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NASA NTRS2008

Radio-sensitization of Prostate Cancer Cells by Monensin Treatment and its associated Gene Expression Profiling Changes

Radio-resistant or recurrent prostate cancer represents a serious health risk for approximately 20%-30% of patients treated with primary radiation therapy for clinically localized prostate cancer. Here, we investigated the effect of monensin on sensitizing radiation mediated cell killing of two radio-resistant prostate cell lines Lncap (P53+ and AR+) and PC3 (P53- and AR-). Treatment with monensin alone (5 micromoles-20 micromoles) showed a significant direct cell killing of Lncap (10-30%), but not PC3 cells. Monensin was also shown to successfully sensitize Lncap cells to X-ray radiation (2Gy-10Gy) mediated cell death, up to 50% of killing with the combined treatment. To better understand the mechanisms of radio-resistance of these two cell lines and their different response to monensin, the apoptosis related gene expression profiles in both cell lines were analyzed using cDNA PCR array. Without any treatment, PC3 showed a much higher expression level of antiapoptosis genes than Lncap in the BCL2 family, the caspase/card family and the TNF ligand/receptor family. At 2 hr after 20 micormolar monensin treatment alone, only the TRAF and CIDE family showed a greater induction in Lncap cells than in PC3. Exposures to 10 Gy X-rays alone of Lncap cells significantly induced gene expression levels in the death and death receptor domain family, the TNF ligand and receptor family, and apoptotic group of BCL2 family; whereas exposures of PC3 induced only the expression of genes in the anti-apoptosis group of CASP and CARD family. Furthermore, we selectively suppressed the expression of several anti-apoptosis genes (BCL-xl, Bcl2A1, BIRC2, BIRC3 and CASP2) in PC3 cells by using the siRNA treatment. Exposure to 10Gy X-rays alone showed an enhanced cell killing (about 15%) in BCL-x1 silenced cells, but not in cells with siRNA treatment targeting other anti-apoptosis genes. We also exposed PC3 cells to protons in the Bragg peak region to compare the effectiveness of cell killing of X-rays. Interestingly, in comparison to X-rays, protons significantly reduced the gene expression in the anti-apoptosis family, suggesting that proton treatment may be more effective for PC3 cells. As a conclusion, monensin was found to sensitize Lncap cells, but not PC3, and over-expression of Bcl-xl cells may be responsible for the radio- or chemo-resistance characteristics of PC3 cells.

Zhang Ye↗
NASA NTRS2006

The Solar Wind and Geomagnetic Activity as a Function of Time Relative to Corotating Interaction Regions

Corotating interaction regions during the declining phase of the solar cycle are the cause of recurrent geomagnetic storms and are responsible for the generation of high fluxes of relativistic electrons. These regions are produced by the collision of a high-speed stream of solar wind with a slow-speed stream. The interface between the two streams is easily identified with plasma and field data from a solar wind monitor upstream of the Earth. The properties of the solar wind and interplanetary magnetic field are systematic functions of time relative to the stream interface. Consequently the coupling of the solar wind to the Earth's magnetosphere produces a predictable sequence of events. Because the streams persist for many solar rotations it should be possible to use terrestrial observations of past magnetic activity to predict future activity. Also the high-speed streams are produced by large unipolar magnetic regions on the Sun so that empirical models can be used to predict the velocity profile of a stream expected at the Earth. In either case knowledge of the statistical properties of the solar wind and geomagnetic activity as a function of time relative to a stream interface provides the basis for medium term forecasting of geomagnetic activity. In this report we use lists of stream interfaces identified in solar wind data during the years 1995 and 2004 to develop probability distribution functions for a variety of different variables as a function of time relative to the interface. The results are presented as temporal profiles of the quartiles of the cumulative probability distributions of these variables. We demonstrate that the storms produced by these interaction regions are generally very weak. Despite this the fluxes of relativistic electrons produced during those storms are the highest seen in the solar cycle. We attribute this to the specific sequence of events produced by the organization of the solar wind relative to the stream interfaces. We also show that there are large quantitative differences in various parameters between the two cycles.

McPherron, Robert L.↗
NASA NTRS2008

Solar-Terrestrial Coupling Evidenced by Periodic Behavior in Geomagnetic Indexes and the Infrared Energy Budget of the Thermosphere

We examine time series of the daily global power (W) radiated by carbon dioxide (at 15 microns) and by nitric oxide (at 5.3 microns) from the Earth s thermosphere between 100 km and 200 km altitude. Also examined is a time series of the daily absorbed solar ultraviolet power in the same altitude region in the wavelength span 0 to 175 nm. The infrared data are derived from the SABER instrument and the solar data are derived from the SEE instrument, both on the NASA TIMED satellite. The time series cover nearly 5 years from 2002 through 2006. The infrared and solar time series exhibit a decrease in radiated and absorbed power consistent with the declining phase of the current 11-year solar cycle. The infrared time series also exhibits high frequency variations that are not evident in the solar power time series. Spectral analysis shows a statistically significant 9-day periodicity in the infrared data but not in the solar data. A very strong 9-day periodicity is also found to exist in the time series of daily A(sub p) and K(sub p) geomagnetic indexes. These 9-day periodicities are linked to the recurrence of coronal holes on the Sun. These results demonstrate a direct coupling between the upper atmosphere of the Sun and the infrared energy budget of the thermosphere.

Mlynczak, Martin G.↗
NASA NTRS2008

On the Relationship Between Solar Wind Speed, Geomagnetic Activity, and the Solar Cycle Using Annual Values

The aa index can be decomposed into two separate components: the leading sporadic component due to solar activity as measured by sunspot number and the residual or recurrent component due to interplanetary disturbances, such as coronal holes. For the interval 1964-2006, a highly statistically important correlation (r = 0.749) is found between annual averages of the aa index and the solar wind speed (especially between the residual component of aa and the solar wind speed, r = 0.865). Because cyclic averages of aa (and the residual component) have trended upward during cycles 11-23, cyclic averages of solar wind speed are inferred to have also trended upward.

Wilson, Robert M.↗
NASA NTRS2008

On the Relationship between Solar Wind Speed, Earthward-Directed Coronal Mass Ejections, Geomagnetic Activity, and the Sunspot Cycle Using 12-Month Moving Averages

For 1996 .2006 (cycle 23), 12-month moving averages of the aa geomagnetic index strongly correlate (r = 0.92) with 12-month moving averages of solar wind speed, and 12-month moving averages of the number of coronal mass ejections (CMEs) (halo and partial halo events) strongly correlate (r = 0.87) with 12-month moving averages of sunspot number. In particular, the minimum (15.8, September/October 1997) and maximum (38.0, August 2003) values of the aa geomagnetic index occur simultaneously with the minimum (376 km/s) and maximum (547 km/s) solar wind speeds, both being strongly correlated with the following recurrent component (due to high-speed streams). The large peak of aa geomagnetic activity in cycle 23, the largest on record, spans the interval late 2002 to mid 2004 and is associated with a decreased number of halo and partial halo CMEs, whereas the smaller secondary peak of early 2005 seems to be associated with a slight rebound in the number of halo and partial halo CMEs. Based on the observed aaM during the declining portion of cycle 23, RM for cycle 24 is predicted to be larger than average, being about 168+/-60 (the 90% prediction interval), whereas based on the expected aam for cycle 24 (greater than or equal to 14.6), RM for cycle 24 should measure greater than or equal to 118+/-30, yielding an overlap of about 128+/-20.

Wilson, Robert M.↗
NASA NTRS2007

OMI Tropospheric NO2 from Lightning in Observed Convective Events

Lightning is responsible for an estimated 10-20% of NO(x) emissions in the troposphere. In this study, we present evidence of lightning-generated NO2 (LNO2) using data from the Ozone Monitoring Instrument (OMI), which has observed tropospheric NO2 since its launch in 2004. Although LNO2 has been also reported in previous satellite studies from the Global Ozone Monitoring Experiment (GOME) and SCIAMACHY, OMI is better suited for such measurements by virtue of its higher resolution and daily global coverage. The LNO2 signal is clearly seen in OMI data on two days over and downwind of convective systems in the US Midwest in 2006. We also present an analysis of OMI data over northern Australia during the SCOUT-O3/ACTIVE field campaigns in November and December 2005. Both single- and multi-day averages are presented to examine possible LNO2 signals from individual diurnally recurrent convective events. In these events we compare the OMI signals with aircraft observations from the storm anvils.

Pickering, Kenneth↗
NASA NTRS2008

An Estimate of North Atlantic Basin Tropical Cyclone Activity for 2008

The statistics of North Atlantic basin tropical cyclones for the interval 1945-2007 are examined and estimates are given for the frequencies of occurrence of the number of tropical cyclones, number of hurricanes, number of major hurricanes, number of category 4/5 hurricanes, and number of U.S. land-falling hurricanes for the 2008 hurricane season. Also examined are the variations of peak wind speed, average peak wind speed per storm, lowest pressure, average lowest pressure per storm, recurrence rate and duration of extreme events (El Nino and La Nina), the variation of 10-yr moving averages of parametric first differences, and the association of decadal averages of frequencies of occurrence of North Atlantic basin tropical cyclones against decadal averages of Armagh Observatory, Northern Ireland, annual mean temperature (found to be extremely important for number of tropical cyclones and number of hurricanes). Because the 2008 hurricane season seems destined to be one that is non-El Nino-related and is a post-1995 season, estimates of the frequencies of occurrence for the various subsets of storms should be above long-term averages.

Wilson, Robert M.↗
NASA NTRS2008

Use of Potassium Citrate to Reduce the Risk of Renal Stone Formation During Spaceflight

Introduction: NASA s Vision for Space Exploration centers on exploration class missions including the goals of returning to the moon and landing on Mars. One of NASA s objectives is to focus research on astronaut health and the development of countermeasures that will protect crewmembers during long duration voyages. Exposure to microgravity affects human physiology and results in changes in the urinary chemical composition favoring urinary supersaturation and an increased risk of stone formation. Nephrolithiasis is a multifactorial disease and development of a renal stone is significantly influenced by both dietary and environmental factors. Previous results from long duration Mir and short duration Shuttle missions have shown decreased urine volume, pH, and citrate levels and increased calcium. Citrate, an important inhibitor of calcium-containing stones, binds with urinary calcium reducing the amount of calcium available to form stones. Citrate inhibits renal stone recurrence by preventing crystal growth, aggregation, and nucleation and is one of the most common therapeutic agents used to prevent stone formation. Methods: Thirty long duration crewmembers (29 male, 1 female) participated in this study. 24-hour urines were collected and dietary monitoring was performed pre, in, and postflight. Crewmembers in the treatment group received two potassium citrate (KCIT) pills, 10 mEq/pill, ingested daily beginning 3 days before launch, all inflight days and through 14 days postflight. Urinary biochemical and dietary analyses were completed. Results: KCIT treated subjects exhibited decreased urinary calcium excretion and maintained the levels of calcium oxalate supersaturation risk at their preflight levels. The increased urinary pH levels in these subjects reduced the risk of uric acid stones. Discussion: The current study investigated the use of potassium citrate as a countermeasure to minimize the risk of stone formation during ISS missions. Results suggest that supplementation with potassium citrate decreases the risk of stone formation during and immediately after spaceflight.

Whitson, P. A.↗
NASA NTRS2009

Changes of Gene Expression in the Apoptosis Pathway in Lncap and PC3 Cells Exposed to X-Rays or Protons

Radio-resistant or recurrent prostate cancer represents a serious health risk for approximately 20%-30% of patients treated with primary radiation therapy for clinically localized prostate cancer. In our current studies, we investigated the expressions of apoptosis related gene expression profile (84 genes) in two distinct prostate cell lines Lncap (P53+ and AR+) and PC3 (P53- and AR-) before and after exposure to X-rays or protons, using cDNA PCR arrays. In Lncap cells, 10Gy X-ray radiation significantly induced the expression of 19 out of 84 genes at 4h after irradiation. The changed genes were mostly in death and death receptor domain families, TNF ligand and receptor families, and apoptotic group of the BCL2 family, especially in P53 related genes, such as FAS, BAX, BAK1 and GADD45A. In PC3, X-rays only induced the expression of 3 genes, including an increased expression of BIRC3. There was no difference of the X-ray mediated cell killing in both cell lines using the cell cycle analysis. However, these X-ray-induced gene expression differences between PC3 and Lncap may explain the phenotype of PC3 cells that shows more tolerant not only to radiation, but also to other apoptosis inducing and sensitizing reagents. To compare the effectiveness of cell killing with X-rays, we also exposed PC3 cells to 10Gy protons at the Bragg peak region. Protons did not induce more apoptosis than X-rays for the same dose. In comparison to X-rays, protons significantly altered expressions of 13 genes in PC3, which included decreased expressions of anti-apoptosis genes (BCL2 and BCL2L2), and increased expressions of death and death receptor domain family genes, TNF ligand and receptor family and several kinases (FAS, DAPK1 and RIPK2). These data suggest that proton treatment is more effective in influencing the apoptosis pathways in PC3 cells than X-rays, thus protons may be more effective in the treatment of specific prostate tumor.

Zhang, Ye↗
NASA NTRS2009

Renal Stone Risk during Spaceflight: Assessment and Countermeasure Validation

NASA's Vision for Space Exploration centers on exploration class missions including the goals of returning to the moon and landing on Mars. One of NASA's objectives is to focus research on astronaut health and the development of countermeasures that will protect crewmembers during long duration voyages. Exposure to microgravity affects human physiology and results in changes in the urinary chemical composition favoring urinary supersaturation and an increased risk of stone formation. Nephrolithiasis is a multifactorial disease and development of a renal stone is significantly influenced by both dietary and environmental factors. Previous results from long duration Mir and short duration Shuttle missions have shown decreased urine volume, pH, and citrate levels and increased calcium. Citrate, an important inhibitor of calcium-containing stones, binds with urinary calcium reducing the amount of calcium available to form stones. Citrate inhibits renal stone recurrence by preventing crystal growth, aggregation, and nucleation and is one of the most common therapeutic agents used to prevent stone formation. Methods: Thirty long duration crewmembers (29 male, 1 female) participated in this study. 24-hour urines were collected and dietary monitoring was performed pre-, in-, and postflight. Crewmembers in the treatment group received two potassium citrate (KCIT) pills, 10 mEq/pill, ingested daily beginning 3 days before launch, all in-flight days and through 14 days postflight. Urinary biochemical and dietary analyses were completed. Results: KCIT treated subjects exhibited decreased urinary calcium excretion and maintained the levels of calcium oxalate supersaturation risk at their preflight levels. The increased urinary pH levels in these subjects reduced the risk of uric acid stones. Discussion: The current study investigated the use of potassium citrate as a countermeasure to minimize the risk of stone formation during ISS missions. Results suggest that supplementation with potassium citrate decreases the risk of stone formation during and immediately after spaceflight.

Whitson, Peggy A.↗
NASA NTRS2008

Broadband X-Ray Spectra of GX 339-4 and the Geometry of Accreting Black Holes in the Hard State

A major question in the study of black hole binaries involves our understanding of the accretion geometry when the sources are in the "hard" state. In this state, the X-ray energy spectrum is dominated by a hard power-law component and radio observations indicate the presence of a steady and powerful "compact" jet. Although the common hard state picture is that the accretion disk is truncated, perhaps at hundreds of gravitational radii (R(sub g)) from the black hole, recent results for the recurrent transient GX 339-4 by Miller and co-workers show evidence for optically thick material very close to the black hole's innermost stable circular orbit. That work focused on an observation of GX 339-4 at a luminosity of about 5% of the Eddington limit (L(sub Edd)) and used parameters from a relativistic reflection model and the presence of a soft, thermal component as diagnostics. In this work, we use similar diagnostics, but extend the study to lower luminosities (2.3% and 0.8% L(sub Edd)) using Swift and RXTE observations of GX 339-4. We detect a thermal component with an inner disk temperature of approx.0.2 keV at 2.3% L(sub Edd). At 0.8% L(sub Edd), the spectrum is consistent with the presence of such a component, but the component is not required with high confidence. At both luminosities, we detect broad features due to iron Ka that are likely related to reflection of hard X-rays off the optically thick material. If these features are broadened by relativistic effects, they indicate that optically thick material resides within 10 R(sub g) down to 0.8% L(sub Edd), and the measurements are consistent with the inner radius of the disk remaining at approx.4 R(sub g) down to this level. However, we also discuss an alternative model for the broadening, and we note that the evolution of the thermal component is not entirely consistent with the constant inner radius interpretation. Finally, we discuss the results in terms of recent theoretical work by Liu and co-workers on the possibility that material may condense out of an Advection-Dominated Accretion Flow to maintain an inner optically thick disk.

Tomsick↗
NASA NTRS2009

Perl Modules for Constructing Iterators

The Iterator Perl Module provides a general-purpose framework for constructing iterator objects within Perl, and a standard API for interacting with those objects. Iterators are an object-oriented design pattern where a description of a series of values is used in a constructor. Subsequent queries can request values in that series. These Perl modules build on the standard Iterator framework and provide iterators for some other types of values. Iterator::DateTime constructs iterators from DateTime objects or Date::Parse descriptions and ICal/RFC 2445 style re-currence descriptions. It supports a variety of input parameters, including a start to the sequence, an end to the sequence, an Ical/RFC 2445 recurrence describing the frequency of the values in the series, and a format description that can refine the presentation manner of the DateTime. Iterator::String constructs iterators from string representations. This module is useful in contexts where the API consists of supplying a string and getting back an iterator where the specific iteration desired is opaque to the caller. It is of particular value to the Iterator::Hash module which provides nested iterations. Iterator::Hash constructs iterators from Perl hashes that can include multiple iterators. The constructed iterators will return all the permutations of the iterations of the hash by nested iteration of embedded iterators. A hash simply includes a set of keys mapped to values. It is a very common data structure used throughout Perl programming. The Iterator:: Hash module allows a hash to include strings defining iterators (parsed and dispatched with Iterator::String) that are used to construct an overall series of hash values.

Tilmes, Curt↗
NASA NTRS2009

Shuttle Ground Support Equipment (GSE) T-0 Umbilical to Space Shuttle Program (SSP) Flight Elements Consultation

The NASA Engineering and Safety Center (NESC) was tasked with assessing the validity of an alternate opinion that surfaced during the investigation of recurrent failures at the Space Shuttle T-0 umbilical interface. The most visible problem occurred during the Space Transportation System (STS)-112 launch when pyrotechnics used to separate Solid Rocket Booster (SRB) Hold-Down Post (HDP) frangible nuts failed to fire. Subsequent investigations recommended several improvements to the Ground Support Equipment (GSE) and processing changes were implemented, including replacement of ground-half cables and connectors between flights, along with wiring modifications to make critical circuits quad-redundant across the interface. The alternate opinions maintained that insufficient data existed to exonerate the design, that additional data needed to be gathered under launch conditions, and that the interface should be further modified to ensure additional margin existed to preclude failure. The results of the assessment are contained in this report.

Wilson, Timmy R.↗
NASA NTRS2008

Broadband X-Ray Spectra of GX 339-4 and the Geometry of Accreting Black Holes in the Hard State

A major question in the study of black hole binaries involves our understanding of the accretion geometry when the sources are in the "hard" state, with an X-ray energy spectrum dominated by a hard power-law component and radio emission coming from a steady "compact" jet. Although the common hard state picture is that the accretion disk is truncated, perhaps at hundreds of gravitational radii (Rg) from the black hole, recent results for the recurrent transient GX 339-4 by Miller and coworkers show evidence for disk material very close to the black hole's innermost stable circular orbit. That work studied GX 339-4 at a luminosity of approximately 5% of the Eddington limit (L(sub Edd) and used parameters from a relativistic reflection model and the presence of a thermal component as diagnostics. Here we use similar diagnostics but extend the study to lower luminosities (2.3% and 0.8% L(sub Edd)) using Swift and RXTE observations of GX 339-4. We detect a thermal component with an inner disk temperature of approximately 0.2 keV at 2.3% L (sub Edd). At both luminosities, we detect broad features due to iron K-alpha that are likely related to reflection of hard X-rays off disk material. If these features are broadened by relativistic effects, they indicate that the material resides within 10 Rg, and the measurements are consistent with the disk's inner radius remaining at approximately 4 Rg down to 0.8% L(sub Edd). However, we also discuss an alternative model for the broadening, and we note that the evolution of the thermal component is not entirely consistent with the constant inner radius interpretation. Finally, we discuss the results in terms of recent theoretical work by Liu and co-workers on the possibility that material may condense out of an Advection-Dominated Accretion Flow to maintain an inner optically thick disk.

Tomsick, John A.↗
NASA NTRS2010

An Examination of the Hadley Sea-Surface Temperature Time Series for the Nino 3.4 Region

The Hadley sea-surface temperature (HadSST) dataset is investigated for the interval 1871-2008. The purpose of this investigation is to determine the degree of success in identifying and characterizing El Nino (EN) southern (ENSO) extreme events, both EN and La Nina (LN) events. Comparisons are made against both the Southern Oscillation Index for the same time interval and with published values of the Oceanic Nino Index for the interval since 1950. Some 60 ENSO extreme events are identified in the HadSST dataset, consisting of 33 EN and 27 LN events. Also, preferential associations are found to exist between the duration of ENSO extreme events and their maximum anomalous excursion temperatures and between the recurrence rate for an EN event and the duration of the last known EN event. Because the present ongoing EN is a strong event, it should persist 11 months or longer, inferring that the next EN event should not be expected until June 2012 or later. Furthermore, the decadal sum of EN-related months is found to have increased somewhat steadily since the decade of 1920-1929, suggesting that the present decade (2010-2019) possibly will see about 3-4 EN events, totaling about 37 +/- 3 EN-related months (i.e., months that meet the definition for the occurrence of an EN event).

Wilson, Robert M.↗
NASA NTRS2010

Recent Results on SNRs and PWNe from the Fermi Large Area Telescope

a) Symbiotic Binary System: White dwarf + red giant system. b) Nova: White dwarf builds up mass envelope to the point of thermonuclear fusion. c) Dramatic increase in visual magnitude. d) Recurrent Nova? e) Hints but no strong confirmation of previous nova f) Pre-nova activity: 1) White dwarf shows ongoing variability at level of several in magnitude. 2) V407 Cyg companion is a Mira star showing variability at level of several in magnitude. g) Origin of the gamma rays? 1) Strong shock propagating into dense medium around giant star land stellar wind. 2) Pion decay or electron processes?

Hays, Elizabeth↗
NASA NTRS2010

Comparative Spectra of Oxygen-Rich Versus Carbon-Rich Circumstellar Shells: VY Canis Majoris and IRC(plus)10216 at 215-285 GHz

A sensitive (1sigma rms at 1 MHz resolution approx.3 mK) 1 mm spectral line survey (214.5-285.5 GHz) of VY Canis Majoris (VY CMa) and IRC +10216 has been conducted to compare the chemistries of oxygen- and carbon-rich circumstellar envelopes. This study was carried out using the Submillimeter Telescope of the Arizona Radio Observatory with a new Atacama Large Millimeter Array type receiver. This survey is the first to chemically characterize an O-rich circumstellar shell at millimeter wavelengths. In VY CMa, 128 emission features were detected arising from 18 different molecules; and in IRC +10216, 720 lines were observed, assigned to 32 different species. The 1 mm spectrum of VY CMa is dominated by SO, and SiS; in IRC +10216, C4H and SiC2 are the most recurrent species. Ten molecules were common to both sources: CO, SiS, SiO, CS, CN, HCN, HNC, NaCl, PN, and HCO(+). Sulfur plays an important role in VY CMa, but saturated/ unsaturated carbon dominates the molecular content of IRC +102.16, producing CH2NH, for example. Although the molecular complexity of IRC +10216 is greater, VY CMa supports a unique "inorganic" chemistry leading to the oxides PO, AlO, and AlOH. Only diatomic and triatomic compounds were observed in VY CMa, while species with four or more atoms are common in IRC +10216, reflecting carbon's ability to form multiple strong bonds, unlike oxygen. In VY CMa, a new water maser (v2 = 2) has been found, as well as vibrationally excited NaCl. Toward IRC +10216, vibrationally excited CCH was detected for the first time.

Tenebaum, E. D.↗
NASA NTRS2010

Evaluating Descent and Ascent Trajectories Near Non-Spherical Bodies

Spacecraft landing on small bodies pass through regions where conventional gravitation formulations using exterior spherical harmonics are inaccurate. An investigation shows that a formulation using interior solid spherical harmonics might be satisfactory. Interior spherical harmonic expansions are usable inside an imaginary, empty sphere. For this application, such a sphere could be positioned in empty space above the intended landing site and rotating with the body. When the spacecraft is inside this sphere, the interior harmonic expansion would be used instead of the conventional, exterior harmonic expansion. Coefficients can be determined by a least-squares fit to gravitation measurements synthesized from conventional formulations. Due to their unfamiliarity, recurrences for interior, as well as exterior, expansions are derived. Hotine's technique for partial derivatives of exterior spherical harmonics is extended to interior harmonics.

Werner, Robert A.↗