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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.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Root Characteristics Vary with Depth Across Four Lowland Seasonal Tropical Forests

Fine roots are key to ecosystem-scale nutrient, carbon (C), and water cycling, yet our understanding of fine root trait variation within and among tropical forests, one of Earth’s most C-rich ecosystems, is limited. We characterized root biomass, morphology, nutrient content, and arbuscular mycorrhizal fungal (AMF) colonization to 1.2 m depths across four distinct lowland Panamanian forests, and related root characteristics to soil C stocks. We hypothesized that: (H1) Fine root characteristics vary consistently with depth across seasonal tropical forests, with deeper roots exhibiting more exploratory traits, such as for deep water acquisition; (H2) fine root characteristics vary among tropical forests mainly in surface soils, where resource availability also varies. Here we found consistent variation with depth across the four forests, including decreased root biomass, root tissue density, and AMF, and increased specific root length. Among the forests, there was variation in some fine root characteristics, including greater surface root biomass and lower SRL in the wettest forest, and smaller fine root diameter in the driest forest. We also found that root characteristics were related to total soil C stocks, which were positively related to root biomass and negatively related to specific root length. These results indicate emergent properties of root variation with depth across tropical forests, and show site-scale variation in surface root characteristics. Future work could explore the flexibility in root characteristics under changing conditions such as drought.

13C NMR↗
DOE OSTI2025

Data from TropiRoot 1.0 database: tropical root characteristics across environments

TropiRoot 1.0 is a new tropical root database with root characteristics across environment gradients. It has data extracted from 104 new sources, resulting in more than 8000 rows of data (either species or community data). Most of the data in TropiRoot 1.0 includes root characteristics such as root biomass, morphology, root dynamics, mass fraction, architecture, anatomy, physiology and root chemistry. This initiative represents an approximately 30% increase in the currently available data for tropical roots in the Fine Root Ecology Database (FRED). TropiRoot 1.0, contains root characteristics from 25 different countries where seven are located in Asia, six in South America, five in Central America and the Caribbean, four in Africa, two in North America, and 1 in Oceania. Due to the volume of data, when ancillary data was available, including soil data, these data was either extracted and included in the database or their availability was recorded in an additional column. Multiple contributors checked the entries for outliers during the collation process to ensure data quality. For text-based observations, we examined all cells to ensure that their content relates to their specific categories. For numerical observations, we ordered each numerical value from least to greatest and plotted the values, checking apparent outliers against the data in their respective sources and correcting or removing incorrect or impossible values. Some data (soil and aboveground) have different columns for the same variable presented in different units, including originally published units, but root characteristics data had units converted to match the ones reported in FRED. By filling a gap from global databases, TropiRoot 1.0 expands our knowledge of otherwise so far underrepresented regions, and our ability to assess global trends. This advancement can be used to improve tropical forest representation in vegetation models.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Soil properties and root characteristics across four lowland Panamanian forests from 0 - 1 m soil depths

Objectives:Fine roots significantly influence ecosystem-scale cycling of nutrients, carbon (C), and water, yet there is limited understanding of how fine root traits vary across and within tropical forests, some of Earth's most C-rich ecosystems. The biomass of fine roots can impact soil carbon storage, as root mortality is a primary source of new carbon to soils. A positive relationship has been observed between fine root biomass and soil carbon stocks in Panama (Cusack et al 2018). Beyond biomass, root characteristics like specific root length (SRL) could also influence soil carbon, as roots with higher SRL are less dense and thinner, potentially decomposing more easily or promoting soil aggregation. Understanding the effects of root morphology and tissue quality on soil carbon storage and with soil properties in general can improve predictions of landscape-scale carbon patterns. We aggregated new data of root biomass, morphology and nutrient content at 0-10 cm, 10-20 cm, 20-50 cm and 50-100 cm depth increments across four distinct lowland Panamanian forests and paired with already published datasets (Cusack et al 2018; Cusack and Turner 2020) of soil chemistry from the same sites and soil depths to explore relationship between soil carbon stocks and root characteristics.Datasets included:The datasets provided include .csv and .xlsx files for fine root characteristics and soil chemistry from four different forests across 0-10 cm, 10-20 cm, 20-50 cm, and 50-100 cm depth increments. Root characteristics include live fine root biomass, dead fine root biomass, coarse root biomass, specific root length, root diameter, root tissue density, specific root area, root %N, root %C, and root C/N ratio. Soil chemistry data includes total carbon (TC), dissolved organic carbon (DOC), bulk density, total phosphorus (TP), available phosphorus (AEM Pi), and various Mehlich-extractable elements such as aluminum, calcium, iron, potassium, manganese, phosphorus, and zinc. Nitrogen content measures include ammonium, nitrate, total dissolved nitrogen (TDN), dissolved inorganic nitrogen (DIN), and dissolved organic nitrogen (DON). The dataset also includes total exchangeable bases (TEB) and effective cation exchange capacity (ECEC) in both centimoles of charge per kilogram and micromoles of charge per gram. The soil chemistry data was obtained from Cusack et al (2018) and Cusack and Turner (2020) and paired with root characteristics data for the same depth increments and sites. Additionally, a .kml file is provided with coordinates for all 32 plots included in the study across four forests (n = 8 plots per site). Root data was averaged across these 8 plots per site and soil data was collected in one pit in each site. This dataset serves as baseline data before a throughfall exclusion experiment, Panama Rainforest Changes with Experimental Drying (PARCHED), was implemented. No special software is needed to open these files.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

TropiRoot 1.0: Database of tropical root characteristics across environments

Tropical ecosystems contain the world's largest biodiversity of vascular plants. Yet, our understanding of tropical functional diversity and its contribution to global diversity patterns is constrained by data availability. This discrepancy underscores an urgent need to bridge data gaps by incorporating comprehensive tropical root data into global datasets. Here, we provide a database of tropical root characteristics. This new database, TropiRoot 1.0, will be instrumental in evaluating an array of hypotheses pertaining to root functional ecology and plant biogeography, both within the tropics and relative to other global biomes. The data compilation was conducted by the TropiRoot Initiative, in partnership with the Fine-Root Ecology Database (FRED) and the Global Root Trait (GRooT) database, Colorado State University (CSU) and the Smithsonian Tropical Research Institute (STRI). Literature search and data extraction were conducted between 2020 and 2024. Literature was identified using Web of Science, Scopus, and complemented using the expert knowledge of members of TropiRoot. To provide broad environmental and geographical distributions, literature searches included root characteristics (traits) across global change drivers, natural gradients, and from different continents. We adopted FRED standardized data columns and streamlined the format to enhance accessibility for data extraction across various user groups. This optimized framework resulted in a smaller, yet comprehensive datasheet. To make the database compatible with other global root trait initiatives, column identification was standardized following the codes provided by FRED. These efforts culminated in data extracted from 104 new sources, resulting in more than 8000 rows of data (either species or community data). Most of the data in TropiRoot 1.0 include root characteristics such as root biomass, morphology, root dynamics, mass fraction, architecture, anatomy, physiology, and root chemistry. This initiative represents a 30% increase in the currently available data for tropical roots in FRED. TropiRoot 1.0 contains root characteristics from 25 different countries, where seven are located in Asia, six in South America, five in Central America and the Caribbean, four in Africa, two in North America, and 1 in Oceania. Due to the volume of data, when ancillary data were available, including soil data, these data were either extracted and included in the database or its availability was recorded in an additional column. Multiple contributors checked the entries for outliers during the collation process to ensure data quality. For text-based observations, we examined all cells to ensure that their content relates to their specific categories. For numerical observations, we ordered each numerical value from least to greatest and plotted the values, checking apparent outliers against the data in their respective sources and correcting or removing incorrect or impossible values. Some data (soil and aboveground) have different columns for the same variable presented in different units, including originally published units, but root characteristics data had units converted to match those reported in FRED. By filling a gap from global databases, TropiRoot 1.0 expands our knowledge of otherwise so far underrepresented regions and our ability to assess global trends. This advancement can be used to improve tropical forest representation in vegetation models. The data are freely available and should be cited when used.

FRED↗
NASA NTRS1983

Spatial variation of corn canopy temperature as dependent upon soil texture and crop rooting characteristics

A soil plant atmosphere model for corn (Zea mays L.) together with the scaling theory for soil hydraulic heterogeneity are used to study the sensitivity of spatial variation of canopy temperature to field averaged soil texture and crop rooting characteristics. The soil plant atmosphere model explicitly solves a continuity equation for water flux resulting from root water uptake, changes in plant water storage and transpirational flux. Dynamical equations for root zone soil water potential and the plant water storage models the progressive drying of soil, and day time dehydration and night time hydration of the crop. The statistic of scaling parameter which describes the spatial variation of soil hydraulic conductivity and matric potential is assumed to be independent of soil texture class. The field averaged soil hydraulic characteristics are chosen to be representative of loamy sand and clay loam soils. Two rooting characteristics are chosen, one shallow and the other deep rooted. The simulation shows that the range of canopy temperatures in the clayey soil is less than 1K, but for the sandy soil the range is about 2.5 and 5.0 K, respectively, for the shallow and deep rooted crops.

Choudhury, B. J.↗
NASA NTRS1984

STABCAR: A program for finding characteristic root systems having transcendental stability matrices

STABCAR can be used to determine the characteristic roots of flexible, actively controlled aircraft, including the effects of unsteady aerodynamics. A modal formulation and a transfer-matrix representation of the control system are employed. Operable in either a batch or an interactive mode, STABCAR can provide graphical or tabular output of the variation of the roots with velocity, density, altitude, dynamic pressure or feedback gains. Herein the mathematical model, program structure, input requirements, output capabilities, and a series of sample cases are detailed. STABCAR was written for use on CDC CYBER 175 equipment; modification would be required for operation on other machines.

Adams, W. M., Jr.↗
DOE OSTI2026

Transgenic Mixed‐Linkage‐Glucan Enhancement Affects Root Characteristics and Decomposition in Soils of Contrasting Vegetation History

ABSTRACT Development of transgenic bioenergy sorghum [ Sorghum bicolor (L.) Moench] with increased contents of mixed‐linkage (1,3;1,4)‐β‐glucan (MLG) is an important step towards enhancing quality of bioenergy feedstocks. Since MLG‐enhancement leads to greater biomass digestibility, our overarching hypothesis is that root residues of MLG‐enhanced plants may be more readily decomposed in the soil, potentially creating new opportunities for optimizing soil carbon (C) sequestration, nutrient cycling, and overall agricultural sustainability. The study examined morphological, chemical, and enzymatic characteristics of fine and coarse roots of four bioenergy sorghum genotypes. Then, we incubated the roots within soils with contrasting vegetation histories while measuring C mineralization, microbial biomass C (MBC), and activity of hydrolytic enzymes and calculating vector length and vector angle enzymatic stoichiometry parameters. The results indicated that MLG‐enhancing transformations increased root total nitrogen (N) contents, decreased C/N ratios, and were associated with higher MLG concentrations in fine than in coarse roots. Incubations with transgenic roots led to 16%–38% higher MBC and 19%–41% lower microbial metabolic quotient (qCO 2 ). While enzyme activity differed markedly among the studied genotypes, it did not directly respond to MLG levels in root tissues. The increase in MBC without concurrent increases in C mineralization or hydrolytic enzyme activities in transgenic genotypes suggests that MLG enhancement promoted microbial anabolic retention of root‐derived C rather than stimulating catabolic decomposition. Enzymatic vector results indicated that these parameters reflect a variety of drivers behind microbial enzyme production, including availability of specific substrates, such as MLG here, and/or deficiency in specific nutrients, such as phosphorus (P). The study confirms the positive impacts from the roots of engineered MLG‐enhanced bioenergy plants on soil microbial activity and highlights the interactive influences on the MLG‐enhancement effects from root size and inherent soil properties.

Mahmoodabadi, Majid [Department of Plant, Soil, an↗
NASA NTRS1973

Computer Programs for Calculating and Plotting the Stability Characteristics of a Balloon Tethered in a Wind

Computer programs for calculating the stability characteristics of a balloon tethered in a steady wind are presented. Equilibrium conditions, characteristic roots, and modal ratios are calculated for a range of discrete values of velocity for a fixed tether-line length. Separate programs are used: (1) to calculate longitudinal stability characteristics, (2) to calculate lateral stability characteristics, (3) to plot the characteristic roots versus velocity, (4) to plot the characteristic roots in root-locus form, (5) to plot the longitudinal modes of motion, and (6) to plot the lateral modes for motion. The basic equations, program listings, and the input and output data for sample cases are presented, with a brief discussion of the overall operation and limitations. The programs are based on a linearized, stability-derivative type of analysis, including balloon aerodynamics, apparent mass, buoyancy effects, and static forces which result from the tether line.

Bennett, R. M.↗
DOE OSTI2024

Tropical root responses to global changes: A synthesis

Tropical ecosystems face escalating global change. These shifts can disrupt tropical forests' carbon (C) balance and impact root dynamics. Since roots perform essential functions such as resource acquisition and tissue protection, root responses can inform about the strategies and vulnerabilities of ecosystems facing present and future global changes. However, root trait dynamics are poorly understood, especially in tropical ecosystems. We analyzed existing research on tropical root responses to key global change drivers: warming, drought, flooding, cyclones, nitrogen (N) deposition, elevated (e) CO 2 , and fires. Based on tree species- and community-level literature, we obtained 266 root trait observations from 93 studies across 24 tropical countries. We found differences in the proportion of root responsiveness to global change among different global change drivers but not among root categories. In particular, we observed that tropical root systems responded to warming and eCO 2 by increasing root biomass in species-scale studies. Drought increased the root: shoot ratio with no change in root biomass, indicating a decline in aboveground biomass. Despite N deposition being the most studied global change driver, it had some of the most variable effects on root characteristics, with few predictable responses. Episodic disturbances such as cyclones, fires, and flooding consistently resulted in a change in root trait expressions, with cyclones and fires increasing root production, potentially due to shifts in plant community and nutrient inputs, while flooding changed plant regulatory metabolisms due to low oxygen conditions. The data available to date clearly show that tropical forest root characteristics and dynamics are responding to global change, although in ways that are not always predictable. This synthesis indicates the need for replicated studies across root characteristics at species and community scales under different global change factors.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Impoverishing Roots Will Improve Wheat Yield and Profitability Through Increased Water and Nitrogen Use Efficiencies

We report that more than a 60% increase in crop production is required by the 2050's to feed a growing world population. Understanding how plant functional traits and field management affect crop yields has the potential to improve agricultural productivity, minimize economic and environmental losses, and maximize food security. We explored the influence of winter wheat root characteristics and management on winter wheat growth, yield, and profit using a mechanistic and well-tested ecosystem and crop model, ecosys. We applied and further tested ecosys at an agricultural farm growing winter wheat in Ardmore, Oklahoma, United States. The model accurately predicted observed shoot carbon (R 2 = 0.95), soil moisture (R 2 = 0.67), soil temperature (R 2 = 0.91), and yield (percent error = 17%). Numerical optimization experiments were conducted to explore potential improvements of winter wheat yield and profit by modifying root characteristics, including root radius and root:shoot carbon transfer conductance, and fertilizer inputs. Our results show the potential for simultaneously improving winter wheat yields and profits. The optimum conditions were found to be in the range of root radius between 0.1 and 0.3 mm, carbon transfer conductance between 0.004 and 0.01 h -1 , and the currently applied fertilizer rate of 112 kg ha -1 . Under these conditions, improvements in yields and profits of up to approximately 25% and 110%, respectively, were modeled compared to those under baseline root traits. These improvements were achieved by impoverishing root structures, thereby increasing nutrient allocation to grains. Our results also demonstrate and motivate model structures that integrate the complex network of plant physiology, soil nutrient biogeochemistry, hydrology, and management.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Root biomass and traits across four lowland Panamanian forests from 0 - 1.2 m soil depths

Fine roots are key to ecosystem-scale nutrient, carbon (C), and water cycling, yet our understanding of fine root traits variation within and among tropical forests, one of Earth’s most C-rich ecosystems, is limited. We characterized root biomass, morphology, nutrient content, and arbuscular mycorrhizal fungal (AMF) colonization in 10 cm increments to 1.2 m depth across four distinct lowland Panamanian forests. The datasets provided include a .xlsx file for fine root characteristics across 10 cm increment depths to 1.2 m collected from late 2017 to 2018 across four different forests. Root characteristics include live fine root biomass, dead fine root biomass, coarse root biomass, specific root length, root diameter, root tissue density, specific root area, arbuscular mycorrhizal fungi colonization, root chemistry (e.g., organic chemistry), root %N, root %C, root C/N ratio, and root radiocarbon content. This .xlsx file contain four tabs with 1) Dataset; 2) Metadata with information about each column in the dataset; 3) The sampling methods summarized; 4) Sites information. We also provided csv files for each of these tabs. Additionally, a .kml file is provided with coordinates for all 32 plots included in the study across four forests (n = 8 plots per site/forest). This dataset serves as baseline data before a throughfall exclusion experiment, Panama Rainforest Changes with Experimental Drying (PARCHED), was implemented. No special software is needed to open these files.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Radiocarbon of deep fine roots across four in lowland seasonal tropical forests in Panama

Objectives:Fine roots are key to ecosystem-scale nutrient, carbon (C), and water cycling, but our understanding of fine root traits variation within and among tropical forests, one of Earth’s most C-rich ecosystems, is limited. In 2022 and 2023, we aimed to explore differences in deep root characteristics among four lowland tropical forests in Panama, which vary in fertility and mean annual precipitation. We measured radiocarbon content (fraction modern [FM] and Δ14C) and δ13C of live fine roots at depths greater than 80 cm, up to 120 cm. The goal was to understand how deep root characteristics differ across these sites.Datasets included:The datasets provided include .csv and .xlsx files for radiocarbon content (fraction modern [FM] and Δ14C) and δ13C of live fine roots at depths greater than 80 cm, up to 120 cm collected from late 2017 to 2018 across four different forests. Additionally, a .kml file is provided with coordinates for all 32 plots included in the study across four forests (n = 8 plots per site). This dataset serves as baseline data before a throughfall exclusion experiment, Panama Rainforest Changes with Experimental Drying (PARCHED), was implemented. No special software is needed to open these files.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Toward a coordinated understanding of hydro‐biogeochemical root functions in tropical forests for application in vegetation models

Summary Tropical forest root characteristics and resource acquisition strategies are underrepresented in vegetation and global models, hampering the prediction of forest–climate feedbacks for these carbon‐rich ecosystems. Lowland tropical forests often have globally unique combinations of high taxonomic and functional biodiversity, rainfall seasonality, and strongly weathered infertile soils, giving rise to distinct patterns in root traits and functions compared with higher latitude ecosystems. We provide a roadmap for integrating recent advances in our understanding of tropical forest belowground function into vegetation models, focusing on water and nutrient acquisition. We offer comparisons of recent advances in empirical and model understanding of root characteristics that represent important functional processes in tropical forests. We focus on: (1) fine‐root strategies for soil resource exploration, (2) coupling and trade‐offs in fine‐root water vs nutrient acquisition, and (3) aboveground–belowground linkages in plant resource acquisition and use. We suggest avenues for representing these extremely diverse plant communities in computationally manageable and ecologically meaningful groups in models for linked aboveground–belowground hydro‐nutrient functions. Tropical forests are undergoing warming, shifting rainfall regimes, and exacerbation of soil nutrient scarcity caused by elevated atmospheric CO 2 . The accurate model representation of tropical forest functions is crucial for understanding the interactions of this biome with the climate.

Plant Sciences↗
DOE OSTI2021

Consequences of Altered Root Nutrient Uptake for Soil Carbon Stabilization (Final Report)

Objectives: The primary objectives of this project are to improve our understanding of tropical forest belowground processes, and to increase representation of this biome in ecosystem-scale and global C cycle models. Belowground dynamics present a major source of uncertainty inhibiting our ability to predict C cycle responses to climate change. Representation of tropical forests in these models is particularly lacking, even though these ecosystems hold >25% of terrestrial C stocks. While the majority of humid tropical forests exist on relatively infertile soils (i.e. poor in rock-derived nutrients), enormous gradients in soil nutrient availability exist at landscape-scales due to shifts in geology and soil order. Soil fertility is likely to greatly influence how tropical soil C storage will respond to the declines in precipitation predicted for the tropics. In particular, root dynamics vary across soil fertility gradients, with roots representing the major input of C to subsurface soils. Root characteristics related to soil fertility include biomass, turnover, C exudates, tissue chemistry, and nutrient uptake rates, with each of these likely sensitive to changes in moisture. The proposed project will fully integrate field research with an existing ecosystem model to assess potential effects of drying on belowground tropical C dynamics across a range of soil fertilities. Project Description:. This project will identify key linkages among soil fertility, root nutrient uptake, root dynamics, and soil C storage. The hypothesized framework for understanding these linkages is: Tropical rainforest roots in relatively fertile soils have lesser root biomass, faster turnover, fewer exudates, and improved root tissue quality relative to infertile sites, all resulting from increased root nutrient uptake. Ultimately, these root characteristics in fertile soils promote proportionally greater long-term soil C storage in organo-mineral associations. Specifically, lower root exudates and improved root tissue quality in fertile soils reduce microbial respiration of extant soil C (i.e. “priming) and increase microbial C use efficiency, leading to sorption of protein-rich microbial and plant organic matter on soil mineral surfaces. However, the larger pools of mineral-associated C in fertile tropical soils are more vulnerable to decreased rainfall than smaller C pools in infertile soils, in part because of the lesser root biomass, faster turnover, and softer, more labile root tissues. Also, the shallower lateral rooting structures typical in fertile soils experience greater desiccation and death than deeper roots in infertile sites, with little short-term adaptation to drying.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1976

Complex symmetric root square locus with an application to a spinning drag-free satellite

The parameters and relations associated with optimal systems are examined, taking into account a quadratic performance index and a root square locus plot, including the characteristic roots of the optimal system and its adjoint system as a function of the cost function weights. The calculation of the locus is described and the employment of the considered relations in studies of a drag-free satellite is discussed. Attention is given to weights regarding the initial states, questions of rotating integral control, approaches for experimental verification, and the performance of various methods for the reduction of fuel consumption due to center of spin offsets.

Tashker, M. G.↗
DOE OSTI2025

Higher levels of mixed-linkage (1,3;1,4)-β-glucan in transgenic grasses may impact soil C processing

Carbohydrates, including mixed-linkage glucan (MLG), in grass cell walls make them a valuable potential feedstock for biofuel production. Hence, the development of transgenic grasses with elevated levels of MLG is being actively pursued worldwide. Changes in chemical and physical root characteristics of MLG-overproducing transgenic plants can affect processing of the root-derived carbon (C) by soil microorganisms, impacting soil C cycling. Here, this study is the first attempt to elucidate the impact of MLG-related genetic modifications on root traits, root decomposition, and soil C processing. We explored four genotypes of Brachypodium ( Brachypodium distachyon ): a wildtype, a loss-of-function mutant with low MLG, an MLG overexpressing line, and a line lacking MLG hydrolase (with high MLG), incubating their roots in soils of two contrasting vegetation histories: monoculture switchgrass and polyculture restored prairie. The four genotypes exhibited contrasting root MLG and soluble sugar concentrations, and different growth phenotypes. Roots with the highest MLG content resulted in a ∼55 % increase in microbial biomass C compared to wildtype in both soils. However, the genotype effects on C mineralization rates were influenced by the vegetation history, with significant effects observed only in the soil from switchgrass but not prairie origin. While further work is required to understand the full impact of MLG-overproducing plants on soil C accrual, our findings suggest that their influence on soil C processes cannot be discounted.

Brachypodium distachyon↗
NASA NTRS1989

Thrust law effects on the long-period modes of aerospace craft

An analytical study is presented of the longitudinal long-period dynamics of an aerospace craft in a nearly circular orbit, with a thrust law depending arbitrarily on the speed and altitude. A plane of engine possibilities is first defined, with points corresponding to propulsion systems having prescribed thrust slopes with respect to speed and altitude. Approximate expressions for the characteristic roots and times are obtained by first identifying a small quantity in the coefficients of the characteristic equation, and then expanding in a perturbation series about the origin of the plane of engine possibilities, for which the solution is always known. These expressions agree very well with the exact solutions over a wide range of altitudes and thrust laws. The period of the oscillatory translational mode (phugoid) is found to be independent to first order of the thrust law, generalizing results found by previous investigators for specific thrust laws. The results apply to the speed range from hypersonic to orbital.

Markopoulos, Nikos↗