engine was built in Germany in 1963, by Ludwig Boelkow.


title: Rocket engine

text: Hydrogen peroxide / kerosene fuelled engines such as the British Gamma of the 1950s used a closed-cycle process by catalytically decomposing the peroxide to drive turbines "before" combustion with the kerosene in the combustion chamber proper. This gave the efficiency advantages of staged combustion, without the major engineering problems.


title: Rocket engine

text: Liquid hydrogen engines were first successfully developed in America: the RL-10 engine first flew in 1962. Its successor, the Rocketdyne J-2, was used in the Apollo program's Saturn V rocket to send humans to the Moon. The high specific impulse and low density of liquid hydrogen lowered the upper stage mass and the overall size and cost of the vehicle.


title: Unfriended: Dark Web

text: Unfriended: Dark Web is a 2018 American computer screen horror film written and directed by Stephen Susco in his directorial debut. Shot as a computer screen film, it stars Colin Woodell, Rebecca Rittenhouse, Betty Gabriel, Connor Del Rio, Andrew Lees, Stephanie Nogueras, and Savira Windyani. The film is a sequel to the 2015 film "Unfriended", and follows a group of friends who find a laptop that has access to the dark web, only to realize they are being watched by the original owners, a group of cybercriminal hackers.


title: Unfriended: Dark Web

text: The film had its world premiere at the South by Southwest festival on March 9, 2018, and was theatrically released in the United States on July 20, 2018, by Universal Pictures' OTL Releasing and Blumhouse Productions' BH Tilt. The film received mixed reviews from critics and has grossed $16 million worldwide, against a production budget of $1 million.


title: Unfriended: Dark Web

text: A man named Matias finds a laptop at a cyber café lost-and-found and takes it home. The laptop originally belonged to someone called Norah C. IV. Matias is working with a sign language app titled "Papaya" for his deaf girlfriend, Amaya. However, Amaya becomes upset because the app only makes it convenient for her to understand him, but not for him to understand her. Matias keeps getting messages on the laptop for Norah from someone named Erica. He logs onto a Skype video call with his friends — Damon, A.J., Lexx, Serena, and Nari — but becomes frustrated when the computer keeps freezing up. Matias discovers that Erica is Norah when she demands her laptop back. He decides to return it to the cyber café, but before he can, he sees a message for Norah about the payment they received for a video. Curious, he converses with the mystery person, Charon68. Matias' friends watch his actions, and A.J. recognizes that Matias is connected to the dark web. When Charon68 mentions trepanation, a disturbed Matias ends the chat.


title: Unfriended: Dark Web

text: Damon realizes that "Norah C." is "Charon" spelled backward, as Matias begins to find hidden videos on the laptop. He looks up a street address listed in one of them and sees that Erica Dunne went missing from there; the same Erica that Norah was impersonating. Matias receives a video call from Amaya, only to find that it is Norah C., also known as Charon IV, continuing to demand his laptop back, or else he'll kill Amaya. When Nari tries to call the police, Matias panics and explains that it was all part of an alternate reality game he is developing. His friends are relieved, but Nari remains suspicious. Matias tries to convince Amaya to come over, but she's still upset with him, believing he doesn't want to put in the effort to learn sign language for her. Matias promises he will try and convinces her to come over, though Charon IV secretly follows her. To ensure Charon IV complies, Matias transfers his cryptocurrency from her profile into his. Norah chides Matias for this, but he says he'll return the money and laptop in exchange for Amaya and Erica's safety.


title: Unfriended: Dark Web

text: Matias directs Amaya to the subway so that she will lose connection - once her signal is lost, he tells his friends that the entire thing was real and explains that they are all in danger. Matias suddenly receives messages from a group called "The Circle," who reveal that they are aware of Matias' actions, and dozens of Charon accounts join the chat. One of them posts a video of Lexx being thrown off a roof while another doctors videos of A.J. planning to attack a shopping mall. As police officers storm A.J.'s house, the Circle triggers the sound effect of a shotgun loading on his computer, causing the police to assume he is armed and shoot him to death. Next, they show Nari at the subway and Serena's mother on life support, forcing Serena to save one of them. When she can't make the decision, the Circle kills all three. Matias goes to get Amaya, leaving the laptop open so Damon can copy its files. Unbeknownst to Matias, another Charon brings Erica into his closet. When a Charon takes control of the laptop, Damon tells the Circle he was recording everything for the police to find. To his horror, he realizes that the Circle had planned for Matias to find the laptop so they could frame him and his friends for their crimes. Moments later, a Charon hangs Damon by his closet door. Another Charon writes a fake confession and suicide note, implying everyone killed themselves out of guilt. Matias gets a call from Amaya and realizes that the Circle hacked his messages to lead her astray. He helplessly watches as Amaya is attacked by a Charon, most likely to be part of their videos. Broken, he asks the Circle why they did all of this. They respond with a clip of him saying it was "game night," revealing that this is simply entertainment for them.


title: Unfriended: Dark Web

text: The Circle launches a poll to decide Matias' fate; as they vote, he laughs, no longer caring about his safety. Erica wakes up in Matias' apartment and goes to the computer begging for help. The video cuts out as she finds a hole in her skull. In the street, the poll reaches its end before a Charon in a van fatally strikes Matias, killing him. With their job complete and their crimes pinned on Matias and his friends, the Circle gather in front of the cameras and celebrate while another Charon watches from a camera.


title: Unfriended: Dark Web

text: There have been three alternate endings: The first one shows Matias riding his bike after telling Amaya to meet where they had their first kiss. When Matias gets there, he sees an open coffin when he is knocked out by a Charon with a shovel. Damon is shown seeing Erica waking up in Matias’ apartment and finding a hole in her head; then being cut out. Damon cries as he sees someone in his house with a Charon killing him by hanging him while leaving a suicide note in his laptop (shown in all three alternate endings). A few minutes later, Amaya gets to the place where they had their first kiss while Matias wakes up in a coffin and realizes he was buried alive. As Matias is FaceTiming her, the Circle hacked him by censoring his mouth. When he texts Amaya, the Circle hack his phone so his phone will say “I wish I could sign better”. Since Amaya is deaf, she can’t hear Matias screaming for help. The Circle gather around the cameras to celebrate while a Charon is watching Matias from a camera as he slowly dies.


title: Unfriended: Dark Web

text: In April 2015, the month of "Unfriended"'s release, it was announced that Universal Pictures had greenlit a sequel, tentatively titled "Unfriended 2", with Nelson Greaves writing and Jason Blum and Timur Bekmambetov producing, and a release date then set for the spring of 2016.


title: Unfriended: Dark Web

text: On October 3, 2017, it was reported that "The Grudge" and "Texas Chainsaw 3D" writer Stephen Susco had taken over the project as writer and director. Susco shot the film in secret over one week in late 2016, under the working title "Unfriended: Game Night".


title: Unfriended: Dark Web

text: In March 2018, Blumhouse officially revealed the film as "Unfriended: Dark Web" at SXSW 2018, under the distribution of Universal's OTL Releasing and Blumhouse's BH Tilt.


title: Unfriended: Dark Web

text: "Unfriended: Dark Web" was released in the United States on July 20, 2018. It had a surprise premiere at South by Southwest in March 2018, and then in April 2018 was screened at the Overlook Film Festival with an entirely different ending. It was then revealed in July 2018 that the film would be sent to theaters with the two different endings, which would be played at random, similar to 1985's "Clue", which had multiple endings.


title: Unfriended: Dark Web

text: However, the director has stated that there is only one ending, and that the rumor came from re-editing the film after various test screenings. Still, multiple movie theatre projectionists have confirmed that they were given two copies of the film, each with a different ending, and were told to wait until further instruction on which version to play at their specific location.


title: Unfriended: Dark Web

text: In the United States and Canada, "Unfriended: Dark Web" was released alongside 2 other sequels, "Mamma Mia! Here We Go Again" and "The Equalizer 2", and was initially projected to gross $6–8 million from 1,543 theaters in its opening weekend. However, after making $1.42222 million on its first day, including $350,000 from Thursday night previews, estimates were lowered to $3 million. The film went on to debut to $3.5 million, finishing ninth at the box office. By the end of its theatrical run, the film grossed $15.1 million worldwide, including $8.8 million domestically, less than the $15 million the first film made in its opening weekend alone.


title: Unfriended: Dark Web

text: On review aggregation website Rotten Tomatoes, the film holds an approval rating of based on reviews and an average rating of . The website's critical consensus reads, ""Unfriended: Dark Web" is more interested in chills than an exploration of its timely themes, but horror fans should still find this sequel to be steadily, undeniably effective." On Metacritic, the film has a weighted average score of 53 out of 100, based on reviews from 26 critics, indicating "mixed or average reviews". Audiences polled by CinemaScore gave the film an average grade of "C" on an A+ to F scale, the same score as the first film, while PostTrak reported filmgoers gave it a "low" 59% overall positive score.


title: Unfriended: Dark Web

text: In a positive review in "Vanity Fair", K. Austin Collins wrote that the film "couldn’t seem more topical or relevant," and that it "leaves discerning viewers as entertained and skeptical as they are shaken." Geoffrey Macnab of "The Independent" wrote that "Unfriended: Dark Web" "may be an exploitation movie but it’s an ingeniously made one with a highly original storytelling style which reflects perfectly the screen-dominated lives and leisure habits of its young protagonists."


title: Unfriended: Dark Web

text: In a negative review for RogerEbert.com, Nick Allen wrote "Curiosity killed the dumb horror character, as we know from cinematic death-traps structured just like this one, but "Unfriended: Dark Web" stretches this conceit until it snaps, which happens about 15 minutes in." He added, "good poutine gravy, are these characters dumb, and the movie even more so."


title: Unfriended: Dark Web

text: Susco has teased his interest in a third "Unfriended" film, while also expanding upon Bekmambetov's Screenlife genre that "Dark Web" is a part of. He stated that it is possible that a third film might be made and that it is important that people have a great time watching it and that it leaves them with questions. But while Susco is noncommittal regarding a third "Unfriended", he did go into detail about the Screenlife genre, consisting of films that unfold entirely on computer screens, and has been masterminded by "Unfriended" producer Bekmambetov.


title: Those Were the Days (song)

text: "Those Were the Days" is a song credited to Gene Raskin, who put a new English lyric to the Russian romance song "" (literally "By the long road"), composed by Boris Fomin (1900–1948) with words by the poet Konstantin Podrevsky. It deals with reminiscence upon youth and romantic idealism. It also deals with tavern activities, which include drinking, singing and dancing.


title: Those Were the Days (song)

text: Mary Hopkin's 1968 debut single of "Those Were the Days", which was produced by Paul McCartney of the Beatles, and arranged by Richard Hewson, became a number one hit on the UK Singles Chart and on the Canadian "RPM" Magazine charts. The song also reached number two on the "Billboard" Hot 100, behind "Hey Jude" by the Beatles. It was number one in the first edition of the French National Hit Parade launched by the Centre d'Information et de Documentation du Disque. The song was featured on her debut album "Post Card".


title: Those Were the Days (song)

text: Georgian singer Tamara Tsereteli (1900–1968) and Russian singer Alexander Vertinsky made what were probably the earliest recordings of the song, in 1925 and 1926 respectively.


title: Those Were the Days (song)

text: The song appears in the 1953 British/French movie "Innocents in Paris", in which it was sung with its original Russian lyrics by the Russian Tzigane chanteuse Ludmila Lopato. Mary Hopkin's 1968 recording of it with Gene Raskin's lyric was a chart-topping hit in much of the Northern Hemisphere. On most recordings of the song, Raskin is credited as the sole writer, even though he wrote only the later English lyrics (which are not an English translation of the Russian lyrics) and not the music.


title: Those Were the Days (song)

text: In the early 1960s Raskin, with his wife Francesca, played folk music around Greenwich Village in New York, including White Horse Tavern. Raskin, who had grown up hearing the song, wrote with his wife, new English lyrics to the old Russian music and then copyrighted both music and lyrics in his own name. The Limeliters subsequently released a recording of the song on their 1962 LP "Folk Matinee". The Raskins were international performers and had played London's "Blue Angel" every year, always closing their show with the song. Paul McCartney frequented the club and, being quite taken with the song, he attempted to get several singers or groups (including the early Moody Blues) to record it. Failing at that, after the formation of the Beatles' own Apple Records label, McCartney immediately recorded Mary Hopkin performing the song at Abbey Road Studios in London. He later said "I thought it was very catchy, it had something, it was a good treatment of nostalgia... (Hopkin) picked it up very easily, as if she'd known it for years." The song was eventually recorded in over twenty languages and by many different artists, including Gene and Francesca.


title: Those Were the Days (song)

text: Hopkin's recording was produced by Paul McCartney with an arrangement by Richard Hewson and became a number-one hit on the UK Singles Chart. In the United States, Hopkin's recording reached number two on the "Billboard" Hot 100 (held out of the top spot for three weeks by "Hey Jude" by The Beatles) and topped the Billboard Easy Listening charts for six weeks. In the Netherlands, it topped the charts for two consecutive weeks. The Russian origin of the melody was accentuated by an instrumentation that was unusual for a top-ten pop record, including balalaika, clarinet, hammered dulcimer or cimbalom, tenor banjo and children's chorus, giving a klezmer feel to the song. Mary Hopkin played acoustic guitar on the recording, and Paul McCartney also played acoustic guitar and possibly percussion. The cimbalom was played by Gilbert Webster.


title: Those Were the Days (song)

text: McCartney also recorded Hopkin singing "Those Were The Days" in other languages for release in their respective countries:


title: Those Were the Days (song)

text: The non-English sets of lyrics were also recorded by Dalida and Sandie Shaw, with Shaw recording the English lyrics as well.


title: Those Were the Days (song)

text: The UK and United States recording's B-side was Pete Seeger's "Turn! Turn! Turn!", which had been a United States number-one hit for The Byrds in 1965.


title: Those Were the Days (song)

text: "Those Were the Days" was catalogue number APPLE 2. (The APPLE 1 number had been given to an unreleased version of Frank Sinatra's "The Lady Is a Tramp", recorded specially in 1968, for Maureen Starkey's 22nd birthday, as a gift from Ringo Starr, under the name of "The Lady is a Champ".) It was the second single to be released on the Apple label, the first— "Hey Jude" by the Beatles—had retained the sequential catalogue numbers used by Parlophone (in the UK) and Capitol (in the US).


title: Those Were the Days (song)

text: Hopkin's version was released on the back of her success on the television talent show "Opportunity Knocks" and, around the time of its release, popular singer Sandie Shaw was also asked to record the song by her management, feeling that it should be done by a "real" singer. Shaw's version was released as a single, but did not match the success of Hopkin's version.


title: Those Were the Days (song)

text: At the peak of the song's success, a New York company used the melody in a commercial for Rokeach gefilte fish, arguing that the tune was an old Russian folk-tune and thus in the public domain. (The commercial included the line "The perfect dish, Rokeach Gefilte Fish" where the English-language song would go "Those were the days, oh yes, those were the days.") Raskin successfully sued and won a settlement, since he had slightly altered the tune to fit his lyrics and had taken out the valid new copyright.


title: Those Were the Days (song)

text: In the mid-1970s, after Hopkin's contract with Apple ended, "Those Were the Days" and "Goodbye" were re-recorded with producer Tony Visconti, whom she had married in 1971. These re-recorded versions can be found on music compilations.


title: Those Were the Days (song)

text: On 25 October 2010, Apple Records released "", which included the original recordings of "Those Were the Days" and "Goodbye". The greatest hits compilation album contained songs by artists signed to the Beatles' Apple record label between 1968 and 1973, the first such multi-artist Apple compilation.


title: Those Were the Days (song)

text: On Christmas 1969, the President of Equatorial Guinea, Francisco Macías Nguema, had 150 alleged coup plotters executed in the national stadium while the amplifier system played the Mary Hopkins' recording of "Those Were the Days".


title: Those Were the Days (song)

text: The tune of "Those Were the Days" is used for the Republic of Ireland football chant "Come On You Boys in Green".


title: Those Were the Days (song)

text: In 2011, Hopkin's version of the song was used by Nando's South Africa in a satirical advert featuring Robert Mugabe as the 'Last Dictator Standing'. The advert was axed quickly, due to controversy and condemnation from pro-Mugabe loyalists.


title: Hilbert's paradox of the Grand Hotel

text: Hilbert's paradox of the Grand Hotel (colloquial: Infinite Hotel Paradox or Hilbert's Hotel) is a thought experiment which illustrates a counterintuitive property of infinite sets. It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. The idea was introduced by David Hilbert in a 1924 lecture "Über das Unendliche", reprinted in , and was popularized through George Gamow's 1947 book "One Two Three... Infinity".


title: Hilbert's paradox of the Grand Hotel

text: Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms, where the pigeonhole principle would apply.


title: Hilbert's paradox of the Grand Hotel

text: Suppose a new guest arrives and wishes to be accommodated in the hotel. We can (simultaneously) move the guest currently in room 1 to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from their current room "n" to room "n"+1. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests. In general, assume that "k" guests seek a room. We can apply the same procedure and move every guest from room "n" to room "n + k". In a similar manner, if "k" guests wished to leave the hotel, every guest moves from room "n" to room "n − k".


title: Hilbert's paradox of the Grand Hotel

text: It is also possible to accommodate a "countably infinite" number of new guests: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room "n" to room 2"n" (2 times "n"), and all the odd-numbered rooms (which are countably infinite) will be free for the new guests.


title: Hilbert's paradox of the Grand Hotel

text: It is possible to accommodate countably infinitely many coachloads of countably infinite passengers each, by several different methods. Most methods depend on the seats in the coaches being already numbered (or use the axiom of countable choice). In general any pairing function can be used to solve this problem. For each of these methods, consider a passenger's seat number on a coach to be formula_1, and their coach number to be formula_2, and the numbers formula_1 and formula_2 are then fed into the two arguments of the pairing function.


title: Hilbert's paradox of the Grand Hotel

text: Send the guest in room formula_5 to room formula_6, then put the first coach's load in rooms formula_7, the second coach's load in rooms formula_8; for coach number formula_2 we use the rooms formula_10 where formula_11 is the formula_2th odd prime number. This solution leaves certain rooms empty (which may or may not be useful to the hotel); specifically, all numbers that are not prime powers, such as 15 or 847, will no longer be occupied. (So, strictly speaking, this shows that the number of arrivals is "less than or equal to" the number of vacancies created. It is easier to show, by an independent means, that the number of arrivals is also "greater than or equal to" the number of vacancies, and thus that they are "equal", than to modify the algorithm to an exact fit.) (The algorithm works equally well if one interchanges formula_1 and formula_2, but whichever choice is made, it must be applied uniformly throughout.)


title: Hilbert's paradox of the Grand Hotel

text: Each person of a certain seat formula_15 and coach formula_2 can be put into room formula_17 (presuming "c"=0 for the people already in the hotel, 1 for the first coach, etc.). Because every number has a unique prime factorization, it is easy to see all people will have a room, while no two people will end up in the same room. For example, the person in room 2592 (formula_18) was sitting in on the 4th coach, on the 5th seat. Like the prime powers method, this solution leaves certain rooms empty.


title: Hilbert's paradox of the Grand Hotel

text: For each passenger, compare the lengths of formula_1 and formula_2 as written in any positional numeral system, such as decimal. (Treat each hotel resident as being in coach #0.) If either number is shorter, add leading zeroes to it until both values have the same number of digits. Interleave the digits to produce a room number: its digits will be [first digit of coach number]-[first digit of seat number]-[second digit of coach number]-[second digit of seat number]-etc. The hotel (coach #0) guest in room number 1729 moves to room 01070209 (i.e., room 1,070,209). The passenger on seat 1234 of coach 789 goes to room 01728394 (i.e., room 1,728,394).


title: Hilbert's paradox of the Grand Hotel

text: Unlike the prime powers solution, this one fills the hotel completely, and we can reconstruct a guest's original coach and seat by reversing the interleaving process. First add a leading zero if the room has an odd number of digits. Then de-interleave the number into two numbers: the coach number consists of the odd-numbered digits and the seat number is the even-numbered ones. Of course, the original encoding is arbitrary, and the roles of the two numbers can be reversed (seat-odd and coach-even), so long as it is applied consistently.


title: Hilbert's paradox of the Grand Hotel

text: Those already in the hotel will be moved to room formula_22, or the formula_1th triangular number. Those in a coach will be in room formula_24, or the formula_25 triangular number plus formula_1. In this way all the rooms will be filled by one, and only one, guest.


title: Hilbert's paradox of the Grand Hotel

text: This pairing function can be demonstrated visually by structuring the hotel as a one-room-deep, infinitely tall pyramid. The pyramid's topmost row is a single room: room 1; its second row is rooms 2 and 3; and so on. The column formed by the set of rightmost rooms will correspond to the triangular numbers. Once they are filled (by the hotel's redistributed occupants), the remaining empty rooms form the shape of a pyramid exactly identical to the original shape. Thus, the process can be repeated for each infinite set. Doing this one at a time for each coach would require an infinite number of steps, but by using the prior formulas, a guest can determine what his room "will be" once his coach has been reached in the process, and can simply go there immediately.


title: Hilbert's paradox of the Grand Hotel

text: Let formula_27. formula_28 is countable since formula_29 is countable, hence we may enumerate its elements formula_30. Now if formula_31, assign the formula_32th guest of the formula_33th coach to the formula_1th room (consider the guests already in the hotel as guests of the formula_35th coach). Thus we have a function assigning each person to a room; furthermore, this assignment does not skip over any rooms.


title: Hilbert's paradox of the Grand Hotel

text: Suppose the hotel is next to an ocean, and an infinite number of car ferries arrive, each bearing an infinite number of coaches, each with an infinite number of passengers. This is a situation involving three "levels" of infinity, and it can be solved by extensions of any of the previous solutions.


title: Hilbert's paradox of the Grand Hotel

text: The prime factorization method can be applied by adding a new prime number for every additional layer of infinity ( formula_36, with formula_37 the ferry).


title: Hilbert's paradox of the Grand Hotel

text: The prime power solution can be applied with further exponentiation of prime numbers, resulting in very large room numbers even given small inputs. For example, the passenger in the second seat of the third bus on the second ferry (address 2-3-2) would raise the 2nd odd prime (5) to 49, which is the result of the 3rd odd prime (7) being raised to the power of his seat number (2). This room number would have over thirty decimal digits.


title: Hilbert's paradox of the Grand Hotel

text: The interleaving method can be used with three interleaved "strands" instead of two. The passenger with the address 2-3-2 would go to room 232, while the one with the address 4935-198-82217 would go to room #008,402,912,391,587 (the leading zeroes can be removed).


title: Hilbert's paradox of the Grand Hotel

text: Anticipating the possibility of any number of layers of infinite guests, the hotel may wish to assign rooms such that no guest will need to move, no matter how many guests arrive afterward. One solution is to convert each arrival's address into a binary number in which ones are used as separators at the start of each layer, while a number within a given layer (such as a guest's coach number) is represented with that many zeroes. Thus, a guest with the prior address 2-5-1-3-1 (five infinite layers) would go to room 10010000010100010 (decimal 295458).


title: Hilbert's paradox of the Grand Hotel

text: As an added step in this process, one zero can be removed from each section of the number; in this example, the guest's new room is 101000011001 (decimal 2585). This ensures that every room could be filled by a hypothetical guest. If no infinite sets of guests arrive, then only rooms that are a power of two will be occupied.


title: Hilbert's paradox of the Grand Hotel

text: Although a room can be found for any finite number of nested infinities of people, the same is not always true for an infinite number of layers, even if a finite number of elements exists at each layer.


title: Hilbert's paradox of the Grand Hotel

text: Hilbert's paradox is a veridical paradox: it leads to a counter-intuitive result that is provably true. The statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms.


title: Hilbert's paradox of the Grand Hotel

text: Initially, this state of affairs might seem to be counter-intuitive. The properties of infinite collections of things are quite different from those of finite collections of things. The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of transfinite numbers. Thus, in an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total "number" of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets (sets with the same cardinality as the natural numbers) this cardinality is formula_38.


title: Hilbert's paradox of the Grand Hotel

text: Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers—those numbers which can be written as a quotient of integers—contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: there is a bijection from the naturals to the rationals.


title: Barbora Krejčíková

text: Barbora Krejčíková (; born 18 December 1995) is a Czech professional tennis player. She has a career-high singles ranking of world No. 2, achieved on 28 February 2022, and on 22 October 2018, she became world No. 1 in doubles.


title: Barbora Krejčíková

text: Krejčíková has won one Grand Slam singles title and nine major doubles titles, including the career Super Slam in doubles. She won the 2021 French Open singles title, the women's doubles titles at the 2018 French Open, 2018 Wimbledon Championships, 2021 French Open, 2022 Australian Open, 2022 Wimbledon Championships and 2022 US Open partnering Kateřina Siniaková, and the mixed doubles titles at the 2019 and 2021 Australian Opens partnering Rajeev Ram, and in 2020 with Nikola Mektić. She is one of only three active female players to have won a major title in all three disciplines, after Venus Williams and Samantha Stosur.


title: Barbora Krejčíková

text: Krejčíková has won five singles and fourteen doubles titles on the WTA Tour, including two at the WTA 1000-level in doubles. Alongside her major titles, Krejčíková won the 2021 WTA Finals and a gold medal at the 2020 Tokyo Olympics in women's doubles, both partnering Siniaková, and was part of the Czech team that won the 2018 Fed Cup. The pair also finished runners-up at the 2018 WTA Finals, 2021 Australian Open, and 2022 WTA Finals.


title: Barbora Krejčíková

text: Krejčíková was ranked junior world No. 3 in October 2013. In that year, she won the girls' doubles titles at the French Open, at Wimbledon and the US Open with fellow Czech Kateřina Siniaková. Alongside Oleksandra Korashvili from Ukraine, she also reached the final of the Australian Open, falling one match shy of completing the calendar-year Grand Slam. The same year, she also won the European Junior Championship U18 in Klosters, Switzerland in singles and doubles.


title: Barbora Krejčíková

text: In October 2020, she reached the fourth round of the French Open tournament, which helped her attain a new best year-end ranking of No. 65 in November.


title: Barbora Krejčíková

text: In March 2021, she made progress to her first singles final of a WTA 1000 tournament at the Dubai Championships, eventually losing to Garbiñe Muguruza. As a result, she climbed to a new career-high of No. 38 in the singles rankings.


title: Barbora Krejčíková

text: In May 2021, she won her first WTA singles title at Strasbourg the day before the 2021 French Open started. Krejčíková went on to win the subsequent French Open, her first Grand Slam final in singles. As a result, she climbed to a career-high of No. 15 in the singles rankings. She won her third WTA singles title at Prague, and reached a career high singles ranking of world number 2 in February 2022 after reaching the final at Sydney and the quarterfinals of the 2022 Australian Open. After injuries, Krejčíková would win two further singles titles in 2022: at Tallinn, and at Ostrava.


title: Barbora Krejčíková

text: Together with compatriot Kateřina Siniaková, she won the doubles titles at the 2018 French Open and 2018 Wimbledon Championships.


title: Barbora Krejčíková

text: Krejcikova also won the 2021 French Open Women’s Doubles title alongside Siniaková, and they added the Australian Open Women’s doubles 2022 title in January 2022.


title: Barbora Krejčíková

text: The pair added the 2022 Wimbledon Championships title and completed the career grand slam with the doubles title in New York at the US Open


title: Barbora Krejčíková

text: Krejcikova has won 3 Grand Slam Mixed Doubles titles, winning the Australian Open for three years in a row from 2019–2021. With American partner Rajeev Ram, she won the mixed-doubles competition of the 2019 Australian Open. She succeeded in defending her title the following year alongside Nikola Mektić, and in 2021 again, with Rajeev Ram.


title: Barbora Krejčíková

text: Additionally, Krejčíková has won six more doubles titles on the WTA Tour, one WTA 125K series doubles title, as well as 19 doubles titles on the ITF Circuit.


title: Barbora Krejčíková

text: On 22 October 2018, she reached No. 1 in the doubles rankings, jointly with Kateřina Siniaková. Together they were the fifth and sixth Czechs ranked No. 1 since 1975, and the 14th pair that attained the No. 1 together.


title: Barbora Krejčíková

text: In August 2021, Krejcikova and Siniaková won the gold medal at the Tokyo Olympics in the women’s doubles.


title: Barbora Krejčíková

text: In November 2021, she won the 2021 WTA Finals doubles title with Siniaková, their first year end championships title.


title: Barbora Krejčíková

text: Krejčíková made her WTA Tour debut at the Gastein Ladies, where she and partner Kateřina Siniaková lost in the first round. In singles, her debut was at the Tournoi de Québec, where she qualified for the main draw and reached the second round, losing to Lucie Hradecká. In doubles there, she reached the semifinals. At the Luxembourg Open, she reached her first WTA Tour doubles final pairing Hradecká. However, they were defeated by Timea Bacsinszky and Kristina Barrois.


title: Barbora Krejčíková

text: In 2015, Krejčíková only played one WTA Tour main-draw singles event, at the Tournoi de Québec, where she was defeated in the first round. At all four Grand Slam championships, she failed to qualify for the main draw, losing in the first or second rounds of them all.


title: Barbora Krejčíková

text: In doubles, she had more success, reaching the semifinals at the Diamond Games, debuting in the main draw of the French Open, and winning her first title at the Tournoi de Québec. In November, she won her first WTA 125K title at the Open de Limoges, partnering Mandy Minella.


title: Barbora Krejčíková

text: In singles, Krejčíková mainly played on the ITF Circuit. However, she took part in a few WTA tournaments but lost in qualifying or in early rounds of the main draw. She lost in the first round of qualifying at the Qatar Open, Australian Open, and Wimbledon.


title: Barbora Krejčíková

text: Krejčíková started the year well in doubles, reaching the semifinals at the Auckland Open. Next, she played for the first time at the Australian Open, reaching the second round. In February, she played her first Premier final in doubles at the St. Petersburg Trophy. At the Qatar Open, she made her first appearance at a Premier 5/Premier Mandatory tournament, where she lost in the second round. Her breakthrough came at the French Open, where she, with her doubles partner Siniaková, reached the semifinals, losing to Elena Vesnina and Ekaterina Makarova. This result ranked her in the top 50 for the first time, at No. 34. At Wimbledon, she lost in the first round. At the US Open, she reached her second Grand Slam quarterfinal alongside Siniaková, but lost there to Martina Hingis and CoCo Vandeweghe. She entered the top 30 in rankings for the first time in her career.