Observation of Mercury's sodium exosphere during the transit on November 9, 2006

A rare, but normal, astronomical event occurred on November 9th 2006 (JST) as Mercury passed in front of the Sun from the perspective of the Earth. The abundance of the sodium vapor above the planet limb was observed by detecting an excess absorption in the solar sodium line D₁ during this event. The observation was performed with a 10-m spectrograph of Czerny-Turnar system at Domeless Solar Tower Telescope at the Hida Observatory in Japan. The excess absorption was red-shifted by 10 pm relative to the solar line, and was measured at the dawnside (eastside) and duskside (westside) of Mercury. Between the dawn and dusksides, an asymmetry of total sodium abundance was clearly identified. At the dawnside, the total sodium column density was 6.1×10¹⁰ Na atoms/cm², while it was 4.1×10¹⁰ Na atoms/cm² at the duskside. The investigation of dawn-dusk asymmetry of the sodium exosphere of Mercury is a clue to understand the release mechanism of sodium from the surface rock. Our result suggests that a thermal desorption is a main source process for sodium vapor in the vicinity of Mercury.     

Solar radiation acceleration effects on Mercury sodium emission

A set of Mercury sodium emission data collected over a range of true anomaly angles during 1997-2003 was used to analyze the effect of solar radiation acceleration on sodium emissions. The variation of emission intensity with changing Doppler velocities throughout the orbit was minimized by normalizing the intensities to a constant true anomaly angle. The normalized intensities should be independent of orbital position if sodium density is constant. Plots of the normalized intensities against solar radiation acceleration showed very considerable scatter. However, the scatter was not random, but the result of a systematic variation, such that the normalized emission at a particular value of radiation acceleration took one or the other of two values, depending on the value of the true anomaly angle. We propose that this was the result of solar radiation acceleration changing the velocity of the sodium atoms, and consequently changing the solar continuum seen by the atoms. There is a positive feedback loop in the “out” leg of the orbit, such that radiation acceleration increases the solar continuum intensity seen by the atoms, and a negative feedback loop in the “in” leg of the orbit, such radiation acceleration decreases the continuum intensity. The observations could be approximately fit by assuming that sodium atoms are exposed to sunlight for an average of 1700 s. The emission values corrected for this effect showed much less scatter, with a general trend of about 30% to lower values from minimum to maximum radiation acceleration. The corrected emissions were used to calculate average column densities, and the result compared with the predictions of Smyth and Marconi [Smyth, W.H., Marconi, M.L., 1995. Astrophys. J. 441, 839-864] for the variation of column density with true anomaly angle. The comparison suggests that sodium atoms interact weakly with the surface. The effect of radiation acceleration on emission intensities should be taken into account if column densities are to be calculated from emission intensities.     

Orbital effects on Mercury’s escaping sodium exosphere

We present results from coronagraphic imaging of Mercury’s sodium tail over a 7° field of view. Several sets of observations made at the McDonald Observatory since May 2007 show a tail of neutral sodium atoms stretching more than 1000 Mercury radii (R_m) in length, or a full degree of sky. However, no tail was observed extending beyond 120 R_m during the January 2008 MESSENGER fly-by period, or during a similar orbital phase of Mercury in July 2008. Large changes in Mercury’s heliocentric radial velocity cause Doppler shifts about the Fraunhofer absorption features; the resultant change in solar flux and radiation pressure is the primary cause of the observed variation in tail brightness. Smaller fluctuations in brightness may exist due to changing source rates at the surface, but we have no explicit evidence for such changes in this data set. The effects of radiation pressure on Mercury’s escaping atmosphere are investigated using seven observations spanning different orbital phases. Total escape rates of atmospheric sodium are estimated to be between 5 and 13 × 10²³ atoms/s and show a correlation to radiation pressure. Candidate sources of Mercury’s sodium exosphere include desorption by UV sunlight, thermal desorption, solar wind channeled along Mercury’s magnetic field lines, and micro-meteor impacts. Wide-angle observations of the full extent of Mercury’s sodium tail offer opportunities to enhance our understanding of the time histories of these source rates.     

Exospheric perturbations by radiation pressure: II. Solution for orbits in the ecliptic plane

An earlier paper gave solutions for the mean time rates of change of orbital elements of satellite atoms in an exosphere influenced by solar radiation pressure. Each element was assumet to beahve independently. Here the instantaneous rates of change for three elements $\text{(e, ω,}\text{and}\text{θ = ω + Ω)}$ are integrated simultaneously for the case of the inclination i = 0. The results (a) confirm the validity of using mean rates when the orbits are tightly bound to the planet and (b) serve as examples to be reproduced by the complicated numerical solutions required for arbitrary inclination. Strongly bound hydrogen atoms perturbed in Earth orbit by radiation pressure do not seem a likely cause of the geotail extending in the anti-Sun direction. Instead, radiation pressure wil cause those particles' orbits to form a broad fan-shaped tail and to deteriorate into the Earth's atmosphere. Whether loosely bound H atoms are plentiful enough to create the geotail depends on their source function versusr; that question is beyond the scope of this paper.     

Source rates and ion recycling rates for Na and K in Mercury's atmosphere

The supply rates of Na and K to the atmosphere of Mercury by processes acting on the extreme surface—thermal vaporization, photon-stimulated desorption (PSD), and ion-sputtering—are limited by the rates at which atoms can be supplied to the extreme surface by diffusion from inside the regolith grains. Supply rates to the atmosphere are further regulated by ion retention and by gardening rates that supply new grains to the surface. We consider the limits on supply of sodium and potassium atoms to the atmosphere, and rates of photoion recycling to the surface. Thermal vaporization rates are severely limited by the ability of atoms to diffuse to the surface of the grain. Therefore, the diffusion-limited thermal vaporization rates on Mercury's surface are comparable to or less than the PSD rates. Ion sputtering is primarily due to highly ionized heavy ions, even though they represent a small fraction of the solar wind. We have shown that up to 60% of the Na photoions are deposited on the surface of Mercury. Ion recycling to the surface can have a long-term effect on the regolith abundance if an average recycling pattern persists such that more ions return to a particular area than are launched there. It is unknown whether the formation of latitude bands of >100% ion retention persist on average despite a rapidly changing magnetosphere. The total exospheric column of sodium observed at Mercury between 1997 to 2003 varied by a factor of 2-3 from perihelion to aphelion.     

The sodium tail of the Moon

During the few days centered about new Moon, the lunar surface is optically hidden from Earth-based observers. However, the Moon still offers an observable: an extended sodium tail. The lunar sodium tail is the escaping “hot” component of a coma-like exosphere of sodium generated by photon-stimulated desorption, solar wind sputtering and meteoroid impact. Neutral sodium atoms escaping lunar gravity experience solar radiation pressure that drives them into the anti-solar direction forming a comet-like tail. During new Moon time, the geometry of the Sun, Moon and Earth is such that the anti-sunward sodium flux is perturbed by the terrestrial gravitational field resulting in its focusing into a dense core that extends beyond the Earth. An all-sky camera situated at the El Leoncito Observatory (CASLEO) in Argentina has been successfully imaging this tail through a sodium filter at each lunation since April 2006. This paper reports on the results of the brightness of the lunar sodium tail spanning 31 lunations between April 2006 and September 2008. Brightness variability trends are compared with both sporadic and shower meteor activity, solar wind proton energy flux and solar near ultra violet (NUV) patterns for possible correlations. Results suggest minimal variability in the brightness of the observed lunar sodium tail, generally uncorrelated with any single source, yet consistent with a multi-year period of minimal solar activity and non-intense meteoric fluxes.     

The sodium tail of Mercury

Abstract— Mercury is difficult to observe because it is so close to the Sun. However, when the angle of the ecliptic is near maximum in the northern hemisphere, and Mercury is near its greatest eastern elongation, it can be seen against the western sky for about a half hour after sunset. During these times, we were able to map sodium D₂ emission streaming from the planet, forming a long comet‐like tail. On 2001 May 26 (U.T.) we mapped the tail downstream to a distance of ?40 000 km. Sodium velocities in the tail increased to ?11 km s^?1 at 40 000 km as the result of radiation pressure acceleration. On 2000 June 5 (U.T.) we mapped the cross‐sectional extent of the tail at a distance of ?17 500 km downstream. At this distance, the half‐power full‐width of the emission was ?20 000 km. We estimated the transverse velocity of sodium in the tail to range from 2 to 4 km s^?1. The velocities we observed imply source velocities from the planet surface of the order of 5 km s^?1, or 4 eV. Particle sputtering is a likely candidate for production of sodium atoms at these velocities. The total flux of sodium in the tail was ?1 times 10²³ atoms s^?1, which corresponds to 1 to 10% of the estimated total production rate of sodium on the planet.     

Mercury's sodium exosphere

Mercury's neutral sodium exosphere is simulated using a comprehensive 3D Monte Carlo model following sodium atoms ejected from Mercury's surface by thermal desorption, photon stimulated desorption, micro-meteoroid vaporization and solar wind sputtering. The evolution of the sodium surface density with respect to Mercury's rotation and its motion around the Sun is taken into account by considering enrichment processes due to surface trapping of neutrals and ions and depletion of the sodium available for ejection from the surfaces of grains. The change in the sodium exosphere is calculated during one Mercury year taking into account the variations in the solar radiation pressure, the photo-ionization frequency, the solar wind density, the photon and meteoroid flux intensities, and the surface temperature. Line-of-sight column densities at different phase angles, the supply rate of new sodium, average neutral and ion losses over a Mercury year, surface density distribution and the importance of the different processes of ejection are discussed in this paper. The sodium surface density distribution is found to become significantly nonuniform from day to night sides, from low to high latitudes and from morning to afternoon because of rapid depletion of sodium atoms in the surfaces of grains mainly driven by thermal depletion. The shape of the exosphere, as it would be seen from the Earth, changes drastically with respect to Mercury's heliocentric position. High latitude column density maxima are related to maxima in the sodium surface concentration at high latitudes in Mercury's surface and are not necessarily due to solar wind sputtering. The ratio between the sodium column density on the morning side of Mercury's exosphere and the sodium column density on the afternoon side is consistent with the conclusions of Sprague et al. (1997, Icarus 129, 506-527). The model, which has no fitting parameters, shows surprisingly good agreement with recent observations of Potter et al. (2002, Meteor. Planet. Sci. 8, 3357-3374) successfully explaining their velocity and column density profiles vs. heliocentric distance. Comparison with this data allows us to constrain the supply rate of new sodium atoms to the surface. We also discuss the possible origins of the strong high latitude emissions (Potter and Morgan, 1990, Science 248, 835-838; 1997a, Adv. Space Res. 19, 1571-1576; 1997b, Planet. Space Sci. 45, 95-100; Sprague et al., 1998, Icarus 135, 60-68) and the strong variations of the total content of the sodium exosphere on short (Potter et al., 1999, Planet. Space Sci. 47, 1441-1449) and long time scales (Sprague et al., 1997, Icarus 129, 506-527).     

A general model of the planetary radiation pressure on a satellite with a complex shape

The purpose of this paper is to present a general model for the acceleration exerted on a spacecraft by the radiation coming from a planet. Both the solar radiation reflected by the planet and the thermal emission associated with its temperature are considered. The planet albedo and the planet emissive power are expanded in spherical harmonics with respect to an equatorial reference frame attached to the planet. The satellite external surface is assumed to consist of a juxtaposition of planar surfaces. A particular choice of variables allows to reduce the surface integrals over the lit portion of the planet visible to the satellite to one-dimension integrals.     

Practical SPH models for major planets

Modelling planets is done for two main reasons – the first to further understanding of their internal structure and the second to provide models to explore astrophysical situations in which planets play a role. For the latter reason, the requirements on accuracy are less severe, although the planet must be realistic in its major features. A numerical model of a layered giant planet is developed with an iron core, a silicate mantle, an ice region and a hydrogen–helium atmosphere. The Tillotson equation of state is used and examples of two model planets are given, one reproducing the mass and radius of Jupiter quite closely and the other with two Jupiter masses. Transferring these results into a smoothed particle hydrodynamics (SPH) model presents two main difficulties. A uniform distribution of SPH points leads to too few points representing the non-atmospheric component. It is shown that using a distorted lattice enables the core + silicate + ice to be represented by several hundred points so that the evolution of these regions can be followed in detail. Another difficulty concerns the density discontinuities attendant on a layered structure. Density estimates of SPH points are either too large or too small near material interfaces leading to unrealistic pressure gradients and, consequently, to large and unphysical local forces. Algorithms are described for avoiding this difficulty both at material interfaces and near the surface of the planet. In some astrophysical situations involving SPH-modelled planets, the main bulk of the planet is so opaque that internal heat transfer can be neglected. However, surface regions should radiate and a convenient way for including radiation from a planetary surface is described.     

Low energy neutral atoms imaging of the Moon

Imaging of low-energy neutral atoms (LENAs) in the vicinity of the Moon can provide wide knowledge of the Moon from the viewpoint of plasma physics and planetary physics. At the surface of the Moon, neutral atoms are mainly generated by photon-stimulated desorption, micrometeorite vaporization and sputtering by solar wind protons. LENAs, the energetic neutral atoms with energy range of 10-500 eV, are mainly created by sputtering of solar wind particles. We have made quantitative estimates of sputtered LENAs from the Moon surface. The results indicate that LENAs can be detected by a realistic instrument and that the measurement will provide the global element maps of sputtered particles, which substantially reflect the surface composition, and the magnetic anomalies. We have also found that LENAs around dark regions, such as the permanent shadow inside craters in the pole region, can be imaged. This is because the solar wind ions can penetrate shaded regions due to their finite gyro-radius and the pressure gradient between the solar wind and the wake region. LENAs also extend our knowledge about the magnetic anomalies and associated mini-magnetosphere systems, which are the smallest magnetospheres as far as one knows. It is thought that no LENAs are generated from mini-magnetosphere regions because no solar wind may penetrate inside them. Imaging such void areas of LENAs will provide another map of lunar magnetic anomalies.     

The effects of reduced land fraction and solar forcing on the general circulation: results from the NCAR CCM

Land fraction and the solar energy at the top of the atmosphere (solar constant) may have been significantly lower early in Earth's history. It is likely that both of these factors played some important role in the climate of the early earth. The climate changes associated with a global ocean(i.e. no continents) and reduced solar constant are examined with a general circulation model and compared with the present-day climate simulation. The general circulation model used in the study is the NCAR CCM with a swamp ocean surface. First, all land points are removed in the model and then the solar constant is reduced by 10% for this global ocean case.Results indicate that a 4 K increase in air temperature occurs with global ocean simulation compared to the control. When solar constant is reduced by 10% under global ocean conditions a 23 K decrease in air temperature is noted. The global ocean warms much of the troposphere and stratosphere, while a reduction in the solar constant cools the troposphere and stratosphere. The largest cooling occurs near the surface with the lower solar constant.Global mean values of evaporation, water vapor amounts, absorbed solar radiation and the downward longwave radiation are increased under global ocean conditions, while all are reduced when the solar constant is lowered. The global ocean simulation produces sea ice only in the highest latitudes. A frozen planet does not occur when the solar constant is reduced—rather, the ice line settles near 30° of latitude. It is near this latitude that transient eddies transport large amounts of sensible heat across the ice line acting as a negative feedback under lower solar constant conditions keeping sea ice from migrating to even lower latitudes.Clouds, under lower solar forcing, also act as a negative feedback because they are reduced in higher latitudes with colder atmospheric temperatures allowing additional solar radiation to reach the surface. The overall effect of clouds in the global ocean is to act as a positive feedback because they are slightly reduced thereby allowing additional solar radiation to reach the surface and increase the warming caused by the removal of land. The relevance of the results to the “Faint-Young Sun Paradox” indicates that reduced land fraction and solar forcing affect dynamics, heat transport, and clouds. Therefore the associated feedbacks should be taken into account in order to understand their roles in resolving the “Faint-Young Sun Paradox”.     

On the optical studies of the atmospheric water vapour from the surface of Mars

Remote observations of the atmospheric water vapour from the Mars orbit were usually carried out to study its global distribution and variability. Measurements of the water vapour abundance onboard the landers have recently become an important complement to the orbital sounding. Narrow-band filter photometry and spectroscopy of the solar radiation from the surface of the planet proved to be a powerful tool in the study of atmospheric water. The Imager for Mars Pathfinder (IMP) was the first instrument to measure its amount from the surface. The Surface Stereo Imager (SSI) onboard the Mars Polar Lander (MPL) was to follow but the spacecraft was lost at landing. Nevertheless significant expertise in the optical measurements of atmospheric H₂O was gained during these missions. This paper summarizes this experience emphasizing the radiative transfer aspects of the problem. The results of this study could be of importance for future missions to Mars.     

Loss of water from Mars:: Implications for the oxidation of the soil

The evolution of the martian atmosphere with regard to its H₂O inventory is influenced by thermal loss processes of H, H₂, nonthermal atmospheric loss processes of H⁺, H₂⁺, O, O⁺, CO₂, and O₂⁺ into space, as well as by chemical weathering of the surface soil. The evolution of thermal and nonthermal escape processes depend on the history of the intensity of the solar XUV radiation and the solar wind density. Thus, we use actual data from the observation of solar proxies with different ages from the Sun in Time program for reconstructing the Sun's radiation and particle environment from the present to 3.5 Gyr ago. The correlation between mass loss and X-ray surface flux of solar proxies follows a power law relationship, which indicates a solar wind density up to 1000 times higher at the beginning of the Sun's main sequence lifetime. For the study of various atmospheric escape processes we used a gas dynamic test particle model for the estimation of the pick up ion loss rates and considered pick up ion sputtering, as well as dissociative recombination. The loss of H₂O from Mars over the last 3.5 Gyr was estimated to be equivalent to a global martian H₂O ocean with a depth of about 12 m, which is smaller than the values reported by previous studies. If ion momentum transport, a process studied in detail by Mars Express is significant on Mars, the water loss may be enhanced by a factor of about 2. In our investigation we found that the sum of thermal and nonthermal atmospheric loss rates of H and all nonthermal escape processes of O to space are not compatible with a ratio of 2:1, and is currently close to about 20:1. Escape to space cannot therefore be the only sink for oxygen on Mars. Our results suggest that the missing oxygen (needed for the validation of the 2:1 ratio between H and O) can be explained by the incorporation into the martian surface by chemical weathering processes since the onset of intense oxidation about 2 Gyr ago. Based on the evolution of the atmosphere-surface-interaction on Mars, an overall global surface sink of about 2×10⁴² oxygen particles in the regolith can be expected. Because of the intense oxidation of inorganic matter, this process may have led to the formation of considerable amounts of sulfates and ferric oxides on Mars. To model this effect we consider several factors: (1) the amount of incorporated oxygen, (2) the inorganic composition of the martian soil and (3) meteoritic gardening. We show that the oxygen incorporation has also implications for the oxidant extinction depth, which is an important parameter to determine required sampling depths on Mars aimed at finding putative organic material. We found that the oxidant extinction depth is expected to lie in a range between 2 and 5 m for global mean values.     

Magnetic states of the ionosphere of Venus observed by Venus Express

Strong ultraviolet radiation from the Sun ionizes the upper atmosphere of Venus, creating a dense ionosphere on the dayside of the planet. In contrast to Earth, the ionosphere of Venus is not protected against the solar wind by a magnetic field. However, the interaction between charged ionospheric particles and the solar wind dynamic and magnetic pressure creates a pseudo-magnetosphere which deflects the solar wind flow around the planet (Schunk and Nagy, 1980). The combination of changing solar radiation and solar wind intensities leads to a highly variable structure and plasma composition of the ionosphere. The instrumentation of the Venus Express spacecraft allows to measure the magnetic field (MAG experiment) as well as the electron energy spectrum and the ion composition (ASPERA-4 experiment) of the upper ionosphere and ionopause. In contrast to the earlier Pioneer Venus Orbiter (PVO) measurements which were conducted during solar maximum, the solar activity was very low in the period 2006-2009. A comparison with PVO allows for an investigation of ionospheric properties under different solar wind and EUV radiation conditions. Observations of MAG and ASPERA have been analyzed to determine the positions of the photoelectron boundary (PEB) and the “magnetopause” and their dependence on the solar zenith angle (SZA). The PEB was determined using the ELS observations of ionospheric photoelectrons, which can be identified by their specific energy range. It is of particular interest to explore the different magnetic states of the ionosphere, since these influence the local plasma conductivity, currents and probably the escape of electrons and ions. The penetration of magnetic fields into the ionosphere depends on the external conditions as well as on the ionospheric properties. By analyzing a large number of orbits, using a combination of two different methods, we define criteria to distinguish between the so-called magnetized and unmagnetized ionospheric states. Furthermore, we confirm that the average magnetic field inside the ionosphere shows a linear dependence on the magnetic field in the region directly above the PEB.     

The solar radiation incident at the top of the atmospheres of Uranus and Neptune

The latitudinal and seasonal variation of the direct solar radiation incident at the top of the atmosphere of Uranus and Neptune has been recalculated by use of updated values for the period of axial rotation and the oblateness. Values for the solar radiation are given in Watt per square meter instead of the unit used in earlier papers (calories per square centimeter per planetary day). The solar radiation averaged over a season and a year as a function of planetocentric latitude has also been reviewed. In addition, attention is made to the ratio of the solar radiation incident on an oblate planet to that incident on a spherical planet.     

Bicentennial decrease of the solar constant leads to the Earth’s unbalanced heat budget and deep climate cooling

Long-wave energy emitted by the Earth-atmosphere into space is characterized by changes in power over time that always lag behind the changes in power of the absorbed solar radiation due to slow variation in enthalpy of the Earth-atmosphere system. Long-term variation of the solar energy radiation absorbed by the Earth remains uncompensated by the energy radiated into space over the interval of time that is determined by the thermal inertia. The basic state of the climate system is when the debit and credit sides in the Earth’s global annual mean energy budget (including the air and water envelopes) are almost always unbalanced. The annual mean balance of the heat budget of the Earth-atmosphere over a long time period will reliably define the behavior and magnitude of the energy excess accumulated by the Earth or energy deficit to allow us to determine adequately and to predict beforehand the trend and amplitude of the forthcoming climate change using the prognosis of variations in the total solar irradiance (solar constant). The decrease in solar constant has been observed since the early 1990s. The Earth as a planet will have a negative balance in the energy budget in the future as well, because the Sun is entering the decline phase of the bicentennial luminosity changes. This will lead to a drop in temperature in approximately 2014. The increase in albedo and decrease in greenhouse gas concentration in the atmosphere will result in the additional decrease in absorbed portion of the solar energy and reduced greenhouse effect. The additional drop in temperature exceeding the effect of decreased solar constant can occur as a result of successive feedback effects. A deep bicentennial minimum in solar constant is to be anticipated in 2042 ± 11 and the 19th Little Ice Age (for the last 7500 years) may occur in 2055 ± 11.     

Source dependency of exospheric sodium on Mercury

Due to a large solar radiation effect, the sodium exosphere exhibits many interesting effects, including the formation of an extended corona and a tail-like structure. The current suite of observations allows us to study some physical properties of the sodium exosphere, such as the source rates and the interaction with the surface, both experimentally and theoretically. In order to quantify the complex variations in the sodium exosphere in more detail, we use an exospheric model with the Monte-Carlo method to examine the surface interactions of a sodium atom, including the surface thermal accommodation rate and the sticking coefficient. The source rates from different components, such as the photon stimulated desorption (PSD), the meteoroid impact vaporization (MIV), and the solar wind ion sputtering (IS), can be constrained by comparing our exospheric model calculations with the published observational data. The detected terminator to limb (TL) ratio on the disk and the tail production rate can be explained with no sticking effect and small thermal accommodation rates. We also examine the best fit of the MIV source evolution, through comparison with the disk-averaged emission. The resultant discrepancy between the observations and the model fit may reflect the surface variation in the sodium abundance. A comprehensive mapping of the surface geochemical composition of the surface by the MESSENGER and Bepi-Colombo missions should give us more information about the nature of this surface-bound exosphere.     

Synchrotron and thermal X-ray emission from supernova remnants. Low radiation losses of electrons

A model is proposed for the nonthermal synchrotron emission from supernova remnants in the uniform interstellar medium. Some characteristics of nonthermal and thermal emission (luminosity and surface brightness distribution) are compared. The conditions when the nonthermal component can be prominent in the X-ray spectrum are specified. We point out some observational tests which will allow a number of parameters characterizing the cosmic ray injection on supernova remnant shocks to be estimated. The cases when electron radiation losses may be neglected are considered.     

A new treatment of the albedo radiation pressure in the case of a uniform albedo and of a spherical satellite

The purpose of this paper is to present a model for the radiation pressure acceleration of a spherical satellite, due to the radiation reflected by a planet with a uniform albedo. A particular choice of variables allows one to reduce the surface integrals over the lit portion of the planet visible to the satellite to one-dimensional integrals. Exact analytical expressions are found for the integrals corresponding to the case where the spacecraft does not "see" the terminator. The other integrals can be computed either numerically, or analytically in an approximate form. The results are compared with those of Lochry (1966). The model is applied to Magellan, a spacecraft orbiting Venus.