Letter Folding for Envelopes

Posted on 18-November-2015 by mathscinotes

Quote of the Day

I am a kind of paranoid in reverse. I suspect people of plotting to make me happy.

— J.D. Salinger

Introduction

Figure 1: Standard #10 Business Envelope.

Figure 1: Standard #10 Business Envelope (Source).

I still occasionally write paper letters. In fact, I had some letters to write the other day, and I realized that was doing a bit of math when I folded the letters for placement into standard business envelopes that was worth discussing here. Figure 1 shows the business envelope that I normally use.

For years (i.e. decades), I folded letters into thirds by estimating where the one-third point was on the letter, folding, and hoping for the best. Sometimes, I folded letters by rolling them into a tube-shape that could be flattened for placement into an envelope (Figure 2). I still did not always get this exactly right – remember, my life is a celebration of mind-numbing detail.

Figure 1: Folding a Letter Into Thirds Using the "Tube Method".

Figure 2: Folding a Letter Into Thirds Using the "Tube Method" (Wikihow).

My Approach

I now use one of three methods:

  • For letter material that I created, I use a Word template that has very faint grey markers in margins where the fold lines need to go. Here is a link that tells you how to do this. I know that people may feel that the faint gray lines are unprofessional, but they end up in the crease and are very difficult to see (example). Here is my personal letter template.
  • For letter material that I am including in my note, I determine my fold lines one of two ways:
    • Template
      I  keep a reference sheet of paper in my cube that is exactly 1/3 the height of a sheet of letter-sized paper.
    • Origami
      I occasionally use an origami-based method (see Method 4 on this web page). It is a standard origami technique. Figure 3 proves why the origami-based method works.
Figure 4: Geometry Behind Origami Method.

Figure 3: Geometry Behind Origami Method.

 

Posted in General Mathematics, Origami | 3 Comments

How I Orient My Electrical Outlets

Posted on 16-November-2015 by mathscinotes

Quote of the Day

The most beautiful people we have known are those who have known defeat, known suffering, known struggle, known loss, and have found their way out of the depths. These persons have an appreciation, a sensitivity, and an understanding of life that fills them with compassion, gentleness, and a deep loving concern. Beautiful people do not just happen.

— Elisabeth Kubler-Ross

For years, I have mounted most my electrical outlets "upside down" (see Figure 1) – by upside-down I mean the ground slot is up. Recently, I have started to see more folks orienting their outlets this way. I see that Finehomebuilding Magazine and the Journal of Light Construction (JLC Field Guide) recommend this approach as well. However, it is not required by the National Electrical Code.

Figure 1: FineHomebuilding Magazine Forum Answer On Outlet Orientation.

Figure 1: FineHomebuilding Magazine Forum Answer On Outlet Orientation.

I started doing this when I saw a child drop a metal object onto a loosely plugged in cord. With power and neutral wired as the top slots, the metal object shorted and created a spectacular flash. Fortunately, no one was hurt but the outlet and plug were both damaged. Having the ground slot at the top means that a metal object dropped on a plug is far less likely to contact the power contact. Watching what happened to this child drove me to reorient many of the outlets in my home.

Note that some products (e.g. my refrigerator) have cords that tend to pull out of outlets that are wired with the ground-slot up. These outlets should then be wired in the conventional manner (i.e. ground-slot down). These outlets usually are in places where metal things are not likely to fall on cords plugged into them.

The same logic applies to sideways-mounted outlets (Figure 2). In this case, the neutral slot is mounted up.

Figure 2: Sideways-Oriented Outlet.

Figure 2: Sideways-Oriented Outlet (Source).

Posted in Construction | 2 Comments

My Cub Scout Training Solves Test Equipment Corrosion Problem

Posted on 14-November-2015 by mathscinotes

Quote of the Day

A man's intelligence does not increase as he acquires power. What does increase is the difficulty in telling him so.

— D. Southerland

Figure 1: Connectors Damaged Because of Water Ingress.

Figure 1: Connectors Damaged Because of Water Ingress.

We have recently experienced some laboratory failures during humidity testing that were due to corroded connectors (Figure 1). The connectors had corroded after they become wet from condensation that accumulated on the cables and rolled down to the lowest point on the cable – where the connectors were.

Our testing is performed in environmental test chambers – essentially a combination of oven and refrigerator. We can adjust the both the temperature and humidity within the chambers. To connect the electronics inside the chamber to external test equipment, the cables pass through a hole in the chamber wall (Figures 2 and 3) called the pass-through. The condensation occurs when air leaking through the pass-through hole encounters the outside air.

While we try to seal the cable pass-through, some water inevitably condenses on the cables. Since the cable are hanging, the condensed water runs down the cables until it met a connector and corrosion can start.

Figure 2: Example of an Environmental Chamber with Side Cable Access. Figure 3: Typical Internal Cable Configuration.

This situation reminds me of what I had to deal with as a Cub Scout when we would use  ropes tied to trees to hold up tarps and hammocks (Figure 4). When it was raining, water would run down the ropes and make the hammocks and tarps wet. The solution was to tie a drip line onto the ropes to interrupt the flow of water. It looks like the same solution will work for our environmental test cables.

Figure 4: Cub Scout Drip Line.

Figure 4: Cub Scout Drip Line (source).

 

Posted in Electronics | 1 Comment

Venus-Like Exoplanet In the Neighborhood

Posted on 13-November-2015 by mathscinotes

Quote of the Day

We live in a world where not everyone has the urge to help others… It is OK to encourage others to pull themselves up by the bootstraps, but if you do, just remember that some people have no boots.

— Neil deGrasse Tyson. My father used to make a similar statement. He always wanted to help others.

Introduction

Figure 1: Typical News Report of Venus-Like Exoplanet.

Figure 1: Typical News Report of Venus-Like Exoplanet (Source).

I have been reading a number of articles that are reporting on a Venus-like planet (GJ 1132b) recently discovered in a nearby red dwarf star system (Gliese 1132, 12.0 parsecs away). I like to work a bit with the numbers reported in these articles to determine if I actually understand what is being reported. I have to admit that I also like to imagine the day when astronomers are studying Earth-like planets around other stars. I definitely see that day coming. Discoveries like GJ 1132b are particularly interesting because astronomers for a long time did not think red dwarf stars were promising for Earth-like planets.

All the data being reported in the new reports appears to come from a single journal article. The articles have all made statements similar to those in this list:

  • The planet has an  average surface temperature of 450 °F.
  • The planet has a  diameter of 16% larger than the Earth (9200 miles)
  • The planet has a density of 6 grams/cm3.
  • The planet was detected using the transit photometry, which detected a 0.26% drop in the light output of the star when the planet passed in front of the star.

My goal here is to understand how these estimates were based on the observations the astronomers actually made.

Background

The star, Gliese 1132, is a red dwarf star that I will assume to have the following characteristics:

  • 20% of our Sun's luminosity
  • 20.7% of our Sun's diameter
  • 18.1% of our Sun's mass

Astronomers have well-established methods for determining these parameters, which I will not cover in this post – they will be fodder for a future post.

There really are only two measurements associated with the planet itself.

  • The star had a radial velocity of 2.7 m/s

    The radial velocity is the speed that object moves away from or approaches the Earth. We can measure this speed very accurately using the Doppler shift. This radial velocity is related to the momentum of planet that is orbiting the star, which you can see in Figure 2.

    Figure M: Illustration of Star's Radial Velocity.

    Figure 2: Illustration of Star's Radial Velocity. The star is the large body in the center (source).

  • There was dip of 0.26% in the star's light level that occurred with a 1.63 day period.

    We can illustrate the dip in light level as shown in Figure 3.

    Figure 3: Illustration of Light Level Dip Due to Planet Transit.

    Figure 3: Illustration of Light Level Dip Due to Planet Transit.

The original paper included error ranges about each measurement, which I will ignore for my rough work here. As I show below, these measurements can be used to learn much about the planet that circles this star.

Analysis

Planet Orbital Radius and Mass

Figure 4 shows my calculations for the planet's orbital radius and mass. The approach I used is described in the Wikipedia and this document, so I will not go through the formulas further. I will say that if you look carefully, you will see Kepler's 3rd Law, i.e.  \displaystyle {{R}^{3}}\propto {{T}^{2}} .  As I show in Figure 2, my calculations are almost identical with those reported in the original paper.

Figure 2: Planet Orbital Radius and Mass.

Figure 4: Planet Orbital Radius and Mass.

In Figure 4, I determine the planet's mass using the fact that the star's momentum and the planet's momentum must be equal. Figure 5 illustrates this fact.

Figure 5: Illustration of Star and Planet Momentum, Which are Equal.

Figure 5: Illustration of Star and Planet Momentum, Which are Equal.

Planet Diameter

Figure 6 shows how you can compute the diameter of the planet using the measured dip in the Gliese 1132's light output as the planet passes between Earth and Gliese 1132, an approach that is known as the Transit Method. The diameter estimate assumes the light occulation is proportional to the circular area ratio of the planet to the star Gliese 1132.

Figure 3: Computing the Diameter of the Planet.

Figure 6: Computing the Diameter of the Planet.

Planet Density

Since we have the planet's mass and diameter, we can compute the density of the planet as shown in Figure 7.

FIgure 4: Calculation of the Planet's Density.

Figure 7: Calculation of the Planet's Density.

Planet Temperature

Figure 8 shows how we can compute the temperature of the planet, assuming no greenhouse effect. The technique is described well in the Wikipedia and I will not say more here.

You can see that the temperature estimate is strongly affected by the assumptions made about the planet's bond albedo – the fraction of power in the total electromagnetic radiation incident on an astronomical body that is scattered back out into space. My results are virtually identical to those reported in the original paper, which assumes a range of bond albedo's from 0 to 0.75. The news articles all chose different temperature values within the range reported by the original paper.

Figure 5: Estimate of the Planet's Temperature.

Figure 8: Estimate of the Planet's Temperature.

I should mention that the same calculation performed on the Earth would show that our average temperature should be around -17 °C (see Appendix A). However, the actual mean temperature of the Earth is ~16 °C (source). The difference is because of the greenhouse effect.

 Conclusion

With a bit of basic physics, I was able to reconstruct the derived exoplanet characteristics presented in the original paper. I always find it amazing how much information we can derive about a planet with just a few numbers and no direct visual observations.

Appendix A: Calculation of Earth's Temperature Assuming No Greenhouse Effect.

You can find the Earth's albedo value here. Figure 9 shows the calculation.

Figure 6: Earth's Mean Temperature Assuming No Greenhouse Effect.

Figure 9: Earth's Mean Temperature Assuming No Greenhouse Effect.

The actual mean temperature of the Earth is ~16 °C. The difference of 33 °C is because of the greenhouse effect.

 

Posted in Astronomy | 1 Comment

Screwed Wood Joint That is Totally Concealed

Posted on 12-November-2015 by mathscinotes

Quote of the Day

It is often argued that religion is valuable because it makes men good, but even if this were true it would not be a proof that religion is true. That would be an extension of pragmatism beyond endurance. Santa Claus makes children good in precisely the same way, and yet no one would argue seriously that the fact proves his existence.

— H.L. Menken

This is the first woodworking tool that I have seen that uses a two-part electric motor. I have encountered this type of two-part electric motor before in situations where some mechanical object needed to be rotated while contained in a completely sealed environment. In this case, the electric motor's armature has a screw thread that pulls the wood joint tight. See this web page for more details. I certainly can see several applications for this type of connection.

Posted in Construction | 2 Comments

WW2 Reticles

Posted on 5-November-2015 by mathscinotes

Quote of the Day

Old men want to date a nurse with a purse.

— Petra from Texas, a retired woman I met on my recent Alaskan cruise. She was discussing her opinion of dating older men. Basically, older men want a woman with money and will care for them.

Introduction

Figure 1: Illustration Showing Common Type of WW2 Reticle.

Figure 1: Illustration Showing
Common Type of WW2 Reticle
(Source).

I have watched a lot of old WW2 combat footage, and I have noticed that many of the machine guns and fighter planes had similar reticles. A reticle is a fine-grid of lines used in conjunction with an eyepiece to assist in taking measurements or with accurately pointing an instrument. Figure 1 shows a reticle similar to what I have seen in numerous combat scenes.

I became curious when I saw that there were numerous variations this basic design for these reticles. To get some insight into these reticles, I began reading some old WW2 manuals I found on the Internet. During my reading, I discovered that  reticles are earlier versions of modern telescopic used by target shooters and hunters today (link). The ultimate realization of this type of reticle is made by Horus.

This post will summarize the results of my reading into a short explanation of how these reticles were used.

Background

These reticles are designed to serve two purposes:

  • measure the range to the target

    The reticles of optical sights have long been used to estimate the range to a target using stadiametric methods, which I have covered in other posts

  • determine the required lead for a target of a given speed.

    If you assume the target and projectile move at constant speeds, you can use the reticle to estimate the amount of deflection (aka lead) required. This method of lead determination is the same as what I have discussed with WW2 straight-running torpedoes. The Wikipedia also has an excellent discussion of stadiametric ranging.

Use Cases

Example Sight

I found a close up of a common WW2 reticle called the Mk 10 in this manual.  The sight is composed of the following parts:

  • An outer ring of 4 inch diameter called the 200 knot ring.
  • An inner ring of 2 inch diameter called the 100 knot ring.
  • A tiny circular ring in the middle that is used to center the front sight.
Figure M: Mark 10 Reticle.

Figure 2: Mark 10 Reticle (Source).

You can also find reticles that have a third ring called "300 knot ring." Figure 3 shows a 20 mm cannon with a 3-ring reticle.

Figure M: 20 mm Oelikon Gun.

Figure 3: 20 mm Oelikon Gun (source).

Manual Ranging Example

Figure 4 shows how the reticle was used to estimate the range to an aircraft, in this case an HE 177, which measured 50 feet between the wingtip and fuselage centerline. The gunner needed to know the type of airplane and its dimensions in order to estimate the range to the target.

Figure 3: Manual Example of Range Measurement.

Figure 4: Manual Example of Range Measurement (Source).

Lead Determination Example

Figure 5 illustrates the reasoning behind calling the rings "100 knot" and "200 knot". If we assume that the target and projectile move at constant speed, the amount of deflection θ required to hit the target is constant. We can show that θ is constant by observing that for any time of projectile flight, the ranges covered by target and the projectile form similar triangles. This means that the the deflection angle θ will be constant.

This means we can construct reticle rings that will provide the gunner with the amount of deflection required to hit a moving target of a given speed.

Figure M: Demonstration of Rationale Behind 200 Knot and 100 Knot Rings.

Figure 5: Demonstration of Rationale Behind 100 Knot and 200 Knot Rings.

Conclusion

The effective use of these rings requires a very well trained gunner. The gunner must be able to:

  • estimate the target speed
  • identify the dimensions of the specific target types
  • compensate for the projectiles drop under gravity (something I have not covered in this post)

During WW2, the gunners spent much time practicing with the reticle so that using the reticle became a reflex action, i.e. done without conscious thought.

Posted in Ballistics, Military History | Comments Off on WW2 Reticles

Gift Wrapping Math

Posted on 3-November-2015 by mathscinotes

Quote of the Day

A lawyer is either a social engineer or a parasite on society. A social engineer is a highly skilled, perceptive, sensitive lawyer who [understands] the Constitution of the United States and [knows] how to explore its uses in the solving of problems of local communities and in bettering conditions of the underprivileged.

— Charles Hamilton Houston, NAACP Litigation Director.

Introduction

Figure 1: Examples of Christmas Gift Wrapping.

Figure 1: Examples of Christmas Gift
Wrapping (Source).

My sister works as an event planner/wedding planner. She wrote me an email today with the following gift wrapping question.

I need to wrap 375 boxes. 16x14x6. The wrapping paper is 30 inches wide. I say I need about 1100 feet. People are telling me I only need half that.

She then referred me to a website that gives a formula for the amount of wrapping paper for a given prismatic box. Her team was not familiar with evaluating formulas, and she asked me if I could do the calculations.

This formula was supposedly created by Sara Santos, a well-known applied mathematician. However, when I looked at the equation (Equation 1), I knew something was wrong. I am sure Sara derived it correctly – the problem is one of getting it on the page correctly. I still remember a conversation I had years ago with a typesetter who complained about setting mathematical type because it was "fussy" – he charged extra for it because it had to be done EXACTLY right.

Here is a video that provides a good background on the formula and how it is used.

For those who don't want the joy of computing, I include an Excel spreadsheet here to help you. No macros, just a simple calculator.

Background

Equation as Stated

Equation 1 is supposed to give the area of wrapping paper required for a single box.

Eq. 1 GiftWrap

The presence of the equality is what confused me. However, I have figured out what is going on and the equality should not be there. The first term \frac{1}{2}\cdot {{\left( {d+2\cdot h+w} \right)}^{2}}  is for rectangular boxes and the second term \displaystyle 2\cdot {{\left( {w+h} \right)}^{2}} is for square boxes – the square box term is a special case of the rectangular term with d = w.

The gift wrapping formula does not care which numbers are assigned to d, w, or h. If you have two dimensions that are equal, then make those d and w because you have a box with a square side.

Analysis

Derivation

The derivation is straightforward given two cases: (1) square box, and (2) rectangular box. Figure 3 provides my drawings of these cases. The required wrapping paper area follows directly from these drawings.

DerivationSquare Rectangular
Figure 3(a): Square-Base Box. Figure 3(b): Rectangular-Base Box.

Now I understand that Equation 1 should not contain an equality. Instead, there are two separate cases covered. Really, the square-base box case is just a special case of the rectangular-box case.

Wrapping Paper Constraint

The web page did not mention that the derivation makes an assumption that the wrapping paper is wide enough to support this approach to wrapping the box. To wrap a box this way requires that the wrapping paper meet the following width constraint (Equation 2).

Eq. 2 \displaystyle {{w}_{{Wrap}}}=\sqrt{2}\cdot \left( {w+h} \right) square-base box
\displaystyle {{w}_{{Wrap}}}=\frac{{w+d+2\cdot h}}{{\sqrt{2}}} rectangular-base box

Solution to My Sister's Problem

My sister needed to know how much 30-inch wide wrapping paper to buy. Figure 4 shows my answer.

Figure 4: My Answer to My Sister's Problem.

Figure 4: My Answer to My Sister's Problem.

1100 feet of wrapping paper is a lot of paper.

Conclusion

I love to work geometric problems, and my sister gave me a practical one – everyone has to wrap packages. This problem also fits in nicely with my interests in origami and paper engineering.

Save

Save

Posted in Geometry, Paper Machines | 21 Comments

Index/Instrument Error Correction in Sextant Measurements

Posted on 31-October-2015 by mathscinotes

Quote of the Day

No scientific discovery is named after its original discoverer.

— Stigler's Law of Eponymy

Introduction

Figure 1: Illustration of Index Error.

Figure 1: Illustration of Index Error.

Sextants make measurements that are subject to systematic errors – all instruments are subject to systematic errors. Alas, much of my career has been spent on calibrating out systematic instrument errors. Even with all my efforts, residual systematic errors remain.

Figure 1 illustrates a common situation.
Ideally, an angle measurement from my sextant would exactly equal the actual altitude of the celestial object I am observing (green line). Unfortunately, there is always some error intrinsic to the sextant itself. In the case of a sextant, the errors are usually modeled as being composed as a fixed offset plus a angle offset that is a function of the angle measurement. This type of error modeling is common – I basically use the same error modeling approach with the angle settings on my compound miter saw.

In my opinion, the index and instrument errors are the easiest of all the altitude measurement errors to understand. In this post, I will define the terms and show how these errors are used.

Background

Error Model

The errors related to the sextant itself are usually modeled as shown in Equation 1. Some people also add a term called Personal Error (PE), which models errors introduced by the sextant operator. The PE usually not known, but can be determined. I will ignore it for this post.

Eq. 1 \displaystyle TE=I+IE

where

  • TE is the total sextant measurement error attributable to the instrument itself.
  • I is the fixed instrument error – it is intrinsic to the instrument as it was constructed. This error is documented by the sextant manufacturer on a calibration certificate shipped with the sextant. In Figure 1, I is represented by the gold-colored line. Figure 3 shows two sextant calibration certificates. As you can see in Figure 3, the fixed error varies with the angle being measured.
  • IE is the called the Index Error, which consists of the sum of a number of correctable (adjustable) instrument error sources that add an offset to any angle measured (see Figure 1). A properly calibrated sextant should have an IE of 0. However, it can be time-consuming to adjust this error to exactly 0°, so people often use a computation correction for small IEs. It is measured by observing the horizon, bringing the reflected and direct images of the horizon into coincidence, and then reading the sextant angle – which ideally should be zero, but usually isn't.

Definitions

Correctable Instrument Error Sources

An error source is said the be correctable when the navigator can adjust his sextant to eliminate them. Here are the key sources of error that generally can be removed by making adjustments to the instrument.

Error of perpendicularity
The index mirror must be perpendicular to the sextant frame, otherwise an error is introduced.
Side Error
The horizon glass must be perpendicular to the frame of the sextant, otherwise an error is introduced.
Index Error
The horizon glass must be parallel to the index mirror when the sextant is set on zero, otherwise an error is introduced.
Collimation Error
The telescope must be parallel to the frame, otherwise an error is introduced.

The sum of these errors is usually referred to as the index error – technically, this is not correct but it is common practice.

These sum of these errors results in a measurement that is biased by a fixed value. Figure 2 shows two views of a horizon with a sextant set to 0° – Figure 2(a) shows the horizon through a well-calibrated sextant, and Figure 2(b) shows the horizon through a sextant with a small index error.

NoIndexError FiniteIndexError
Figure 2(a): Sextant Horizon View with No Index Error. Sextant is reading 0° (Source). Figure 2(b): Sextant Horizon View with Index Error. Sextant is reading 0° (Source).

Fixed Instrument Error Sources

An instrument error source is uncorrectable when the sextant provides no means of adjusting the instrument to eliminate the error source. The sextant manufacturer must measure the sum of these errors and inform the navigator of the error values, which they do with a calibration certificate (see Figure 3). I see these sorts of certificates on instruments all the time – I jokingly call them birth certificates.

Graduation Error
Imperfections introduced in the manufacture of the graduations of the sextant's angle scale. As a side note, one of the most interesting books I have read is a history of graduating sextants and surveying instruments called The Divided Circle. I was amazed at how much history there is on the subject.
Prismatic Error
The error introduced when the planes of a mirror are not exactly parallel.
Centering Error
The error introduced when the index arm does not pivoted at the exact center of arc's curvature.

Figure 3 shows two examples of sextant calibration certificates. Note how the errors can be positive or negative.

TamayaCertificate NavalObservatory
Figure 3(a): Sextant Horizon View with No Index Error. Sextant is reading 0° (Source). Figure 3(b): Sextant Horizon View with Index Error. Sextant is reading 0° (Source).

Conclusion

The nature of the sextants instrument and index errors are such that they just end up being corrections directly applied to any measurement taken by a specific sextant – pretty straightforward.

Posted in Naval History, Navigation | 2 Comments

Refraction Error Correction in Sextant Measurements

Posted on 27-October-2015 by mathscinotes

Quote of the Day

We must teach our children to dream with their eyes open.

— Harry Edwards

Introduction

Figure 1: Illustration of Light Refraction Through the Atmosphere.

Figure 1: Illustration of Light Refraction
Through the Atmosphere (source).

Refraction is probably the most difficult to understand of all the altitude observation corrections. It is also the most difficult to estimate accurately because it depends so strongly on atmospheric conditions, particularly the rate of temperature variation with altitude (see lapse rate). I will derive in this post a commonly used expression for the refraction correction required for a celestial object with an altitude greater than or equal to 15°. The accuracy of this expression degrades significantly for objects below 15°.

Formulas exist for refraction correction on objects with altitudes lower than 15°, but they are complex and are subject to substantial uncertainties. Since my goal here is provide some insight into how refraction is modeled, I will keep things simple and assume that we are working with objects relatively high in the sky.

Background

Definitions

refraction
Refraction is the change in direction of propagation of a wave due to a change in its transmission medium (source).
index of refraction
The index of refraction is a parameter that relates the speed of light in the material to the speed of light in a vacuum, i.e. v=\frac{c}{n}, where c is the speed of light in a vacuum, n is the index of refraction, and v is the speed light in the material with index n. I discuss how the index of refraction varies with air's density in Appendix A.
angle of incidence (θ1 in Figure 2)
The angle of incidence is the angle between a ray incident on a surface and the line perpendicular to the surface at the point of incidence, called the normal (source).
angle of refraction (θ2 in Figure 2)
The angle formed by a refracted ray or wave and a line perpendicular to the refracting surface at the point of refraction (source).

Index of Refraction

Model

Figure 2 shows how we usually model refraction at the interface between two materials through which the light passes.

Figure M: Refraction Model (Wikipedia).

Figure 2: Refraction Model (Wikipedia).

Refraction effects are zero for celestial objects at our zenith and a maximum for objects on the horizon. Refraction causes celestial objects to have an apparent altitude that is higher than their true altitude. Thus, refraction corrections always involve subtracting a correction term from a measured altitude.

Snell's Law

Snell's law relates the angle of incidence to the angle of refraction, which is usually expressed as shown in Equation 1.

Eq. 1 \displaystyle \frac{{\sin {{\theta }_{1}}}}{{\sin {{\theta }_{2}}}}=\frac{{{{v}_{1}}}}{{{{v}_{2}}}}=\frac{{{{n}_{2}}}}{{{{n}_{1}}}}

where

  • θ1 is the light's angle of incidence.
  • θ2 is the light's angle of refraction.
  • v1 is the speed of light in material 1.
  • v2 is the speed of light in material 2.
  • n1 is the index of refraction of in material 1.
  • n2 is the index of refraction of in material 2.

Atmospheric Assumption

For celestial bodies with altitudes greater than 15°, we can model refraction with sufficient accuracy for navigational purposes by assuming that the atmosphere is series of flat layers with different densities. This assumption is not true for celestial objects with altitudes lower than 15° because the light from these objects will pass through atmospheric layers that are significantly curved.

To keep this discussion simple, I will only model refraction for objects with altitudes greater than 15°.

Analysis

Key Result

Eq. 2 \displaystyle R =\left( {{{n}_{{Air}}}-1} \right)\cdot \cot \left( \theta \right)=0.974\text{ arcmin}\cdot \text{cot}\left( \theta \right)=58\text{ arcsec}\cdot\text{cot}\left( \theta \right)

where

  • nAir is the index of refraction at the Earth's surface with a pressure of 760 mmHg and a temperature of 10 °C. It is usually assumed to have a mean value of 1.0002835. The reference temperature and pressure varies with the person doing the modeling. I often see 1010 millibars and 0 °C used as well.
  • θ is the altitude of the celestial object.
  • R is the angular measure of the amount of refraction.

Derivation

For a very detailed version of this discussion, see the excellent work in  "Textbook on Spherical Astronomy" by Smart, which is my favorite text on the subject of astronomical measurements. Fortunately, this section of the text on refraction is available in its Google Books preview on page 58 (at least it was when I looked).

We begin deriving Equation 2 by assuming the atmosphere is composed of layers of air with different refractive indexes as shown in Figure 3. We can relate the angle of incidence in space (z) to the angle of incidence on the ground (ζ) using the argument shown in Figure 4. The idea here is is to look at the angles of incidence and refraction for each layer. As is shown in the Figure 4, all the intermediate indexes of refraction can be eliminated, leaving only the index of refraction of space (n = 1) and at the observer. The angles of incidence and refraction will be converted to altitudes in Figure 5.

Figure 4: Key Relationship Between True and Observed Altitude.

Figure 4: Key Relationship Between True and Observed Altitude.

Given the relationship (yellow highlight) shown in Figure 4, we can now derive a commonly seen approximation for the refraction correction as a function of altitude (Figure 5).

Figure 5: Derivation of Equation 2.

Figure 5: Derivation of Equation 2.

Figure 5 shows how to derive Equation 2 and presents it in the two forms that it is usually seen – one for arcminutes and the other for arcseconds. As an example of the use of these expressions, I am including a link to a surveying text that uses the arcsecond form.

Conclusion

Equation 2 is a commonly seen expression for the effect of refraction on celestial objects with relatively high altitudes (>15°).  We can correct a sextant reading for refraction by subtracting the value generated by Equation 2 from the altitude reading we obtained from our sextant.

My next post will discuss index error, which is an error that is much simpler to describe than refraction.

Appendix A: Atmospheric Refraction Index

We need to have a model of the air's index of refraction in order to create an effective model refraction on sextant measurements. I will model the air's index of refraction using Equation 3.

Eq. 3 \displaystyle {{n}_{{Air}}}=1+2.9\cdot {{10}^{{-4}}}\cdot \frac{P}{{{{P}_{0}}}}\cdot \frac{{{{T}_{0}}}}{T}

where

  • P0 at the reference pressure, which is assumed to be 760 mmHg.
  • T0 is the reference temperature of T0 = 0°C. I see different references temperatures used.
  • 2.9E-4 is the index of refraction air at a pressure of 760 mmHg and temperature of 0 °C.

I show my derivation for Equation 3 in Figure 6.

Figure M: Air Index Linear Model Variation with Pressure and Temperature.

Figure 6: Air Index Linear Model Variation with Pressure and Temperature.

I can graph Equation 3 and compare it other models for air's index of refraction as shown in Figure 7. The agreement is good.

Figure M: Comparison of the Linear Model to Model Found on Web.

Figure 7: Comparison of the Linear Model to Model Found on Web.

Web index reference
Posted in Astronomy, Naval History, Navigation | 3 Comments

Correcting for Sextant Parallax Error

Posted on 24-October-2015 by mathscinotes

Quote of the Day

If you think adventure is dangerous, try routine; it is lethal.

— Paulo Coelho

Introduction

Figure 1: Common Example of Parallax.

Figure 1: Common Example of Parallax (source).

Navigators use the altitudes of solar system objects to assist them with determining their positions. The most commonly used solar system objects are the Sun, Moon, Venus, and Mars. There is a small error caused by the fact that navigators are making their sextant measurements from the surface of the ocean and not from the center of the Earth, which is the reference point used by nautical almanacs. Parallax has no practical significance when measuring the positions of stars because they are so far away relative to the radius of the Earth.

Solar system objects have been used by navigators for other purposes. For example, Galileo recommended using the Jovian moons to determine Greenwich Mean Time (GMT). Before Harrison and the marine chronometer, this was one of the few means by which navigators could determine GMT at far distant points around the world.

Background

parallax
Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines.
Parallax Error (P)
The very small angle (fraction of an arcmin) between the observer and the center of the Earth (see Figure 3), as seen from the celestial object of interest (Sun, Moon, Venus, or Mars). This angle is maximum when the celestial object is low on the horizon and nil when it is overhead.
Horizontal Parallax (HP)
This is the maximum possible parallax at a specific time for a given lunar distance from Earth. It is computed assuming that the center of the Moon is on the horizon. Almanac's list HP at hourly intervals. The actual parallax reduces as the altitude of Moon moves above the horizon.

Analysis

Parallax Equation

Equation 1 shows a commonly used expression for the parallax error present in a celestial body measurement.

Eq. 1 \displaystyle P=\arcsin \left( {\sin \left( {{{H}_{P}}} \right)\cdot \cos \left( {{{H}_{T}}} \right)} \right)

where

  • P is angle of parallax correction.
  • HP is maximum parallax, which occurs when the celestial object is on the horizon.
  • Ha is apparent altitude of the celestial object.

The value of HP is given by Equation 2.

Eq. 2 \displaystyle H_P=\arcsin \left( {\frac{{{{r}_{E}}}}{R}} \right)

where

  • rE is radius of the Earth.
  • R is distance from the center of the Earth to the center of the celestial body.

The closest navigational celestial object is the Moon and it has the greatest parallax variation. Stars are so far away that we cannot measure their parallax using navigational instruments – stellar parallax does exist, however. For more on stellar parallax and its use in determining the distances to nearby stars, see this Wikipedia article.

The dynamic range of the Moon's HP is fairly limited, as I will show here.  The radius of the Earth is constant, but the distance between the Earth and the Moon varies because the Moon's orbit is an ellipse. The following  Wikipedia quote gives the range of variation.

The actual distance varies over the course of the orbit of the moon, from 363,104 km (225,622 mi) at the perigee and 405,696 km (252,088 mi) at apogee, resulting in a differential range of 42,592 km (26,465 mi).

Figure 2 shows that range of values that the Moon's HP can take as its distance from Earth varies.

Figure 3: Range of HP.

Figure 2: Range of Lunar HP Values.

Derivation

Figure 3 shows how you can derive the Equation 1 using the Law of Sines. The derivation assume that their is negligible difference in the distance to the celestial object from the observer (R′) and the center of the Earth (R).

Figure 2: Tutorial Image on Parallax (<a href="https://mathscinotes.com/wp-content/uploads/2015/10/Chapter2-1.pdf" target="_blank">source</a>).

Figure 3: Tutorial Image on Parallax (source).

As shown in Equation 3, Equation 1 is often simplified to P\doteq \cos \left( {{{H}_{T}}} \right)\cdot {{H}_{P}} by assuming that \sin \left( {{{H}_{P}}} \right)\doteq {{H}_{P}} for HP small.

Eq. 3 P\doteq \cos \left( {{{H}_{T}}} \right)\cdot {{H}_{P}}

Conclusion

Now that I understand how to correct for parallax errors, I will move on to index and refraction errors. Once I have reviewed all the errors, I will then show how to apply them using several examples.

Posted in Astronomy, Naval History, Navigation | 1 Comment