Percentage of Atmosphere Beneath Observatories

Posted on 3-March-2016 by mathscinotes

Quote of the Day

Truth … is much too complicated to allow anything but approximation.

— John von Neumann. It took me a long time to accept that all models are wrong at some level, but that you can use them to produce useful results.

Introduction

Figure 1: Observatories are usually placed on Remote Mountaintops.

Figure 1: Observatories are usually placed on
remote mountaintops. Here is a picture of the
Sphinx Observatory (Source).

I often see popular descriptions of observatories that say things like the observatory "is above 40% of the Earth's atmosphere". I had not thought much about this kind of statement until I saw the Wikipedia's list of the world's highest-altitude observatories, which surprised me as to the height and remoteness of the largest telescopes.  I cannot imagine trying to build on these locations (Figure 1 is an extreme example). In some respects, the construction challenges remind me of what builders must have gone through on some lighthouses.

In this post, I will look at the highest altitude observatories and compute the percentage of atmosphere that they are above.

Background

The atmospheric pressure at a location is a measure of the weight of air above that location. We can determine the percentage of the air column below a given altitude by computing the ratio of p(h)/p(0), which represents the percentage of atmosphere above altitude h , and subtracting that ratio from 100%.

Eq. 1 \displaystyle \%\text{AtmoBelow(h)}=100\%-\frac{{p(h)}}{{p(0)}}

where

  • %AtmoBelow(h) is the percentage of atmosphere below the altitude h.
  • p(h) is the atmospheric pressure at altitude h.
  • p(0) is the atmospheric pressure at sea level, h= 0.

In this post, I will:

  • Verify some general statements about the amount of atmosphere below certain altitudes.
  • Determine the amount of atmosphere below the world's highest observatories.

Analysis

Atmospheric Pressure Curve Fit

To estimate the barometric pressure at different altitudes, I grabbed a table of pressures from the web and did a simple interpolation so that my function, p(h), is continuous (Figure 2).

Figure 2: Mathcad Interpolation of Barometric Pressure.

Figure 2: Mathcad Interpolation of Barometric Pressure.

Web References

Table 1 shows three examples of references to the percentage of atmosphere below that reference level. Note that the reference for Everest (marked in red) got their percentage number wrong by ~10%.

Table 1: Four Examples of Atmosphere Percentage Statements.
Statement Altitude (m) Stated Air % Below My Air % Below
Mauna Kea rises 9,750 meters from the ocean floor to an altitude of 4,205 meters above sea level, which places its summit above 40 percent of the Earth's atmosphere (Source). 4,205 40 40.8%
57.8 percent of the atmosphere is below the summit of Mount Everest (Source). 8,848 57.8 68.9
72 percent of the atmosphere is below the common cruising altitude of commercial airliners (about 10,000 m) (Source). 10,000 72 73.8

Figure 3 shows how the calculation was performed using Equation 1.

Figure M: Example Calculations.

Figure 3: Example Calculations.

World's Highest Observatories

The Wikipedia has a list of the highest observatories in the world. I used Equation 1 to find the percentage of atmosphere below each observatory in Table 2. The University of Tokyo Atacama Observatory is unbelievably high – that has to be a challenge for those who work there.

Table 2: World's Highest Observatories.
Observatory Name Elev.(m)
 Air % Below Observatory Site Location
University of Tokyo Atacama Observatory (TAO) 5,640 51.0% Cerro Chajnantor Atacama Desert, Chile
Chacaltaya Astrophysical Observatory 5,230 48.3% Chacaltaya Andes, Bolivia
James Ax Observatory 5,200 48.0% Cerro Toco Atacama Desert, Chile
Atacama Cosmology Telescope 5,190 48.0% Cerro Toco Atacama Desert, Chile
Llano de Chajnantor Observatory 5,104 47.4% Llano de Chajnantor Atacama Desert, Chile
Shiquanhe Observatory
(NAOC Ali Observatory)		 5,100 47.4% Shiquanhe, Ngari Plateau Tibet Autonomous Region, China
Llano de Chajnantor Observatory 4,800 45.2% Pampa La Bola Atacama Desert, Chile
Large Millimeter Telescope Alfonso Serrano 4,580 43.6% Sierra Negra Puebla, Mexico
Indian Astronomical Observatory 4,500 43.0% Hanle Ladakh, India
Meyer-Womble Observatory 4,312 41.6% Mount Evans Colorado, United States
Yangbajing International Cosmic Ray Observatory 4,300 41.5% Yangbajain Tibet Autonomous Region, China
Mauna Kea Observatory 4,190 40.7% Mauna Kea Hawaii, United States
High-Altitude Water Cherenkov (HAWC) Gamma-Ray Observatory 4,100 40.0% Sierra Negra Puebla, Mexico
Barcroft Observatory 3,890 38.3% White Mountain Peak California, United States
Very Long Baseline Array (VLBA), Mauna Kea Site 3,730 37.0% Mauna Kea Hawaii, United States
Llano del Hato National Astronomical Observatory 3,600 36.0% Llano del Hato Andes, Venezuela
Sphinx Observatory 3,571 35.7% Jungfraujoch Bernese Alps, Switzerland
Mauna Loa Observatory 3,394 34.3% Mauna Loa Hawaii, United States
Magdalena Ridge Observatory 3,230 32.8% South Baldy New Mexico, United States
Mount Graham International Observatory 3,191 32.5% Mount Graham Arizona, United States
Gornergrat Observatory 3,135 32.0% Gornergrat Pennine Alps, Switzerland
European Extremely Large Telescope 3,060 31.4% Cerro Armazones Atacama Desert, Chile
Haleakala Observatory 3,036 31.2% Haleakala Hawaii, United States

Conclusion

Just a quick note to explain where some of these atmosphere percentage numbers come from. I became interested in this after watching the movie Everest, which did an excellent job showing the effect of high altitudes on people.

Posted in Astronomy | 1 Comment

How to Read a Book

Posted on 2-March-2016 by mathscinotes

Quote of the Day

The best software is usually written three times: (1) First, you write the software to prove to yourself (or a client) that the solution is possible. Others may not recognize that this is just a proof-of-concept, but you do; (2) The second time, you make it work; (3) The third time, you make it work right.

— Esther Schindler, software blogger. During my time at HP, they had the concept of an initial "throw away" design that was done to learn about the product. Today, we call them "proof-of-concept" designs.

Figure 1: Mortimer Adler, who wrote the book called "How to Read a Book".

Figure 1: Mortimer Adler, who
wrote the book called How to
Read a Book (Source).

Last week, I was having a conversation with my youngest son about how I read the books that I own – he sees that I vigorously engage with these books, and he was wondering why I read this way. I shared the following story with him, and it may be of interest to some of you.

When my son says that I vigorously engage with a book, he means that I do things like:

  • Read the book in stages.
  • Write up pertinent sections in my own voice.
  • Make notes throughout the book
  • Cross-check the book I am reading with  other sources.
  • On technical books, I write Excel, Mathcad, or Mathematica summaries of my readings. In fact, these summaries grew into this blog.

My approach to reading came from an interview that I saw as a boy on the show Firing Line – yes, my father was a staunch Republican and all his children watched Firing Line on Sunday morning. Normally, I paid no attention to what was being said on the program. However, one episode contained an interview with Mortimer Adler (Figure 1) on how to read a book, and I listened carefully to his every word. I was so impressed with that interview that the next day (Monday) I marched to the Osseo Public Library and requested Adler's How to Read A Book. I would have gone to the library on Sunday, but it was closed that day.

I was highly receptive to his message, and I decided to try his approach. While I was not a good student, the Space Program was an inspiration for me to improve because I wanted to understand what they astronauts were doing. I even secretly hoped to someday participate  – kind of my own October Sky tale. So began my lifelong effort to learn how to learn. I am still working at it and no end is in sight.

Fortunately, Youtube now has many interviews with Adler. Here is a good one.

Figure 2: Youtube Video of a Mortimer Adler Interview on How to Read a Book.

The only problem with my method is that when I am done with a book, it has been worked hard: marked-up, dog-eared, and torn dust cover. I long ago decided that is a small price to pay to learn what is inside the book.

I should mention that the Wikipedia has a good summary of the key points of How to Read a Book.

P.S.

Of course, I treat library books VERY carefully. I am speaking here of books that I own.

Posted in Personal | 3 Comments

Roche Limit Examples

Posted on 1-March-2016 by mathscinotes

Quote of the Day

Whenever you find yourself on the side of the majority, it is time to pause and reflect.

— Mark Twain

Introduction

Figure 1: Image of Phobos, the largest moon of Mars.

Figure 1: Image of Phobos, the
largest moon of Mars (Wikipedia).

While listening to the audio book The Search for Exoplanets: What Astronomers Know, I heard the lecturer (Professor Joshua Winn) mention the Roche limit and gave a simple approximate formula for evaluating it. The Roche limit provides a lower boundary on how close a satellite may revolve around a planet or star. It is based on the idea that the gravitational and centrifugal forces of the planet work to pull a satellite apart, while the self-gravity of the satellite tends to hold it together. The Roche limit is where these forces are in balance – any closer and the satellite's gravity will be weaker than the centrifugal force plus the planet or star's gravity.  Within the Roche limit, the satellite is subject to forces that tend to break it apart. Satellites moving inside the Roche limit are thought to be one way that planetary rings are formed.

You occasionally hear the Roche limit discussed for  satellites in our solar system,  like Phobos of Mars (Figure 1), where the satellite is near the Roche limit and may begin the process of coming apart in as little as 10 million years. There are actually a number of satellites in our solar system that are under threat of being torn apart by the centrifugal and gravitational forces induced by their orbiting close to a more massive companion. At least one comet has been observed breaking up due to gravitational forces (Appendix B).

In this post, I will be reviewing the Wikipedia's article on the Roche limit, and I will derive an interesting approximation that Professor Winn mentioned in his audio lectures.  I find the approximation interesting because it expresses the Roche limit in terms of orbital period rather than distance.

Background

Definitions

Tidal Force
In the context of the Roche limit, a tidal force arises because the gravitational force exerted by one body on another is not constant across it; the nearest side is attracted more strongly than the farthest side. The differential nature of the tidal force tends to distort the shape of the smaller body and in extreme cases can even cause the smaller body to break up. (Source).
Hydrostatic Equilibrium
There are three forces on a satellite that are balanced at the Roche Limit – self-gravity of the body, internal pressure, and gravitational tidal forces (Source –Chapter 14). The Wikipedia has a good discussion of the concept.
Roche Limit
Minimal orbital distance compatible with hydrostatic equilibrium within a planet (Source –Chapter 14).

Roche Limit Statement

You usually see the Roche limit expressed in one of two ways: (1) for a rigid satellite, (2) for a fluid  satellite.  Planetary astronomers focus on the fluid satellite formula, but the rigid satellite formula is much simpler to derive and it illustrates the concepts. The derivation for a fluid satellite is more complex because it assume that the satellite becomes distorted under tidal forces  – two examples of distorted bodies from within our solar system are shown in Appendix A.

Rigid Satellite

The rigid-body Roche limit assumes a spherical satellite – the irregular shapes caused by the tidal deformation of a body are neglected. It is assumed to be in hydrostatic equilibrium. These assumptions, although unrealistic, greatly simplify calculations.

Eq. 1 \displaystyle d_{Rigid}=1.44\cdot {{R}_{M}}\cdot {{\left( {\frac{{{{\rho }_{M}}}}{{{{\rho }_{m}}}}} \right)}^{{\frac{1}{3}}}}

where

  • ρM is the density of of the large body (planet or star).
  • ρm is the density of of the satellite.
  • RM is the radius of the planet or star that the satellite orbits.
  • dRigid is the rigid satellite Roche limit – within this limit, a rigid satellite will experience tidal forces sufficient to tear it apart.

Note that the leading coefficient in the rigid body derivation is sometimes given as 1.26. A more accurate derivation (given in the Analysis portion) gives 1.44 for the leading coefficient.

Fluid Satellite

Equation 2 shows the Roche limit for a fluid body, i.e. a body with no internal binding forces. Note that the Wikipedia has a coefficient of 2.44, which was derived by Roche and is commonly seen. I am using the form presented on the McGill University web site, which has a coefficient of 2.423.

Eq. 2 \displaystyle {{d}_{{Fluid}}}=2.423\cdot {{R}_{M}}\cdot {{\left( {\frac{{{{\rho }_{M}}}}{{{{\rho }_{m}}}}} \right)}^{{\frac{1}{3}}}}

where

  • dFluid is the fluid satellite Roche limit.

Analysis

Rigid Satellite Roche Limit Derivation

Figure 2 shows my derivation of Equation 1. The derivation of the Roche limit for rigid body is much simpler than for a fluid body because the fluid body will undergo major distortion into an ellipsoid shape, which is analytically much more difficult.

Figure M: Derivation of Roche's Rigid Satellite Result.

Figure 2: Derivation of Roche's Rigid Satellite Result.

Fluid Satellite Roche Limit Derivation

The derivation of Equation 2 is much more involved, and I will not go through the details here. The best derivation I found was on the McGill University web site. There is also a good reference on google books (Figure 3).

Figure 3: Google Books Reference on Fluid Satellite Form of Roche Formula.

Application Example

Figure 4 shows an example I copied from the Wikipedia entry on the Roche limit. For my example, I have chosen to use the rigid satellite formula from the Wikipedia (my Equation 1)  and the fluid formula from the McGill University web site (my Equation 2), mainly because I understand the derivations of both formulas. The normalize Roche limit is simply the Roche limit expressed in terms of the radius of the planet or star.

Figure 4: Worked Example from the Wikipedia.

Figure 4: Worked Example.

Interesting Alternative Roche Limit for Fluid Satellites

Derivation

Professor Winn expressed the Roche limit in terms of a minimum  orbital period (Equation 3). The derivation of this form of the Roche limit simply requires applying Kepler's third law.

Eq. 3 \displaystyle {{d}_{{Fluid}}}\approx \frac{{12.6\text{ hr}}}{{\sqrt{{{{\rho }_{M}}}}}}

Figure 5 shows the details of my derivation.

Figure M: Derivation of the Approximation of Equation 3.

Figure 5: Derivation of the Equation 3 Approximation.

Worked Example

Mars' Phobos provides an interesting application example for Equation 3. Figure 6 shows my calculations using Equation 3 and we can see that Phobos nearing the fluid Roche limit.

Figure 6: Phobos Example Using Equation 3.


Figure 6: Phobos Example Using Equation 3.

Since Phobos is in a decaying orbit, some astronomers speculate the Phobos may only have 10 million years of left as an intact body. Here is an interesting quote from this article on the "doomed" satellite Phobos.

But Phobos won't zip around the red planet forever. The doomed moon is spiraling inward at a rate of 1.8 centimeters (seven-tenths of an inch) per year, or 1.8 meters (about 6 feet) each century. Within 50 million years, the moon will either collide with its parent planet or be torn into rubble and scattered as a ring around Mars.

Conclusion

I have been looking for a good excuse to go through the derivation of the Roche limit and the effort was worth it. I was able to duplicate the results shown on the Wikipedia and on a university astronomy department web site. I was also able to derive a useful approximation that I heard in the audio book version of The Search for Exoplanets.

After writing this post, I found a good paper online co-written by Professor Winn on this topic.

Appendix A: Examples of Satellites Possibly Undergoing Tidal Stress.

Figure 7 shows two satellites that some planetary astronomers feel may be undergoing tidal stress.

Figure M: Satellites Atlas (Jupiter) and Pan (Saturn).

Figure 7: Satellites Atlas (Jupiter) and Pan (Saturn) (Source).

The following quote from Quora does a nice job describing why satellites like these are hanging together.

Pan and Metis are held together by tensile forces. Tensile strength of a body is the maximum stress it can withstand before being pulled apart by stretching. Had their core been weaker, they would have disintegrated. The tidal forces affecting the bodies is why both the satellites are irregularly shaped. It is presumed that the primary's tidal forces can actually lift an object off the satellites' surface.  Naturally, since the satellites are so close to their respective planets, there is massive tidal deceleration at play. This means that the satellites are gradually spiraling towards the primaries, owing to the decay of their orbits. The tidal forces are constantly tugging at the satellites. Pan's surface (as we've discovered from Cassini) consists of a large amount of porous material that it has accreted. The  particles are weakly bound by their self-gravity, and had there been no other satellites,  would eventually shear out  forming large clumps before  they disintegrate and the particles join other clumps.

Appendix B: Example of Comet Broken Up By Tidal Forces

Figure 8 shows a picture of comet Shoemaker-Levy 9, which was broken up by Jupiter's gravity in 1992.

Figure M: Shoemaker-Levy Comet Broken Up By Jupiter's Gravity.

Figure 8: Shoemaker-Levy Comet Broken Up By Jupiter's Gravity (Source).

Posted in Astronomy | Comments Off on Roche Limit Examples

Book Review: A Passion for Leadership

Posted on 26-February-2016 by mathscinotes

Quote of the Day

The same boiling water than softens the potato hardens the egg.

— African Proverb

Figure 1: Cover of Gate's A Passion for Leadership.

Figure 1: Cover of Gate's A
Passion for Leadership
 (Source).

I just finished reading Bob Gate's A Passion for Leadership, and I am a bit torn. I regularly read books on management and most of them do not contribute anything to improving management– that is not true for A Passion for Leadership. The book is a well-written memoir in which Gates shows how he applied standard management lessons in difficult circumstances. These standard management lessons are worth repeating. My feelings about the book are torn because there is nothing new here. I understand that one could argue that the principles of good management are timeless, but I tend to like authors who give me a new way to look at things.

Gate's is an old-school manager of the type that I sincerely miss. I have only had a couple of managers of his ilk during my career. These managers inspired loyalty because of the way they balanced the needs of the organization with the needs of their people. Gates did emphasize more than most the need for a vision on where the organization needed to go and the need for building some consensus on how to get there.

Overall, his prescription for management success is timeless:

  • respect your people
  • respect your organization
  • understand how change you are driving will affect your organization and help those you lead to handle the change.
Figure 2: Scott Adams Book.

Figure 2: Scott Adams Book (Source).

Some of Gates' observations were all too familiar. For example, Gates recalled situations where senior government officials tried to blame low-level staff for major system failures (e.g. the debacle at the Walter Reed Medical Center). All too often I have seen upper management claim to be accountable by punishing low-level staff members for major systemic failures for which those staff members had little or no involvement. A supervisor in my group had an expression for this behavior – "Bring me the head of Willie the Mail Boy" – a reference to Scott Adams' Dilbert (Figure 2).

I recall a senior software manager who actually described himself as "The Designated Scapegoat" during the kickoff meeting for a very challenging program before we even had started. Of course, the senior VP in charge of our division immediately leaped to the podium and said that the software manager was joking. However, nine months later the software manager was fired for his failure to execute on a "death march" project – that project was doomed the day it started.

Overall, Gates has put together a good memoir with some useful management lessons. Nothing earth-shattering, but filled with good reminders on the importance of treating both people and organizations with respect.

Posted in Management | Comments Off on Book Review: A Passion for Leadership

A Quick Power Over Ethernet Review

Posted on 24-February-2016 by mathscinotes

Quote of the Day

Until lions have their historians, tales of the hunt shall always glorify the hunter.

— West African proverb. A variation on Winston Churchill's "History is written by the victors" quote.

Introduction

Figure 1: Hookup for a 30 W PoE Type 2 System.

Figure 1: Hookup for a 30W PoE Type 2 System.

I have been asked to write some requirements for an optical product that is powered using Power Over Ethernet (PoE). It has been a few years since I have worked on a PoE-based design, I thought it would be useful to review the standard and ensure that I still understand it. This is a good exercise in basic electrical design and will also illustrate how to design circuits using Mathcad utility functions that I have written over the years.

My objective in this post is to show how a useful amount of power (25.5 W at the load) can be transferred over an Ethernet cable. I will be avoiding discussions on the protocol details associated with PoE because that would result in an enormous post.

Background

Definitions

Power over Ethernet (PoE)
PoE is an IEEE standard for sending power and data over the same category 5e Ethernet cable, which contains four wire-pairs (i.e. 8 wires total). PoE is enormously popular because only one cable is required to network an Ethernet-fed device, which greatly reduces the cost and complexity of networking remote devices, like cameras. For the version of PoE discussed in this post, power is transmitted over two  wire-pairs by applying a DC voltage between each pair (see Figure 1). Superimposing DC on the wire-pairs does not interfere with data transmission because Ethernet uses differential signalling.
Type 2 PoE
Type 2 PoE is an IEEE standard (802.3at) for transferring as much as 25.5 W over an Ethernet cable. The standard is also known as  "PoE+".
Power Supplying Device (PSE)
A PSE is a device that provides power on an Ethernet cable.
Powered Device (PD)
A PD is a device powered by a PSE.

PoE Basics

Here are the key points about a type 2 PoE system discussed in this post:

  • The source power is limited to 30 W.
  • The wire temperature is assumed to be no more than 50°C.
  • All design work will assume category 5e cable, which means 24 AWG wire.
  • I will be using two of the four Ethernet pairs for power transmission.
  • The category 5e cable length is limited to 100 m.
    100 m is also the maximum reach for data transfers on Ethernet. This means that you can use PoE on any Ethernet network.

Analysis

Modeling Resistance of Annealed Copper Wire

Figure 2 shows my linear interpolation of some annealed copper wire resistance data that I found years ago. I believe it was from an old Bell Telephone, but I do not recall the original source – certainly something I googled. I have scanned the original table into an Excel workbook.

Figure M: Linear Interpolation of Copper Resistance Data.

Figure 2: Linear Interpolation of Copper Resistance Data.

One-Way Cable Resistance

Figure 3 shows how to compute the resistance of a 100 m long, 24 AWG, category 5e wire at 50 °C using the functions shown in Figure 2. My calculations show the maximum wire resistance is ~10 Ω, which does not include connector losses. The standard actually assumes 12.5 Ω, which will provide a reasonable amount of margin.

Figure M: One Resistance of a Categrory 5 Cable Wire.

Figure 3: One Resistance of a Category 5e Cable Wire.

Figure 4 shows the basic circuit I am working with here.

Figure 4: Resistive Circuit Model for PoE.

Figure 4: Circuit Model for PoE (Source).

PoE+ Analysis

My intent here is compute a few of the key product parameters of my PoE+ driven system, like maximum input power and internal heat generation (shown shaded green in Figure 5).  To perform my analysis, I need to state a few PoE+ characteristics:

  • The source power, PSource, is limited to 30W.
  • The load power, PLoad, is limited to 25.5 W.
  • The minimum source voltage, VMin, is specified as 50 V.
Figure 5: PoE+ Analysis.

Figure 5: PoE+ Analysis.

Conclusion

This exercise was a useful refresher exercise as to how PoE delivers power over category 5e cables. My application requires 15.3 W, so the 25.5 W capability of PoE+ will provide enough power for my application plus some reserve power in case we want to add more features later.

Posted in Electronics | 1 Comment

45-Star US Flag Heirloom

Posted on 23-February-2016 by mathscinotes

Quote of the Day

If the brain were so simple we could understand it, we would be so simple we couldn't.

— Lyall Watson, biologist

Figure 1: US Flag with 45 Stars.

Figure 1: US Flag with 45 Stars (Source).

A friend showed me a family photo of a 45-star US flag that was purchased in 1898, which was the year their grandfather was born. Figure 1 shows an example of the flag in their photo. The flag, which is quite large, is often used as a backdrop for a family photos. The flag is carefully stored and only taken out for special events, like reunions. I think this is a great use for an old flag.

Since each star represents a state, their flag piqued my curiosity as to how the number of stars changed over time. I found a table of union-entry dates for each state, and I decided to plot the number of stars at the end of each year versus the year. Figure 2 is the result. Observe that the rate of state entry was fairly constant until 1912, which was the year New Mexico and Arizona entered the union. Only Alaska and Hawaii have become states since 1912.

Figure 2: Number of States in the Union Versus Time.

Figure 2: Number of States in the Union Versus Time.

Posted in History Through Spreadsheets, Personal | Comments Off on 45-Star US Flag Heirloom

Moving Lake Ice in Northern Minnesota

Posted on 22-February-2016 by mathscinotes

Quote of the Day

Only put off until tomorrow what you are willing to die having left undone.

— Pablo Picasso

Figure 1: Carving of a Bear Holding a Light.Figure 1: Carving of a Bear Holding a Light.

Figure 1: Carving of a Bear Holding a Light.

I just came back from a weekend visiting friends in Northern Minnesota. The snow is beginning to melt, and this makes everyone excited about the arrival of spring. During my weekend, I spent some time walking the streets around Gull Lake, which is near the city of Nisswa. I often see things that strike my fancy while walking. For example, I am always looking for good ideas to apply to my cabin, which is in the Grand Rapids area. Figure 1 shows a unique street light consisting of a wooden carving of a bear holding an old-style hurricane lamp. Bears and moose are common themes in northern Minnesota lore.

One interesting aspect of the arrival of spring is how the wind pushes the ice around on the lakes. Figure 2 show a common sight on many lakes. The homeowners in this area have put riprap under the water near the beach in order to direct the moving ice sheet up rather than against the beach. This reduces the amount of beach erosion that occurs every spring as the ice sheet slides around on the lakes.

Figure 2: Wind Has Blown the Ice Sheet Up On The Beach.

Figure 2: Wind Has Blown the Ice Sheet Up On The Beach.

Figure 3 is a similar photo, but with a couple of my friends next to the ice so that you can get  better idea of size of the ice projection.

Figure 3: Dave and Ray by the Gull Lake Ice.

Figure 3: Dave and Ray by the Gull Lake Ice.

Posted in Personal | Comments Off on Moving Lake Ice in Northern Minnesota

Challenge of Viewing an Earth-Sized Planet

Posted on 18-February-2016 by mathscinotes

Quote of the Day

Mother: Why do you drink so much?
Brother: I drink to forget.
Mother: What do you need to forget?
Brother: I don't know, I forgot.

— Actual conversation between my mother and one of my brothers on his lifestyle while at university.

Introduction

Figure 1: Geometry of the Observing Earth From Epsilon Eridani.

Figure 1: Geometry of the Observing
Earth From Epsilon Eridani.

I listen to audio books during my nightly walks around a local lake. My current selection, Searching for Exoplanets, is one of the best audiobooks I have listened to. The book consists of a series lectures on the state of the search for exoplanets by MIT Professor Joshua Winn. The lectures provide an excellent summary of how astronomers are using remarkably sensitive methods for indirectly detecting the presence of exoplanets circling remote stars.

The technology for studying exoplanets is still in its infancy, and we are years away from directly seeing the light from an Earth-like planet – the Holy Grail of exoplanet research. However, Professor Winn does include a lecture on imaging exoplanets.  He frames the problem of imaging an exoplanet in terms of the difficulty that an astronomer on an exoplanet would have imaging the Earth. Here is a quote from lecture 20.

For the moment, our goal is to see an exoplanetary system as a series of dots surrounding a star.... The first problem is that planets are very faint. If we were looking at the Earth with a 1 meter telescope from 10 light-years away, say we're astronomers on a planet around Epsilon Eridani, then Earth would deliver fewer than one photon per second to our detector.

My objective in this post is to verify his statement that fewer than one photon per second will be seen by a person trying to observe Earth from Epsilon Eridani. I am treating this as a Fermi problem and all my work will be approximate.

Background

This is a routine calculation. The key concepts to keep in mind are:

Analysis

Figure 1 shows the basic geometry of the observing situation. Figure 2 shows my calculations for replicating the estimate for the number of photons from Earth that would be observable by an astronomer in the Epsilon Eridani system. I compute 0.9 photons per second, which agrees with Professor Winn's statement of less than 1 photon per second.

Figure 2: Photons Per Second When A Remote Observer Attempts to View Earth.

Figure 2: Photons Per Second When A Remote Observer Attempts to View Earth.

Conclusion

I was able to duplicate Professor Winn's estimate with little effort. This example does a good job of showing why astronomers want to build large aperture telescope – they need all the light they can get to drive instruments like spectroscopes and coronagraphs. You can also see that we are a long way from imaging even a relatively close exoplanet.

Posted in Astronomy | 1 Comment

WW2 Submarine Endurance on Batteries

Posted on 16-February-2016 by mathscinotes

Quote of the Day

The only hope of a pure mathematician is to die before their work is applied.

— Pure mathematician stunned to hear that his work found an application in string theory.

Introduction

Figure 1: Running Time Versus Submerged Speed.

Figure 1: Running Time
Versus Submerged Speed
(Source).

I have been reading the book The Bravest Man, a biography of the WW2 exploits of US Navy submarine commander Dick O'Kane. I have not formed an opinion on the book since I just started reading it, but the book does highlight the submerged maneuvering limitations imposed on a ww2 submarine because of its lead-acid battery-based power plant. The book's discussion made me curious about the operational characteristics of a Gato-class submarine when operating submerged on batteries.  In this post, I will be examining the Gato-class submarine's run time versus speed plot (Figure 1).

I should mention that O'Kane is most famous for his work as skipper of the USS Tang, a Balao-class submarine. I currently am reading about O'Kane's time serving as executive officer on the USS Wahoo under Mush Morton. I find their relationship uniquely effective – in part because Morton chose to  conn the submarine while O'Kane man the periscope.  It was an excellent example of teamwork and not letting ego get in the way of one's task.

My goal here is to fit the published Gato battery endurance data to my usual battery endurance models. I will focus on the peace-time operating time curve because the war-time curve reduces the battery's lifetime and my model is not as applicable.

Background

This analysis is similar to that done on this previous blog post. Ultimately, I will use this model in a simple software tool I am creating to help me when playing my favorite submarine simulation, Silent Hunter.

Analysis

I have examined submarine battery power for modern electric submarines on this blog before. I will apply a similar analysis to this class of WW2-era boats.

Submarine Power Model

Figure 2 shows how I usually model submarine battery endurance. The only thing that is a bit unusual is my modeling of the inefficiency of drawing high currents from a battery. I chose to simple set the total battery energy to "1" and then model the inefficiency as a simple power law, i.e. 1-{{k}_{1}}\cdot v_{{Ship}}^{{{{k}_{3}}}}, where k1 and k3 are fitted parameters, and vShip is the speed of the submarine.

Figure 2: Submarine Battery Endurance Model.

Figure 2: Submarine Battery Endurance Model.

Digitize Figure 1 and Fit a Model

I digitized Figure 1 using Dagra and then performed a curve fit to Equation 1 using Mathcad (Figure 3).

Figure 2: Digitize Figure 1 and Fit Model.

Figure 3: Digitize Figure 1 and Fit Model.

Graphical Comparison of Figure 1 and Equation 1

Figure 4 shows the raw data capture and my fitted model – the agreement is excellent. So the Gato data is consistent with other submarine battery data I have seen.

Figure 3: Comparison of Raw Data and Model Fit.

Figure 4: Comparison of Raw Data and Model Fit.

Conclusion

I fitted the running time of the Gato class to my usual model for submarine battery endurance with good results. I will be using this model to build some simple tools to help me with my submarine gaming adventures.

Posted in Naval History | Comments Off on WW2 Submarine Endurance on Batteries

Floating Habitat on Venus

Posted on 13-February-2016 by mathscinotes

Quote of the Day

General, would you rather command an army of slaves?

— Milton Friedman responding to General William Westmoreland, who had said that commanding a volunteer army would be like commanding an army of mercenaries.

Introduction

Figure 1: NASA Conception of a Floating Venus Colony.

Figure 1: NASA Conception of a Floating Venus Colony (Source).

I have been keeping a close eye on the discussions occurring about sending people to Mars on both one-way and two-way trips. You do not hear similar discussions about Venus because its surface temperature (467 °C) and pressure (93 bar) are too extreme to imagine people surviving there.

Interestingly, 50 km above the Venusian surface, the temperature and pressure are comparable to that of the Earth's surface. In fact, one Venus researcher (Geoffrey A. Landis) commented that "At cloud-top level, Venus is the paradise planet.” But how could people live in the clouds? Balloons.

While balloon-based cities may seem like pure science fiction, there are credible proposals for sending explorers to bases borne by balloons in Venus' atmosphere. In this post, I will look at what how a balloon filled with Earth air could be used as a floating residence and laboratory. I will also look at the characteristics of two balloons that the Russians actually have floated in atmosphere of Venus.

Background

Definitions

Archimedes principles
Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object (Archimedes).
Lifting gas
Lifting gases are gases that can be used in lighter-than-air aircraft because they have average densities lower than that of air; thus, they are buoyant in air.
Superpressure Balloon
A superpressure balloon is a style of aerostatic balloon where the volume of the balloon is kept relatively constant in the face of changes in the temperature of the contained lifting gas. This allows the balloon to keep a stable altitude for long periods. This is in contrast with the much more common variable-volume balloons, which are either only partially filled with lifting gas, or made with more elastic materials (Source).

Atmosphere of Venus

Figure 2 shows how atmospheric pressure and temperature vary with altitude on Venus. At an altitude of ~50 km, the air pressure is ~1 bar and the temperature is ~0°C. These conditions are reasonable for human survival – we just need a surface to live on.

Figure 2: Atmospheric Pressure and Temperature and Pressure Versus Altitude.

Figure 2: Atmospheric Pressure and Temperature and Pressure Versus Altitude (Source).

Basic Concept

The following quote does the best job I have seen of explaining how one of these balloon living quarters would work.

Landis has proposed aerostat[airship] habitats followed by floating cities, based on the concept that breathable air (21:79 oxygen/nitrogen mixture) is a lifting gas in the dense carbon dioxide atmosphere, with over 60% of the lifting power that helium has on Earth. In effect, a balloon full of human-breathable air would sustain itself and extra weight (such as a colony) in midair. At an altitude of 50 kilometres (31 mi) above Venerian surface, the environment is the most Earth-like in the solar system – a pressure of approximately 1 bar and temperatures in the 0 °C–50 °C range. Protection against cosmic radiation would be provided by the atmosphere above, with shielding mass equivalent to Earth's.

Here is a good NASA video on an advanced Venus balloon concept.

Figure 3: NASA Video on Venus Balloon Concept.

Precedent

In 1985, the Russians successfully deployed two balloons, at 54 km altitude in the atmosphere of Venus. The balloon-based probes returned data for more than 46 hours.  So we know that a balloon can be successfully deployed and floated in the Venusian atmosphere.

Analysis

Why an Air-Filled Balloon?

I have read about concepts for both air and hydrogen-filled balloons. The advantage of an air balloon is that people could actually live inside of a 1 atmosphere-pressure balloon. The disadvantage of the air-filled balloon is that it must be much larger than a hydrogen-filled balloon with the same lift.

Lift Capacity

Equation 1 shows the formula for the buoyant force per unit volume of balloon's lifting gas. In this calculation, I assume that the atmosphere of Venus is pure CO2 (close to true) and the balloon contains Earth air.

Eq. 1 \displaystyle {{F}_{B}}=\left( {{{\rho }_{{Atmosphere}}}-\text{ }{{\rho }_{{Balloon}}}} \right)\text{ }\cdot \text{ }{{g}_{{Planet}}}

where

  • FB is the buoyant force per unit volume.
  • ρAtmosphere is density of the air surrounding the balloon.
  • ρBalloon is density of the gas in the balloon.
  • gPlanet is the acceleration due to gravity on the planet with the balloon.

Air Balloon Lift on Venus

Figure 3 shows how to compute the lift force per unit volume of an air balloon on Venus.

Figure 3: Air and Hydrogen Lift Forces Per m3 On Venus.

Figure 3: Air and Hydrogen Lift Forces Per m3 On Venus.

Russian Balloon Characteristics

The Russians have deployed two Venus balloons for which I have located the following information on the Wikipedia:

  • Balloon diameter: 3.54 m
  • Total mass:21 kg
  • Lifting gas: He
  • Atmospheric pressure when floating: 535 mbar
  • Temperature when floating: 300K

Given this information, we can approximately calculate the lifting force generated by these balloons (Figure 4). I should mention that this exercise would be a good one for a high-school chemistry or physics class that is learning about the ideal gas law.

Figure 4: Russian Balloon Lifting Force Shown To Be Equal to Payload Weight.

Figure 4: Russian Balloon Lifting Force Shown To Be Equal to Payload Weight.

Conclusion

The most surprising part of this exercise was seeing that there is a place in our solar system besides Earth that has some physical characteristics similar to here – 1 atmosphere pressure, ~20 °C temperature, and protection from space radiation. On the downside, the Venusian atmosphere is filled with corrosive chemicals.

I am not sure of the value in having people floating around Venus, but I could see the concept being very interesting for a robot explorer. I should mention that the Europeans have discussed attempting a Venus balloon mission (see Figure 5).

Figure M: Artist's Conception of the European Venus Explorer (EVE).

Figure 5: Artist's Conception of the European Venus Explorer (EVE) (Source).

Postscript

I just saw on Quora an artist's concept for a cloud city on Venus (Figure 6). The figure dates from 1971.

Figure 6: Artist's Conception of a Floating City on Venus.

Figure 6: Artist's Conception of a Floating City on Venus.

Posted in Astronomy | 6 Comments