Search NASA⌕ Search

SEARCH · Search NASA

Results for “SINGLE STAGE TO ORBIT ROCKETS”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 109 records · Page 6

A real-time guidance algorithm for aerospace plane optimal ascent to low earth orbit

Problems of onboard trajectory optimization and synthesis of suitable guidance laws for ascent to low Earth orbit of an air-breathing, single-stage-to-orbit vehicle are addressed. A multimode propulsion system is assumed which incorporates turbojet, ramjet, Scramjet, and rocket engines. An algorithm for generating fuel-optimal climb profiles is presented. This algorithm results from the application of the minimum principle to a low-order dynamic model that includes angle-of-attack effects and the normal component of thrust. Maximum dynamic pressure and maximum aerodynamic heating rate constraints are considered. Switching conditions are derived which, under appropriate assumptions, govern optimal transition from one propulsion mode to another. A nonlinear transformation technique is employed to derived a feedback controller for tracking the computed trajectory. Numerical results illustrate the nature of the resulting fuel-optimal climb paths.

Calise, A. J.↗

System Sensitivity Analysis Applied to the Conceptual Design of a Dual-Fuel Rocket SSTO

This paper reports the results of initial efforts to apply the System Sensitivity Analysis (SSA) optimization method to the conceptual design of a single-stage-to-orbit (SSTO) launch vehicle. SSA is an efficient, calculus-based MDO technique for generating sensitivity derivatives in a highly multidisciplinary design environment. The method has been successfully applied to conceptual aircraft design and has been proven to have advantages over traditional direct optimization methods. The method is applied to the optimization of an advanced, piloted SSTO design similar to vehicles currently being analyzed by NASA as possible replacements for the Space Shuttle. Powered by a derivative of the Russian RD-701 rocket engine, the vehicle employs a combination of hydrocarbon, hydrogen, and oxygen propellants. Three primary disciplines are included in the design - propulsion, performance, and weights & sizing. A complete, converged vehicle analysis depends on the use of three standalone conceptual analysis computer codes. Efforts to minimize vehicle dry (empty) weight are reported in this paper. The problem consists of six system-level design variables and one system-level constraint. Using SSA in a 'manual' fashion to generate gradient information, six system-level iterations were performed from each of two different starting points. The results showed a good pattern of convergence for both starting points. A discussion of the advantages and disadvantages of the method, possible areas of improvement, and future work is included.

Olds, John R.↗

Linear rocket engines for advanced space transportation systems

The linear rocket engine is shown to be a viable candidate propulsion system for post-Space Shuttle single-stage-to-orbit systems. The linear engine system has been developed and fired demonstrating high performance and long life with firing durations exceeding 500 seconds. The application of the split or dual combustor to the linear engine permits the uses of two different propellant combinations in a single engine system. The split combustor possesses the advantages of the two position extendible bell nozzle in a fixed nozzle configuration. Engine power cycles and applications to typical vehicles are discussed.

Sobin, A. J.↗

Implementation of a Single-Stage-To-Orbit (SSTO) model for stability and control analysis

Three NASA centers: Marshall Space Flight Center (MSFC), Langley Research Center (LaRC), and Johnson Space Center (JSC) are currently involved in studying a family of single-stage- and two-stage-to-orbit (SSTO/TSTO) vehicles to serve as the next generation space transportation system (STS). A rocketed winged-body is the current focus. The configuration (WB001) is a vertically-launched, horizontally-landing system with circular cross-section. Preliminary aerodynamic data was generated by LaRC and is a combination of wind-tunnel data, empirical methods, and Aerodynamic Preliminary Analysis System-(APAS) generated values. JSC's efforts involve descent trajectory design, stability analysis, and flight control system synthesis. Analysis of WB001's static stability indicates instability in 'tuck' (C(sub mu) less than 0: Mach = 0.30, alpha greater than 3.25 deg; Mach = 0.60, alpha greater than 8.04), an unstable dihedral effects (C(sub l(beta)) greater than 0: Mach = 30,alpha less than 12 deg.; Mach = 0.60, alpha less than 10.00 deg.), and, most significantly, an unstable weathercock stability derivative, C(sub n(beta)), at all angles of attack and subsonic Mach numbers. Longitudinal trim solutions for Mach = 0.30 and 0.60 indicate flight path angle possibilities ranging from around 12 (M = 0.30) to slightly over 20 degrees at Mach = 0.60. Trim angles of attack increase from 6.24 at Mach 0.60 and 10,000 feet to 17.7 deg. at Mach 0.30, sea-level. Lateral trim was attempted for a design cross-wind of 25.0 knots. The current vehicle aerodynamic and geometric characteristics will only yield a lateral trim solution at impractical tip-fin deflections (approximately equal to 43 deg.) and bank angles (21 deg.). A study of the lateral control surfaces, tip-fin controllers for WB001, indicate increased surface area would help address these instabilities, particularly the deficiency in C(sub n(beta)), but obviously at the expense of increased vehicle weight. Growth factors of approximately 7 were determined using a design C(sub n(beta)) of 0.100/radian (approximate subsonic values for the orbiter).

Ingalls, Stephen A.↗

Game Plan

Industry proposals for the Crew Exploration Vehicle that NASA plans as a replacement for the space shuttle are due next week, but the agency's new chief says it might be necessary to slow the CEV procurement at first to speed it up later. After a quick trip to Kennedy Space Center for briefings on getting the space shuttle back in operation, Michael D. Griffin sat down with his growing staff last week to begin work on modifying the CEV procurement. "We are going to rethink our entire program in that area," he said during an inaugural press conference Apr. 18. The proposals due May 2 are being prepared in response to NASA's call for a "risk-reduction flight effort" in 2008 that would lead to delivery of a human-rated CEV in 2014. But Griffin was co-leader on an independent study in 2004 that recommended a way to get the CEV flying astronauts in 2010, the year President Bush has set as a deadline for retiring the space shuttle fleet. In that study, produced for The Planetary Society, Griffin and his team called for development of a 13-15-ton "Block 1" CEV limited to low Earth orbit (LEO) that would be launched atop a single space shuttle solid rocket motor (SRM), with a new cryogenic upper stage based on existing rocket engine technology, Under this approach, NASA would develop a "Block 2" CEV later for human exploration beyond LEO.

Morring, Frank, Jr.↗

A Collaborative Analysis Tool for Integrated Hypersonic Aerodynamics, Thermal Protection Systems, and RBCC Engine Performance for Single Stage to Orbit Vehicles

Presented is a computer-based tool that connects several disciplines that are needed in the complex and integrated design of high performance reusable single stage to orbit (SSTO) vehicles. Every system is linked to every other system, as is the case of SSTO vehicles with air breathing propulsion, which is currently being studied by NASA. An RBCC propulsion system integrates airbreathing and rocket propulsion into a single engine assembly enclosed within a cowl or duct. A typical RBCC propulsion system operates as a ducted rocket up to approximately Mach 3. Then there is a transition to a ramjet mode for supersonic-to-hypersonic acceleration. Around Mach 8 the engine transitions to a scramjet mode. During the ramjet and scramjet modes, the integral rockets operate as fuel injectors. Around Mach 10-12 (the actual value depends on vehicle and mission requirements), the inlet is physically closed and the engine transitions to an integral rocket mode for orbit insertion. A common feature of RBCC propelled vehicles is the high degree of integration between the propulsion system and airframe. At high speeds the vehicle forebody is fundamentally part of the engine inlet, providing a compression surface for air flowing into the engine. The compressed air is mixed with fuel and burned. The combusted mixture must be expanded to an area larger than the incoming stream to provide thrust. Since a conventional nozzle would be too large, the entire lower after body of the vehicle is used as an expansion surface. Because of the high external temperatures seen during atmospheric flight, the design of an airbreathing SSTO vehicle requires delicate tradeoffs between engine design, vehicle shape, and thermal protection system (TPS) sizing in order to produce an optimum system in terms of weight (and cost) and maximum performance. To adequately determine the performance of the engine/vehicle, the Hypersonic Flight Inlet Model (HYFIM) module was designed to interface with the RBCC engine model. HYFIM performs the aerodynamic analysis of forebodies and inlet characteristics of RBCC powered SSTO launch vehicles. HYFIM is applicable to the analysis of the ramjet/scramjet engine operations modes (Mach 3-12), and provides estimates of parameters such as air capture area, shock-on-lip Mach number, design Mach number, compression ratio, etc., based on a basic geometry routine for modeling axisymmetric cones, 2-D wedge geometries. HYFIM also estimates the variation of shock layer properties normal to the forebody surface. The thermal protection system (TPS) is directly linked to determination of the vehicle moldline and the shaping of the trajectory. Thermal protection systems to maintain the structural integrity of the vehicle must be able to mitigate the heat transfer to the structure and be lightweight. Herein lies the interdependency, in that as the vehicle's speed increases, the TPS requirements are increased. And as TPS masses increase the effect on the propulsion system and all other systems is compounded. The need to analyze vehicle forebody and engine inlet is critical to be able to design the RBCC vehicle. To adequately determine insulation masses for an RBCC vehicle, the hypersonic aerodynamic environment and aeroheating loads must be calculated and the TPS thicknesses must be calculated for the entire vehicle. To accomplish this an ascent or reentry trajectory is obtained using the computer code Program to Optimize Simulated Trajectories (POST). The trajectory is then used to calculate the convective heat rates on several locations on the vehicles using the Miniature Version of the JA70 Aerodynamic Heating Computer Program (MINIVER). Once the heat rates are defined for each body point on the vehicle, then insulation thicknesses that are required to maintain the vehicle within structural limits are calculated using Systems Improved Numerical Differencing Analyzer (SINDA) models. If the TPS masses are too heavy for the performance of the vehicle the process may be repeated altering the trajectory or some other input to reduce the TPS mass. E-PSURBCC is an "engine performance" model and requires the specification of inlet air static temperature and pressure as well as Mach number (which it pulls from the HYFIM and POST trajectory files), and calculates the corresponding stagnation properties. The engine air flow path geometry includes inlet, a constant area section where the rocket is positioned, a subsonic diffuser, a constant area afterburner, and either a converging nozzle or a converging-diverging nozzle. The current capabilities of E-PSURBCC ejector and ramjet mode treatment indicated that various complex flow phenomena including multiple choking and internal shocks can occur for combinations of geometry/flow conditions. For a given input deck defining geometry/flow conditions, the program first goes through a series of checks to establish whether the input parameters are sound in terms of a solution path. If the vehicle/engine performance fails mission goals, the engineer is able to collaboratively alter the vehicle moldline to change aerodynamics, or trajectory, or some other input to achieve orbit. The problem described is an example of the need for collaborative design and analysis. RECIPE is a cross-platform application capable of hosting a number of engineers and designers across the Internet for distributed and collaborative engineering environments. Such integrated system design environments allow for collaborative team design analysis for performing individual or reduced team studies. To facilitate the larger number of potential runs that may need to be made, RECIPE connects the computer codes that calculate the trajectory data, aerodynamic data based on vehicle geometry, heat rate data, TPS masses, and vehicle and engine performance, so that the output from each tool is easily transferred to the model input files that need it.

Stanley, Thomas Troy↗

Comparison of super-high-energy-propulsion-systems based on metallic hydrogen propellant for ES to LEO space transportation

The application is studied of metallic H2 as a rocket propellant, which contains a specific energy of about 52 kcal/g in theory yielding a maximum specific impulse of 1700 s. With the convincing advantage of having a density 14 times that of conventional liquid H2/liquid O2 propellants, metallic H2 could satisfy the demands of advanced launch vehicle propulsion for the next millennium. Provided that there is an atomic metallic state of H2, and that this state is metastable at ambient pressure, which still is not proven, the results are given of the study of some important areas, which concern the production of metallic H2, the combustion, chamber cooling, and storage. The results show that the use of metallic H2 as rocket propellant could lead to revolutionary changes in space vehicle philosophy toward small size, small weight, and high performance single stage to orbit systems. The use of high metallic H2 mass fractions results in a dramatic reduction of required propellant volume, while gas temperatures in the combustion chamber exceed 5000 K. Furthermore, it follows, that H2 (liquid or slush) is the most favorable candidate as working fluid. Jet generated noise due to high exhaust velocities could be a problem.

Thierschmann, M.↗

Physical Limitations of Nuclear Propulsion for Earth to Orbit

An assessment of current nuclear propulsion technology for application in Earth to Orbit (ETO) missions has been performed. It can be shown that current nuclear thermal rocket motors are not sufficient to provide single stage performance as has been stated by previous studies. Further, when taking a systems level approach, it can be shown that NTRs do not compete well with chemical engines where thrust to weight ratios of greater than I are necessary, except possibly for the hybrid chemical/nuclear LANTR (LOX Augmented Nuclear Thermal Rocket) engine. Also, the ETO mission requires high power reactors and consequently large shielding weights compared to NTR space missions where shadow shielding can be used. In the assessment, a quick look at the conceptual ASPEN vehicle proposed in 1962 in provided. Optimistic NTR designs are considered in the assessment as well as discussion on other conceptual nuclear propulsion systems that have been proposed for ETO. Also, a quick look at the turbulent, convective heat transfer relationships that restrict the exchange of nuclear energy to thermal energy in the working fluid and consequently drive the reactor mass is included.

Blevins, John A.↗

NanoTHOR: Low-Cost Launch of Nanosatellites to Deep Space

The rapid development of high-performance nanosatellite platforms is enabling NASA and commercial ventures to consider performing missions to the asteroids, the Moon, and Mars at lower cost and on shorter timelines than traditional large spacecraft platforms. Currently, however, opportunities to launch secondary payloads to Earth escape are rare, and using chemical rockets to propel secondary payloads from LEO rideshares to escape is problematic due to the risks posed to primary payloads. The NanoTHOR effort has explored the technical feasibility and value proposition for using a simple momentum-exchange tether system to scavenge orbital energy from an upper stage in geostationary transfer orbit in order to boost nanosatellites to Earth escape. A NanoTHOR module will accomplish rapid transfer of a nanosatellite to an escape trajectory by deploying the nanosat at the end of a long, slender, high-strength tether and then using winching in the Earth's gravity!gradient to convert orbital angular momentum into rotational angular momentum. In the Phase I effort, we developed and simulated methods for controlling tether deployment and retraction to spin up a tether system, and these simulations demonstrated the feasibility of providing delta-Vs on the order of 800 m/s with a simple, low-mass tether system. Moreover, the NanoTHOR tether can act as a reusable in-space upper stage, boosting multiple nanosatellites on a single launch and doing so with a mass requirement lower than that of conventional rocket technologies. Serving as an escape-injection stage, NanoTHOR can enable a 6U CubeSat to deliver small payloads to Mars orbit, lunar orbit, and rendezvous with at least 110 of the known near-Earth asteroids. Evaluation of the technology readiness of the component technologies required for NanoTHOR indicate that the hardware required is all mid-TRL, and the lower-TRL controls and integration components can be advanced to mid-TRL with modest investment. By scavenging orbital energy from upper stages without any stored energy or propellant requirements, NanoTHOR permits deep-space nanosat missions to launch on rideshare opportunities, enabling NASA and commercial ventures to conduct affordable and frequent missions to explore deep space destinations.

Hoyt, Robert↗

Evaluation of Proposed Rocket Engines for Earth-to-Orbit Vehicles

The objective is to evaluate recently analyzed rocket engines for advanced Earth-to-orbit vehicles. The engines evaluated are full-flow staged combustion engines and split expander engines, both at mixture ratios at 6 and above with oxygen and hydrogen propellants. The vehicles considered are single-stage and two-stage fully reusable vehicles and the Space Shuttle with liquid rocket boosters. The results indicate that the split expander engine at a mixture ratio of about 7 is competitive with the full-flow staged combustion engine for all three vehicle concepts. A key factor in this result is the capability to increase the chamber pressure for the split expander as the mixture ratio is increased from 6 to 7.

Martin, James A.↗

Human Lunar Mission Capabilities Using SSTO, ISRU and LOX-Augmented NTR Technologies: A Preliminary Assessment

The feasibility of conducting human missions to the Moon is examined assuming the use of three 'high leverage' technologies: (1) a single-stage-to-orbit (SSTO) launch vehicle, (2) 'in-situ' resource utilization (ISRU)--specifically 'lunar-derived' liquid oxygen (LUNOX), and (3) LOX-augmented nuclear thermal rocket (LANTR) propulsion. Lunar transportation system elements consisting of a LANTR-powered lunar transfer vehicle (LTV) and a chemical propulsion lunar landing/Earth return vehicle (LERV) are configured to fit within the 'compact' dimensions of the SSTO cargo bay (diameter: 4.6 m/length: 9.0 m) while satisfying an initial mass in low Earth orbit (IMLEO) limit of approximately 60 t (3 SSTO launches). Using approximately 8 t of LUNOX to 'reoxidize' the LERV for a 'direct return' flight to Earth reduces its size and mass allowing delivery to LEO on a single 20 t SSTO launch. Similarly, the LANTR engine's ability to operate at any oxygen/ hydrogen mixture ratio from 0 to 7 with high specific impulse (approximately 940 to 515 s) is exploited to reduce hydrogen tank volume, thereby improving packaging of the LANTR LTV's 'propulsion' and 'propellant modules'. Expendable and reusable, piloted and cargo missions and vehicle designs are presented along with estimates of LUNOX production required to support the different mission modes. Concluding remarks address the issue of lunar transportation system costs from the launch vehicle perspective.

Borowski, Stanley K.↗

An evaluation of composite propulsion for single-stage-to-orbit vehicles designed for horizontal take-off

Composite propulsion was analyzed for single-stage-to-orbit vehicles designed for horizontal take-off. Trajectory, geometric, and mass analyses were performed to establish the orbital payload capability of six engines. The results indicated that none of the engines performed adequately to deliver payloads to orbit as analyzed. The single-stage turbine and oxidizer-rich gas generator resulted in a low engine specific impulse, and the performance increment of the ejector subsystem was less than that of a separate rocket system with a high combustion pressure. There was a benefit from incorporating a fan into the engine, and removal of the fan from the airstream during the ramjet mode increased the orbital payload capability.

Martin, J. A.↗

Efficient GO2/GH2 Injector Design: A NASA, Industry and University Cooperative Effort

Developing new propulsion components in the face of shrinking budgets presents a significant challenge. The technical, schedule and funding issues common to any design/development program are complicated by the ramifications of the continuing decrease in funding for the aerospace industry. As a result, new working arrangements are evolving in the rocket industry. This paper documents a successful NASA, industry, and university cooperative effort to design efficient high performance GO2/GH2 rocket injector elements in the current budget environment. The NASA Reusable Launch Vehicle (RLV) Program initially consisted of three vehicle/engine concepts targeted at achieving single stage to orbit. One of the Rocketdyne propulsion concepts, the RS 2100 engine, used a full-flow staged-combustion cycle. Therefore, the RS 2100 main injector would combust GO2/GH 2 propellants. Early in the design phase, but after budget levels and contractual arrangements had been set the limitations of the current gas/gas injector database were identified. Most of the relevant information was at least twenty years old. Designing high performance injectors to meet the RS 2100 requirements would require the database to be updated and significantly enhanced. However, there was no funding available to address the need for more data. NASA proposed a teaming arrangement to acquire the updated information without additional funds from the RLV Program. A determination of the types and amounts of data needed was made along with test facilities with capabilities to meet the data requirements, budget constraints, and schedule. After several iterations a program was finalized and a team established to satisfy the program goals. The Gas/Gas Injector Technology (GGIT) Program had the overall goal of increasing the ability of the rocket engine community to design efficient high-performance, durable gas/gas injectors relevant to RLV requirements. First, the program would provide Rocketdyne with data on preliminary gas/gas injector designs which would enable discrimination among candidate injector designs. Secondly, the program would enhance the national gas/gas database by obtaining high-quality data that increases the understanding of gas/gas injector physics and is suitable for computational fluid dynamics (CFD) code validation. The third program objective was to validate CFD codes for future gas/gas injector design in the RLV program.

Tucker, P. K.↗

Minimum fuel trajectory for the aerospace-plane

An overall trajectory for a single-stage-to-orbit vehicle with an initial weight of 234 tons is calculated, and four different propulsion models including turbojet, ramjet, scramjet, and rocket are considered. First, the atmospheric flight in the thicker atmosphere is discussed with emphasis on trajectory optimization, optimization problem, aerodynamic problem, propulsion model, and initial conditions. The performance of turbojet and ramjet-scramjet engines is analyzed; and then the flight to orbit is assessed from the optimization point of view. It is shown that roll modulation saves little during the trajectory, and the combined application of airbreathing propulsion and aerodynamic lift is suggested.

Breakwell, John V.↗

Launch Vehicle Systems Analysis

This report summaries the key accomplishments of Georgia Tech's Space Systems Design Laboratory (SSDL) under NASA Grant NAG8-1302 from NASA - Marshall Space Flight Center. The report consists of this summary white paper, copies of technical papers written under this grant, and several viewgraph-style presentations. During the course of this grant four main tasks were completed: (1)Simulated Combined-Cycle Rocket Engine Analysis Module (SCCREAM), a computer analysis tool for predicting the performance of various RBCC engine configurations; (2) Hyperion, a single stage to orbit vehicle capable of delivering 25,000 pound payloads to the International Space Station Orbit; (3) Bantam-X Support - a small payload mission; (4) International Trajectory Support for interplanetary human Mars missions.

Olds, John R.↗

Exploring Mars: the Ares Payload Service (APS)

In last year's Mars Society convention we introduced the results of five years of studies of space launch capability for the second millennium. We concluded that Single Stage to Orbit (SSTO) vehicles such as the Delta Clipper X33, and X34 cannot make it to orbit from the Earth's surface. Whether taking off vertically or horizontally or landing vertically or horizontally, the rocket equations, the performance of available fuels, and the realities of the weight and strength of materials leave no margin for payload. The promised savings from SSTO systems are illusory. However, a configuration that is able to deliver useful payload to orbit is the Single step to Orbit, SsTO, a rocket plane that is released fully fueled, from 35,000 to 40,000 feet altitude. Three approaches have been proposed. The Hot'l and Molnya Corporation designs carry the fueled rocket plane to altitude on the back of a carrier aircraft. In this design the carrier aircraft is Russia's Antonov 225 the world's largest cargo plane. The rocket plane is a modified version of the Buran, Russia's own space shuttle. Another configuration is Kelly Aviation's concept in which the fully fueled rocket plane is towed to altitude by the cargo plane and then released. A third approach is based on the early "X" planes, which were dropped from the belly of the carrier plane. While the rocket equations indicate that these three concepts can deliver useful payloads, the Stanford review found significant advantages to the approach of Pioneer Rocket, in which the rocket plane flies up to the carrier plane with conventional jet engines, docks, and then loads on the oxidizer for the flight to orbit. This architecture has more reasonable abort modes in case of system failure in either aircraft and can deliver a larger final payload to orbit for a given sized carrier. The Stanford recommendation is that the carrier aircraft be the Antonov 225. A design based on this was presented in a report last year. Refinements to the design notably an improved re-entry cooling system and fueling stability analysis were done this year. More technical detail and a proposed international consortium to develop the SSTO is presented in another session of this year's Mars convention. We believe that there will be no human exploration of Mars based on the Shuttle or Expendable launch vehicles, and no resources available except for a cooperative international program. However, just as the world is learning to cooperate in peacekeeping, we hold out the hope that similar cooperation will develop for Mars exploration. With that in mind, this year we asked the question- "How will the human mission get to Mars if it has to use the SsTO for transportation?"

Bowen, Justin↗

Nozzle selection for optimized single-stage shuttles

A computerized preliminary design procedure has been used to determine rocket-engine bell-nozzle configurations which achieve either minimum gross mass or minimum dry mass on an advanced technology winged single-stage-to-orbit launch vehicle. This procedure evaluates the trade offs between the optimized ascent flight trajectory performance and the flight vehicle sizing driven by the engine mass and propellant requirements. Numerous mission, flight, and vehicle related requirements and constraints are satisfied in this process. Propulsion system configurations which incorporate dual-position nozzles are evaluated and compared to configurations using all fixed nozzles.

Eldred, C. H.↗

Effect of propulsion system characteristics on ascent performance of dual-fueled single-stage earth-to-orbit transports

The results of a parametric study of ascent performance are presented for a vertical take off, horizontal landing, single stage earth-to-orbit transport vehicle. Two dual fueled concepts, series burn and parallel burn, were investigated, both of which utilized dual position rocket nozzles. The analysis was made by systematically varying a set of propulsion similarity parameters, initial thrust-weight ratio, the proportion of the thrust due to dual position nozzle engines, expansion ratios of the rocket nozzle, and the relative split between the two fuels, hydrogen and hydrocarbon. The data are presented as a series of curves of mass ratio plotted against each of the similarity parameters for various combinations of the other similarity parameters.

Rehder, J. J.↗