The steady-state temperature across a plate placed on the surface of a planet is directly proportional to the heat coming out of the planet. With a judicious choice of plate thermal conductivity and surface radiative properties, the temperature across the plate can be maximized for the environmental conditions, and therefore the systematic errors minimized.
Here, I solve the 1-D heat equation to obtain the steady state and transient solutions. The lower boundary condition is assumed to be in perfect contact with the surface, which is at temperature Ts. The upper boundary condition is convective and radiative heat flow due to the temperature difference between the plate and the Venus environment, which is also at temperature Ts.
The temperature field is u(z,t) where the surface of the planet is at z=0. The depth of the plate is a, and the thermal diffusivity is k. The convection coefficient at the top of the plate is h, and the quantity I am solving for is Fg, the geothermal heat flux. The heat equation and boundary conditions are therefore:
Sunday, August 21, 2011
Wednesday, August 17, 2011
Dr. Strangelove at Venus
Introduction
Imagine that you are on a wild ride, clinging to a spacecraft as it plunges down to the surface of Venus. Major scientific advances in understanding the geology of Venus are possible with surface imaging on descent through the atmosphere. The dense atmosphere provides a leisurely descent speed, valuable for allowing downlinking of images before arrival at the surface. Images from a descent probe to Venus would provide crucial information at a scale not yet explored – the 10 m to 10’s of km range. Equally important, these images would be taken at visible and near-infrared wavelengths. Comparison with Magellan radar data in making new, detailed geologic interpretations based on optical wavelengths could be extrapolated to similar regions across the planet.
Here I discuss the unique optical and spacecraft dynamics issues in imaging the surface of Venus from a descent probe, and the scientific advances that are possible with such an experiment. Cameras must have a large dynamic range, high quantum efficiency at 1 mm, and be capable of millisecond exposure times. Nighttime images would have the highest surface contrast, but independent altimetry would be needed to account for variations in flux due to altitude-dependant surface temperature. At night, near-IR images with about the resolution of Magellan SAR images can be obtained from just beneath the clouds. Images on descent during the day would see a very bright sky but their interpretation would be less dependent upon altimetry. Images with enough contrast for scientific investigations could be obtained with CCD cameras below approximately 15 km.
Atmospheric Properties
Below the clouds, the primary challenges to surface imaging are very broad CO2 and H2O absorption bands longward of 0.8 mm, and Rayleigh scattering (Moroz, 2002). Below-cloud hazes may at times also degrade image quality (Esposito et al., 1983). At least five near-infrared windows in the atmospheric absorption spectrum provide access to the surface. Some of these, such as the ones at 1.02, 1.10, and 1.18 mm, have almost no absorption in them except for the far wings of nearby CO2 absorption lines. The calculated radiance from the surface is shown in Fig. 1. Low lying regions produce more flux because they are hotter (red line), while highlands emit less (blue line). Venus’ atmosphere is 60 times more massive than Earth’s, and CO2 has four times the Rayleigh scattering cross-section of N2. The result is that the atmosphere is extremely bright at visible and even near infrared wavelengths. All visible wavelengths scatter appreciably, not just the blue that we are familiar with in Earth’s skies. Since blue solar light is attenuated in the clouds, the sky beneath the clouds is not blue. Due to Rayleigh scattering of longer visible wavelengths, the sky of Venus may be green or yellow.
Visibility through the scattered light can be expressed in terms of transmittance, as in Fig. 2. At 0.65 mm, less than one third of the radiation from the surface gets through unscattered above 3 km. In the 0.85 mm window, however, one third of the surface flux is unscattered at 9 km. For obtaining images that are not overwhelmed by sky scattering, 1.02 mm surface image acquisition should be possible at 30 km and below. At night, the sky is lit by scattering of the glow from the surface. During the day, the surface is both illuminated by sunlight and emits thermal radiation. Because this light is reflected several times between the surface and bottom of the clouds, the daytime sky is ~35 times brighter than the surface, as seen from 45 km.
Visibility through the scattered light can be expressed in terms of transmittance, as in Fig. 2. At 0.65 mm, less than one third of the radiation from the surface gets through unscattered above 3 km. In the 0.85 mm window, however, one third of the surface flux is unscattered at 9 km. For obtaining images that are not overwhelmed by sky scattering, 1.02 mm surface image acquisition should be possible at 30 km and below. At night, the sky is lit by scattering of the glow from the surface. During the day, the surface is both illuminated by sunlight and emits thermal radiation. Because this light is reflected several times between the surface and bottom of the clouds, the daytime sky is ~35 times brighter than the surface, as seen from 45 km.
Figure 1 (top). Synthetic spectrum of the emitted radiance of Venus at the top of the atmosphere, from 1 to 1.4 mm. Figure 2 (bottom). The transmissivity of the Venus atmosphere due to Rayleigh scattering alone, as a function of altitude, in 3 atmospheric windows.
Moroz (2002) calculated the Rayleigh scattering optical depths in the 0.65, 0.86, and 1.02 mm spectral windows. He also used Venera 13 and 14 spectrophotometer data in a scattering radiative transfer model to calculate ‘visibility factors’ for the surface in each of these windows for daytime conditions. His model assumed a surface albedo of 0.1, with a solar zenith angle of 37º – the conditions of the Venera 13 descent. The visibility factor, VF was defined as
where Fus(l,T) is the upward flux emitted from the surface, and Fua(l,T) is the upward flux in the atmosphere. Its reciprocal is thus the ratio of sky brightness to surface brightness. The results of Moroz (2002) for both day and night imaging of the surface are summarized in terms of sky brightness in Table 1.
At night, the sky is 2.6 times brighter than the surface at 45 km. All of the sky flux is from the thermal glow of the surface. Below 8 km, flux from the surface exceeds that from the sky. During the day, however, multiple scattering between the surface and cloud base cause the sky to be 38.5 times brighter than the surface At 2 km, the sky between the probe and surface is still twice as bright as the surface. The situation is worse in the 0.85 mm window, where the sky is 3 times brighter than the surface at 2 km. It is 13 times brighter than the surface as seen in the 0.65 mm window.
Table 1. Ratio of surface to sky brightness as seen by a Venus descent probe during the day and at night, at the 0.65, 0.85, and 1.02 mm windows.
Altitude | 1.02 mm night | 1.02 mm day | 0.85 mm day | 0.65 mm day | ||||
Surface | Sky | Surface | Sky | Surface | Sky | Surface | Sky | |
45 | 1 | 2.6 | 1 | 38.5 | 1 | 232.6 | 1 | 2.6 x 105 |
30 | 1 | 2.3 | 1 | 23.8 | 1 | 196.0 | 1 | 6.7 x 104 |
16 | 1 | 1.4 | 1 | 13.5 | 1 | 138.9 | 1 | 6.7 x 103 |
8 | 1 | 0.7 | 1 | 6.3 | 1 | 33.3 | 1 | 526.3 |
4 | 1 | 0.3 | 1 | 3.2 | 1 | 9.1 | 1 | 66.7 |
2 | 1 | 0.2 | 1 | 2.0 | 1 | 3.1 | 1 | 12.8 |
1 | 1 | 0.1 | 1 | 1.3 | 1 | 1.6 | 1 | 6.4 |
Visibility of the Surface
Nighttime Descent
Fig. 3 is a Synthetic Aperture Radar (SAR) image of Sapas Mons, a large shield volcano on Venus, acquired by NASA’s Magellan spacecraft. The image is 1024 km across and 1024 x 1024 pixels so it has a resolution of 1 km/pixel. The image shows many distinct lava flows, smooth plains, and tectonically disrupted plains (North is up). Because Magellan’s S-band radar had a wavelength of 12.6 cm, backscatter contrast is due largely to surface roughness differences at this scale. Dielectric properties also affect backscatter. Geologic interpretation of the scene more difficult than with optical or near infrared images.
Figure 3 (top). Magellan Synthetic Aperture Radar image of the volcano Sapas Mons on Venus. The image is 1024 x 1024 km. Figure 4 (bottom). SAR image superimposed on Magellan altimetry.
The atmosphere near the surface exhibited an adiabatic lapse rate during the descent of Pioneer Venus and Venera probes (Tomasko, 1983). There is some telescopic evidence for super adiabatic lapse rates in the deep atmosphere at times (Meadows and Crisp, 1996), although the data may also indicate a near-surface haze layer. Adopting a dry adiabat everywhere enabled (Mueller et al., 2008) to extract relative surface emissivities from the Venus Express VIRTIS data, with no apparent systematic errors to indicate the lapse rate was in error.
The dry-atmosphere temperature gradient, Gd, is
where Cp is the specific heat of the atmosphere and g is the gravitational acceleration. The temperature therefore follows
where z is the altitude relative to a reference altitude and To is the temperature at the reference altitude. The emitted flux from the surface, B(z), is given by
where e is the emissivity of the surface. The effect of topographic features is large, with mountain tops emitting half as much flux as the trenches, as shown in Fig. 5.
What a night time descent probe would see, without any interference from the atmosphere, is shown in Fig. 6. Dark regions are cool and the emission is weak compared with the trenches on either side. The patchiness is due to the much lower resolution of the altimetry data. This points to a difficulty in interpreting emission images from a descent probe. The emission from each point on the image depends upon its altitude and emissivity. Without independent altimetry, compositional differences cannot be distinguished from differences in altitude.
Figure 5 (top). Map of the temperature-dependent thermal flux from the surface of Venus due to topography. Figure 6 (bottom). Calculated emitted flux from the surface, ignoring atmospheric effects, as observed at 45 km.
Upward 1.02 mm radiation from the surface is not appreciably absorbed by the atmosphere, but it does undergo Rayleigh scattering. Applying the transmissivities of Fig. 2 to the atmospheric path from surface to camera, Fig. 7 shows what the emission would look like from 45 km above the surface. The image is vignetted by sky brightness – Rayleigh scattered radiation from the surface. From 8 km, the situation is better (Fig. 9), and the edges of the scene can be seen more clearly.
Figure 7 (top). Simulation of the visibility of the surface of Venus through the 1.02 mm atmospheric window on a night time descent vehicle, from just below the clouds at 45 km. Figure 8 (bottom). Simulated visibility at 8 km during the night, at 1.02 mm.
Daytime Descent
The Venera 13 spectrophotometer measured the upward and downward radiances from 0.4 to 1.2 mm as the probe descended on the day side (Ekonomov et al., 1983). The fluxes at 0.65, 0.85, and 1.02 mm are shown in Fig. 9. As shown by Moroz (2002), multiple reflections between the atmosphere and bottom of the clouds results in very high sky brightness, particularly at 0.65 microns. Although sunlight at these three wavelengths does penetrate and reflect off the surface, the added scattering from multiple reflections makes daylight observations even more difficult than night time ones. However, the reflected light from the surface directly shows compositional or size differences, without the need to have altitude information. Observing Venus at 1.02 mm during the day, about half of the flux from the surface is reflected sunlight, and half is thermal emission. However, the sky may be from twice to 40 times brighter than the surface. This argues for a camera with a very large dynamic range and deep electron well depth.
Figure 9. The upward intensity of the Venus sky as a function of altitude for 3 atmospheric windows, from the Venera 13 spectrophotometer (Ekonomov et al., 1983).
Fig. 10 shows the calculated view of the volcano from 45 km, during the day. The image is heavily vignetted, and the dynamic range of the center of the image is small, about 3 bits. The view from 8 km is much better, with variations in the surface showing up due to both altitude and surface contrast. The peaks of the volcano are dark, but not as dark as when seen in pure emission. The reason is because the peaks of the volcano are bright, presumably due to a compositional or textural difference.
Figure 10 (top). Simulation of the visibility of the surface of Venus through the 1.02 mm atmospheric window on a day time descent vehicle, from just below the clouds at 45 km. Figure 11 (bottom). Simulated visibility at 8 km during the day at 1.02 mm.
Descent Vehicle Motion
In addition to the optical challenges of acquiring descent images in Venus’ atmosphere, spacecraft motion places severe constraints on the camera system. Lorenz (2010) reviewed the data on planetary atmospheric probe motion, and found that Venera 11 and the Pioneer Venus Large Probe had rotation rates of about 10º/sec, or 1.7 RPM. Venera 12’s oscillation was more energetic, with a rate of 20º/sec, or 3.3 RPM, in what might have been a circular planing of the entry vehicle, with a period of 1.5 seconds (Table 2).
Table 2. Empirical data on Venus descent probe motion from Lorenz (2010)
Spacecraft | Rotation Rate | RPM | Period | Angle of Attack |
Venera 11 | 10º/s | 1.7 | 2.5 s | 7º |
Venera 12 | 20º/s | 3.3 | 1.5 s | 8º |
PV Large Probe | 10º/s | 1.7 | 1.1 s | 0-8º |
PV Small Probes | 1-2 s |
To avoid blurring, the exposure time must be less than the time it takes for the vehicle to rotate about any axis by the individual pixel field of view (IFOV). For typical values of spacecraft camera IFOVs, the exposure constraints imposed by vehicle motion are shown in Fig. 12. For an IFOV of 1 mrad/pixel for example (green line), a 4 msec exposure is the longest that is possible without blurring, if the spacecraft is rotating about any of its axes at 3.3 RPM.
Figure 12 The maximum rotation rate about any axis of the spacecraft before image smearing occurs, as a function of exposure time, for 1024x1024 pixels, 60º FOV.
References
Tuesday, August 16, 2011
Science and Values
Values were once transmitted mostly through the religion people practiced. I'm sure this is still true to some extent, although large swaths of the populace are now godless. For the most part, we fashion our values from authority. Religious authority is really good at this because God has no boss. But the real authority now, the Holy Ghost that permeates everything, is science.
Every waking (and sleeping) second is total immersion in a manufactured environment. It isn't just the constant connectedness and cars and sounds and advertisement. We can only value what we know, or are told. There isn't much room in our lives for old-fashioned values anymore, so they have been replaced by something else. The problem is that we don't exactly know what has replaced religion, largely because we don't understand our manufactured world. Science is so good at figuring out how nature works that it has enabled the most extraordinary control of our physical world. Almost none of this manipulation is understood by most people, from flipping on a light switch to inserting human genes in a mouse. Manipulating nature is the norm. Light, magnestism, electrons, chemicals - they all do our bidding and in turn shape our world. And our economy, and our values.
Albert Borgmann, in Crossing the Postmodern Divide, laments the impact that technology has on family dynamics. When choosing what to do, he notes, the family is not simply presented with more options if there's a nice big TV. It dominates them, limits them, leaving little room for anything else. The question is no longer whether to watch TV as one of several options. It is, 'What shall we watch?'. Nothing else is on the table.
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