Stud Length Requirements

Posted on 9-May-2016 by mathscinotes

Quote of the Day

The final test of a leader is that he leaves behind him in other men, the conviction and the will to carry on.

— Walter Lippmann. Every good manager plans for succession.

Introduction

Figure M: Typical Wall Framing.

Figure 1: Typical Wall Framing (Source).

I usually use 92-5/8" studs when I am building standard 8' tall walls –  they sit sit in piles at my nearby lumber yards next to piles of 8' long studs. I recently saw (example) that there are three pre-cut sizes available:

  • 92-5/8" : used for 8' ceiling heights
  • 104-5/8" : used for 9' ceiling heights
  •  116-5/8": used for 10' ceiling heights

In this post, I thought I would take a look at these pre-cut sizes to (1) learn why these values are used,  and (2) to decide if any of them would be appropriate for my building activities.

I recently saw one well-known carpenter make a plea for a 94-1/8" long studs. I will also examine this length to see why it may make sense in some circumstances.

Figure 1 shows how a typical wall is framed. Standard building practice today is to use 2–2"x4"s for the top plate and a single 2"x4" for the lower plate. Green building practices are now being introduced that use a single 2"x4" for the top plate and single 2"x4" for the bottom plate. I will show how building practices dictate the particular pre-cut length.

Background

Definitions

Stud
Stud is shortening of the term "wall stud", which is a vertical framing member in a building's wall of smaller cross section than a post. They are a fundamental element in building framing (Source).
In North America, studs are usually specified by their nominal width and thickness (e.g. 2"x4", 2"x6"). The actual dimensions differ from the nominal dimensions (e.g. 1.5"x3.5",1.5"x5.5"). It is a weird way of specifying dimensions, but that is what we do.
Plate
Plate is a shortening of the term "wall plate". A wall plate is a horizontal, structural, load-bearing member in wooden building framing (Source).
Pre-Cut Length
Historically, wall studs were available standard lengths given in an integral number of feet (e.g. 8', 9', 10'). Similarly, home ceiling heights are often given in even number of feet (e.g. 8', 9', 10'). Because the ceiling height is determined by the combined height of studs and plates, achieving a specific ceiling height (e.g. 8') means cutting every standard length stud on site.
Rather than cut each stud on site, it is more efficient to simple purchase the studs pre-cut (i.e. cut at the factory) to the proper length for a ceiling height that is an integral  number of feet.
Subfloor
A rough floor laid on floor joists and serving as a base for the finish floor. It is usually made of plywood and/or particle board.
Strapping
In construction, strapping consists of thin strips of wood or other material used to level or raise surfaces of another material such as to prevent dampness, to make space for insulation, or to level and resurface ceilings or walls (Source).
green building
Green building refers to both a structure and the using of processes that are environmentally responsible and resource-efficient throughout a building's life-cycle: from siting to design, construction, operation, maintenance, renovation, and demolition (Source).

Optional Strapping Use

My current house has wallboard that is secured directly to the ceiling joists. When the ceiling is lit from the side, it is obvious that the wallboard is not flat because I can see light and dark area on the ceiling – shadows caused by the sagging of the wallboard between the studs. I do not like that look, which makes me very interested in doing everything I can to ensure that my ceiling is flat.

One common way to ensure that your ceiling is flat is to attach 1"x3" (actual dimensions: 0.75"x2.5") strapping to the ceiling joists. This approach allows you to reduce the distance between fasteners on the ceiling and make the sagging imperceptibly small. Figure 2 shows an example of ceiling joists covered with strapping.

Figure M: Strapping Used to Provide Support for Sheetrock and Wiring Pass-Throughs.

Figure 2: Strapping Used to Provide Support for Ceiling Wallboard and a Wiring Channel.

Ceiling Height Options

While the most common ceiling height in North America is nominally 8', you will occasionally see ceiling heights of 9' and 10'. Because there are three commonly used ceiling heights, I see three pre-cut lengths.

Fun Wallboard Facts

  • In my region, ceiling wallboard needs to be 5/8" thick because it has a fire rating.
  • Wall board is available in widths of 48" and 54".

    48" and 54" wallboard works perfectly for 8'  tall and 9'  tall ceilings, respectively. I have not seen a consensus on the best way to place wallboard on a 10' ceiling. Most of the pros I have seen use horizontally placed standard width wallboard panels plus a partial piece placed near the floor.

  • Small gaps along the floor are easily hidden with molding and make installation easier.

    Standard practice allows a 1/4" to 1/2" gap between the bottom wallboard edge and the floor. This gap allows you to place wallboard without any need to force a piece into place.

Analysis

Objective

I want to summarize my research on the stud lengths and wallboard widths I need as a function of ceiling height and building practice (i.e. standard or green). For those who like to deal with the raw spreadsheet, you can download it here.

8' Ceilings

The most common ceiling height in North America is 8', and Figure 3 shows how a wall for this ceiling height would be framed. The actual height of an 8' ceiling is 8' 1/2" =  97.125"-0.625", measured from the subfloor. Since the finished floor height is ~0.5" thick, the final height is ~8'.

Standard8ft Strappped8ft
Figure 3(a): 8' Wall/Wallboard Height (No Strapping). Figure 3(b): 8' Wall/Wallboard Height (With Strapping).

9' Ceilings

Another common ceiling height in North America is 9', and Figure 4 shows how a wall for this ceiling height would be framed. The actual height of a 9' ceiling is 9' 1/2" =  109.125"-0.625", measured from the subfloor. Since the finished floor height is ~0.5" thick, the final height is ~9'.

Standard9ft Strapped9ftCeiling
Figure 4(a): 9' Wall/Wallboard Height (No Strapping). Figure 4(b): 9' Wall/Wallboard Height (With Strapping).

10' Ceilings

Occasionally, you see a  ceiling height in North America of 10', and Figure 4 shows how a wall for this ceiling height would be framed. The actual height of an 10' ceiling is 10' 1/2" =  121.125"-0.625", measured from the subfloor. Since the finished floor height is ~0.5" thick, the final height is ~10'.

 
Figure 5(a): 10' Wall/Wallboard Height (No Strapping). Figure 5(b): 10' Wall/Wallboard Height (With Strapping).

An 8' Green Building Example

I read a Q&A column that mentioned the use of green building practices (e.g. one 2"x4" for the top plate) will require a modification of the pre-cut stud lengths. Specifically, the post mentioned a pre-cut stud length 94 1/8" for an 8' ceiling. Figure 6 shows that he is correct.

Green8
Figure 6: 8' Wall/Wallboard Height Using Green Building Practices (No Strapping).

Conclusion

While this post only contains simple bookkeeping math, it shows how buying pre-cut lengths can save you the time and expense of having to trim every stud used when building a house.

Posted in Construction | 6 Comments

Interior Non-Load Bearing Wall Construction Methods

Posted on 6-May-2016 by mathscinotes

Quote of the Day

When people thought the earth was flat, they were wrong. When people thought the earth was spherical, they were wrong. But if you think that thinking the earth is spherical is just as wrong as thinking the earth is flat, then your view is wronger than both of them put together.

- Isaac Asimov. I completely agree – you can be wrong and you can be REALLY wrong.

Introduction

FIgure 1: Putting a Non-Bearing Wall Into Position.

Fig. 1: Wall Lift Into Position (Source).

I am an hobbyist carpenter who is about to do some wall building. I was reading a forum discussion on the best way to build a non-load bearing, interior wall as part of a remodeling project. The forum conversation was very thoughtful, but no real conclusions were reached. This makes sense because each forum contributor was making assumptions about the construction conditions – the construction conditions dictate which method would be "best."

In fact, I have used every construction method mentioned in the forum discussion because different circumstances dictated different methods. I thought I would summarize the discussion here and provide my opinions on what works best – I have tried them all.

The right answer depends on the situation. You need to answer questions like:

  • Do I have room to build a wall on the floor and tip it up?
  • Are the ceiling and floor parallel?
  • Are their soffits to work around?
  • Am I being thrifty and need to minimize the amount of material I am using?

Options

The tradeoffs are most easily described in list form. Again, the forum discussion was excellent on the pros and cons of each approach. At the bottom of the post, I also verify the geometric argument of one builder who "tips" walls into place with the help of a 16 pound "persuader" sledge hammer.

  1. Stick Frame in Place.
    • Method: Put in the plates and studs one-at-a time, and toe-nail it all together (Example).
    • Advantage
      • Can handle any situation, like soffits and non-parallel floor and ceiling.
      • Thrifty with material.
      • Easily done alone.
    • Disadvantage
      • Slow because each piece may need to be cut to size. Some pros dispute this statement, and I am sure they get very fast because they do this sort of thing all the time. However, I am not that proficient with this approach.
      • Toenailing is a pain without a nail gun.
    • I have had to use this method more than you might you think because I have not had room to assemble the wall on the floor.
    • It is my method of last resort.
  2. Build Flat On the Floor and Tip Into Position.
    • Method: Build flat on the floor with studs nailed in from the bottom of the plates. Tip the wall into position.
    • Advantage
      • When applicable, it is the fastest wall framing method available.
      • Can be used when floor and ceiling are not parallel, but you will then need to shim.
      • Thrifty with material.
    • Disadvantage
      • Requires room on the floor to build the wall.
      • The wall is very heavy so it requires a helper or a hand truck (visit our site for these) to lift it.
      • If you do not make the wall slightly short (see here), tipping into position requires lots of pounding. I have seen people pound so much that they cause cracking in the floor above.
      • If you do make the wall slightly short, you need to shim so the wall remains firmly in position. This is what I do when I have a very simple situation.
  3. Frame the Opening and Lift the Wall Into Place.
    • Method: Mount plates on the ceiling and/or floor first, then lift the wall into position (Figure 1 illustrates this case).
    • Advantage
      • After your plates are in place, assembly is simple – just set your wall into frame created by the top and bottom plates.
      • You may be able to get by with just putting in one of the plates.
      • Wall is a bit smaller than #2 and you may be able to build it in another room and move it (by holding vertically) to where you need it.
      • It is quick compared to #1, not as fast as #2.
    • Disadvantage
      • More assembly steps because you have to install the plates and build the wall.
      • Usually requires a helper because the wall is heavy.
      • The ceiling and floor should be fairly parallel. If they are not parallel, you will have shimming to do.
      • Basements often suffer from non-parallel floors and ceilings.
      • Uses more material.
    • I use this approach when I cannot do #2.

How Short To Make A Wall For Easy Tip Up

You only need to trim the stud length by 1/16 inch for standard 8' tall walls. Figure 2 illustrates why trimming is necessary. The diagonal distance from the bottom plate to the top plate is 1/16" longer than the wall height will make tipping the wall into place impossible – at least without a bunch of pounding, which can cause damage.

Of course, cutting the wall short means it likely will need some shimming to mount securely. Some carpenters cut down by more than 1/16 inch so that they can use larger shims.

Figure 2: Diagonal Length of a Stud Wall.

Figure 2: Diagonal Length of a Stud Wall.

Figure 3 shows the algebra.

Figure 3: Calculation of the Diagonal Height of a Standard Wall.

Figure 3: Calculation of the Diagonal Height of a Standard Wall.

Posted in Construction | Comments Off on Interior Non-Load Bearing Wall Construction Methods

All In A Day's Work on Puget Sound

Posted on 5-May-2016 by mathscinotes

Quote of the Day

Give me six hours to chop down a tree and I will spend the first four sharpening the axe.

— Abraham Lincoln. This quote reminds me how important preparation is to the success of any endeavor.

I was cleaning out my basement storage area when I came across some old photos from when I worked on sonar systems back in the early '90s. I thought I would share some of the photos with you. I have been fortunate in that my career has offered me many opportunities to do different things – including working on Puget Sound, which was fantastic! I spent three years traveling out to Seattle to test undersea systems, and I grew to love the Olympic Peninsula area.

I was the only person on the boat who had a camera, so you see few photos of me.

MV_Response SkipperBringingUsIn

 M/V Response: My Office

The Response was an old barge that had been upgraded with an engine. It was  used for sonar testing. We would anchor out in the sound and then ping at various test targets. When the engine was running, the noise and vibration were incredible.

 The Skipper

The skipper would bring us in to work in the morning on a skiff. Some mornings this went better than others – weather sometimes made the boat transfer more difficult. He had many great sea stories, but I do not think I could tell most of them here.
SonarHead TestDirector

The Sonar Unit Under Test

I spent three years working on this sonar head (copper-colored section). I consider it my best piece of work. The black device above the sonar head is a microphone that we used to synchronize our test gear with the emission of a sound pulse.

 The Test Director

He was responsible for all test operations on the boat. He was a very capable organizer. Preparing for test operations on the water is a critical. If you are missing anything, you lose test time, and test time costs money.
SkipperIssuingInstructions PayingOutCable

 Safety First

The skipper kept an eye on all operations to make sure that nobody got hurt. It is very easy to get yourself hurt on the water.

 Deployment

As the sonar head was deployed, we needed to make sure that all the cables moved freely. After a bit, we worked like a well-oiled machine.
DeployingTheSensor CrushedCable

 Over The Side

The sonar head was mounted on a steerable platform that allowed us to point it at various targets. The boom was used to get some separation between the sonar test unit and the boat.

 Effect of Pressure on Cable

Our testing was performed about 200 feet  underwater. While not that deep, you can see that the pressure squeezed the cables pretty hard.
RotatingSonarTestSystem FinishingTheDay

Another View of the Sonar

I recently received a call about this sonar head from an engineer who was tasked with resurrecting the program because this technology was needed for a new application. It was fun recalling all the little details on the project. It was my first use of large FPGA/CPLD technology. I was amazed at the flexibility programmable hardware provided.

 Going Home After a Long Day At Work

We worked long hours everyday, but still would return to our hotel rooms. We would get at most 4 hours of sleep and then be back at it the following day. It was hard work, but I look back on these days fondly.

I have shown the photo below before, but I should mention the rest of the team. These were very good people, and I remain in contact with a number of them to this day.

The Class Photo

Posted in Personal | Comments Off on All In A Day's Work on Puget Sound

Simple Chip Cost Model Using Mathcad and Excel

Posted on 4-May-2016 by mathscinotes

Quote of the Day

I never haggled with the president, I swallowed the little things so that I could go to bat on the big ones. I never handled a matter apologetically, and I was never contentious.

— George Marshall on his relationship with FDR.

Introduction

Figure 1: Typical Integrated Circuit (Wikipedia).

Figure 1: Typical Integrated Circuit (Wikipedia).

I recently was asked by some staff members to describe the key variables that drive integrated circuit costs. To answer the question, I decided that the best teaching approach was to prepare a spreadsheet that includes all the critical parameters along with some computational examples – people can try different parameters and determine the critical cost sensitivities on their own.

The cost model I use is one that I saw in a lecture years ago, and it still gives me useful results. I originally captured the model in Mathcad. Because my audience has expanded a bit, I decided to include an Excel version because more people are familiar with Excel than Mathcad. For complex work, I always initially capture my models in Mathcad because it is much easier to validate an algorithm in Mathcad than Excel (e.g. automatic unit conversion is a lifesaver). I can then use Mathcad to troubleshoot my Excel results.

There really is no analysis in this post. I just captured an existing model in two different formats. If you want background on the integrated circuit cost model, see the following documents:

This Excel workbook is in my standard template for calculators. My Mathcad worksheet is a bit more free-form. Both are in this zip file.

Posted in Electronics | Comments Off on Simple Chip Cost Model Using Mathcad and Excel

Use of AC Voltage Provides Corrosion Resistance

Posted on 3-May-2016 by mathscinotes

Quote of the Day

I will either find a way or make one.

— Hannibal. I have worked with people that were going accomplish a task no matter what. I refer to these folks as a force of nature.

Figure 1: Corroded F-Connectors.

Figure 1: Corroded F-Connectors (Source).

Some of the most vexing problems I have dealt with in my career are related to connector corrosion problems. While corrosion can create a hard failure (easy to find),  more often it creates an intermittent failure (hard to find). Intermittent failures can be very expensive to find. For example, the automotive industry spent years dealing with intermittent connector problems that were caused by fretting.

There are a number of ways to deal with corrosion:

  • Seal the connector to keep moisture out.

    This is hard to do. We frequently required customers to use caulking to seal cable entry points, but overtime the caulking fails. It is also very difficult to ensure that you have adequately sealed the connection.

  • Use gold coatings to ensure the connection does not corrode.

    This is commonly done, but it must be done properly. Many folks try to get by using a gold flash costing, i.e. 2 μinch to 10 μinch thickness. I have found gold flash to be useless in an outdoor telecommunications product. I only approve connectors for outdoor use with 30+ μinches of gold.

  • Use anti-oxidation lubricants/pastes to ensure keep the connection from corroding.

    These certainly can help. For example, you frequently see pastes in home electrical wiring where aluminum and copper wires meet. I have repaired numerous electrical "failures" in my neighborhood by applying anti-oxidant pastes to my neighbors' aluminum service entry cable. For example, half of one neighbor's house would occasionally lose power. It turns out the service entry contact for that phase had corroded.

I had a discussion with a coaxial cable expert last week who mentioned that the cable industry used AC power voltages to prevent corrosion on coaxial cables. I had not thought about this before, but the cable folks have reported that the copper core and aluminum shield in the cable undergo galvanic corrosion in moist environment (Figure 1). Using an AC voltage will eliminate the ion exchange that occurs from having a fixed bias voltage (link). In the case of the cable industry, they pass power over coaxial cable using 60 V or 90 V AC signals.

Corrosion reactions are typically reversible based on the applied voltage. When you apply a DC voltage, the ions (usually metals and salts) continually go in the same direction, which means their effects accumulate. This is why we plate metal with DC voltage – never AC. With respect to corrosion, we do not want any reaction products to accumulate. With AC voltage, the corrosion reaction spends an equal amount of time going one direction as the other. Thus, if we do something bad on one half-cycle of voltage, we will undue the damage on the next half-cycle.

Back when I worked on US Navy systems, I would often see AC voltages being used on connectors where DC would have been more convenient for the designers. I asked an old-timer why and he told me that using AC reduced the effects of saltwater corrosion.

I must admit that the effects of saltwater corrosion were amazing. In one case, we had an undersea sensor that was anchored to the sea bottom. The anchor chain we used was made of  thick steel (~1 inch diameter) coated with a protective paint that was used on sea-based oil rigs to stop corrosion. During the deployment of the sensor, the anchor chain was nicked and the paint was removed from this tiny location. When we came back six months later, the chain was severed by corrosion at the nick.

Posted in Electronics | 3 Comments

Why Positive Ground in Telcom Power Systems?

Posted on 2-May-2016 by mathscinotes

Quote of the Day

Keep the company of those who seek the truth, run from those who have found it.

— Vaclav Havel

Introduction

Figure 1: Positive Ground Reduces Corrosion of Moist Telcom Wiring.

Figure 1: Positive Ground Reduces Corrosion of
Moist Telcom Wiring.

I have been asked a number of times recently why telecommunications systems are powered using negative potentials like -48 V. The rationale behind this choice has to do with how wires corrode in moist environments.

Today, we use wires with excellent insulation and these moisture-induced corrosion arguments carry less weight.  Back in the old days, however, telecom wiring had cloth-based insulation that would get wet – along with the copper wire. When you mix water and electric fields, you get electrochemical action (i.e. electroplating). While the early systems were configured with negative grounds (i.e. ground as the lowest potential in the system), it was soon discovered that these systems suffered from corrosion. Positive ground (i.e. ground is the highest potential in the system) voltages were found to be effective in combating this corrosion.

Figure 1 illustrates what happens with respect to a standard land-line phone, which consists of a wire pair called tip and ring. If the wire pair becomes moist, the E-fields are directed such that copper will move from the local ground rod to the copper wire. The copper wire is preserved, while an insignificant percentage of copper is lost from the relatively large amount of copper in the ground rod. If either of the wires had a positive potential relative to the ground rod, the positive potential wire would lose copper to the ground rod and would eventually fail.

If you want more details on the subject, see this tutorial.

Posted in Electronics, Telephones | 4 Comments

Quick Look at Recent Meteor Events

Posted on 25-April-2016 by mathscinotes

Quote of the Day

Great things are not done by impulse, but by a series of small things brought together.

— Vincent van Gogh. Two of my three brothers that are football coaches. They regularly talk about breaking a complex athletic skill down into its component parts, master the skills associated with each component, then pull all the individual skills together. The same is true for all human skill building.

Introduction

Figure 1: NASA Map of Boliede Events.

Figure 1: NASA Map of Bolide Events (Source).

I filter my news feed for anything related to astronomy, and I have been seeing a number of articles recently about large meteor explosions (called fireballs or bolides) in the atmosphere (example, example). These events are not occurring more frequently than before –  we now have the technology to discover these explosions. The recent spate of bolide burst reports has caused me to  become curious about how frequently these impacts are occurring. It turns out that NASA has an excellent set of web pages on this topic. There have been quite a few large meteor explosions, with the largest being the Chelyabinsk burst in 2013 (blog post on topic). As you can see in Figure 1, the cataloged meteor events are fairly even spread across the Earth.

My plan here is to do some analysis of the NASA meteor data using R and Rstudio. I am scheduled to provide some in-service R training within the next couple of months, and I am looking for some good examples.  This post will summarize my histogram example.

While walking home from religious education one night in my youth, I saw a fireball that dropped fragments down over a nearby town (Appendix A shows a fireball similar to what I saw).

Background

Objective

I believe that R and Rstudio are the finest data analysis platform available today – I have tried a bunch – and they are free. In the past, I would analyze data using Mathcad or Excel, but today I use a combination of all three tools. While each of these tools has overlapping capabilities, they each also  have distinct areas of strength.

In the case of R, the ggplot2 graphics package gives it superior graphics capability to both Excel and Mathcad. My focus will be on using R and Rstudio transform, analyze and present the NASA data.

Definitions

Bolide
A bolide is an extremely bright meteor, especially one that explodes in the atmosphere. In astronomy, it refers to a fireball approximately as bright as the full moon, and it is generally considered a synonym of a fireball (Source).
Radiant Energy
The total luminous energy released by the fireball.
Impact Energy
The estimated total kinetic energy in the meteor. The meteor dissipates its energy in the form of light and by working against the air. NASA has established an empirical relationship between the radiant energy and the impact energy.
Equivalent Spherical Diameter
Small asteroids are usually irregularly shaped because they are not massive enough for gravity to pull them into a spherical shape. However, NASA often presents a diameter that assumes the asteroid was spherical. This helps the public visualize the size of the asteroid.

Impact Energy Versus Radiant Energy

Equation 1 shows the relationship that NASA uses to relate the radiant energy to the impact energy (Source).

Eq. 1 \displaystyle E=8.20508\cdot E_{{_{0}}}^{{0.885}}

where

  • E0 is the radiant energy of the meteor burst.
  • E is the projected impact energy of the meteor if it had hit the ground.

Chelyabinsk Meteor Computation Example

Figure 2 shows my rework of NASA's Chelyabinsk mass estimate using Mathcad (Appendix B has a copy of NASA's work). I will be repeating this analysis for NASA's list of recent bolide's using R. I rounded my mass estimate to two significant digits – just like NASA did.

Figure 2: Mathcad Version of NASA Chelyabinsk Calculations.

Figure 2: Mathcad Version of NASA Chelyabinsk Calculations.

I should comment that the Chelyabinsk meteor burst was unusually large. Quoting NASA,

This was an extraordinarily large fireball, the most energetic impact event recognized since the 1908 Tunguska blast in Russian Siberia.

The Tunguska meteor was in a different class than we have seen recently – it is estimated to have had a mass of ~500K tonne (tonne = metric ton = 1000 kg).

Analysis

Source Material

Data Cleanup/Transformation

The NASA data is available online here. I grabbed the data using Excel's web scraping tools. You can do similar things in R, but I find the Excel tools simple and quick – particularly with respect to web scraping.

Figure 3 is an excerpt of what I imported. The columns represent:

  • alt = altitude, units: km
  • vx = x-component of velocity, units: km/s
  • vy = y-component of velocity, units: km/s
  • vz = y-component of velocity, units: km/s
  • ER = radiant energy, units: Joules
Figure M: NASA Data I Imported.

Figure 3: Display of Six-Items from NASA Data I Imported (Using Head Command in R) .

Data Augmentation

I need to augment my data with the following information:

  • total velocity (name: v_T, units: km/s)

    I computed the Euclidian norm of the three velocity components.

  • impact energy (name: EIkt, units: kilotons)

    Computed using Equation 1.

  • equivalent spherical diameter (name: Dia, units: m)

    Computed using the formula for the volume of a sphere and NASA's density assumption of 3.6 gm/cm3.

Figure 4 shows the top 25 meteors in terms of mass with my augmentations.

Figure M: Top 25 Meteor Bursts Sorted By Meteor Mass.

Figure 4: Top 25 Meteor Bursts Sorted By Meteor Mass.

Visualization

Altitude of Maximum Brightness Distribution

Figure 5 shows my plot of the altitudes at which the maximum meteor brightness occurred.

Figure M: Distribution of Altitudes.

Figure 5: Distribution of Altitudes.

Velocity Distributions

Figure 6 shows the distribution of meteor velocities. Most of the meteors are in the 15 km/s range.

Figure M: Meteor Velocity Distribution.

Figure 6: Meteor Velocity Distribution.

Mass Distribution

Figure 7 shows the mass distribution of the meteors. Notice how Chelyabinsk is an outlier.

Figure M: Meteor Mass Distribution.

Figure 7: Meteor Mass Distribution.

Diameter Distribution

Figure 8 shows the estimated diameters of the meteors. Again, notice how Chelyabinsk is an outlier.

Figure M: Meteor Diameter Distribution.

Figure 8: Meteor Diameter Distribution.

Conclusion

Sorry for the long post. However, I think I have a good classroom example to use for demonstrating R and Rstudio to my coworkers.

Appendix A: Similar to Fireball I Saw in My Youth

Figure 9 shows a recent fireball that is similar to one I saw in my youth near Osseo MN.

Figure M: This Fireball was Spotted Over the UK.

Figure 9: This Fireball was Spotted Over the UK (Source).

 Appendix B: NASA's Chelyabinsk Calculations

Here are some NASA calculations for the Chelyabinsk meteor that I used as my model (Figure 10).

Figure M: NASA Chelyabinsk Calculations.

Figure 10: NASA Chelyabinsk Calculations.

Posted in Astronomy | 1 Comment

Nonlinear Piecewise Function for Stellar Luminosity vs Mass

Posted on 19-April-2016 by mathscinotes

Quote of the Day

Aboard ship there’s a danger in having too much of anything for then one is bound to have too little of something else.

— William Brown, Botanist, Mutiny on the Bounty (1962). Spacecraft designers say similar things. The same holds true for product design – everything must be in balance.

Introduction

Figure 1: Luminosity vs Mass Chart (Source).

Figure 1: Luminosity vs Mass Chart (Source).

A reader mentioned to me that the Wikipedia has a good entry on stellar luminosity versus stellar mass – I can confirm that the entry is a good one. I thought I would compare the empirical relationship shown in the Wikipedia with a couple of different data sets that I found on the web. I was motivated to perform this analysis because:  (1) I have been doing some reading on exoplanets, and luminosity is important when it comes to exoplanet temperature; and (2) I am presenting a seminar on Mathcad to our engineering staff, and this application provides me a nice demonstration on how to compute nonlinear piecewise functions.

One of my goals with these Mathcad training sessions is to show our staff that math can be well documented. This battle for clarity also occurs with software source.

Background

My first comparison data set is shown in Figure 1, which is from an instructor's web page at Case-Western Reserve. I also found a very poor quality copy of a diagram from a research paper by Svechnickov and Bessonova (1984). I digitized both figures as well as I could using Dagra.  I will graphically compare the Wikipedia's luminosity versus mass relationship with the two different sets of raw data. The digitized versions of this data are shown below.

Analysis

Mathcad Model Implementation

Figure 2 shows how I implemented the complex luminosity versus mass model using a simple Mathcad program.

Figure 2: Mathcad Implementation of Luminosity vs Mass Relationship.

Figure 2: Mathcad Implementation of Luminosity vs Mass Relationship.

Wikipedia reference

Wikipedia Model to Data Set  1 (Figure 1)

Figure 3 shows how well the Wikipedia model matches the data from Figure 1. The agreement looks pretty good. Note that the original data was transformed using a base-10 logarithm. For my purposes, I need to undo this transformation. This transform was accomplished efficiently using an array function.

Figure 3: Model vs Data Set 1.

Figure 3: Model vs Data Set 1.

Wikipedia Model to Data Set 2

Figure 4 shows how well the Wikipedia model matches the data from Svechnickov and Bessonova (1984). The agreement also looks pretty good.

Figure 4: Model vs Data Set 2.

Figure 4: Model vs Data Set 2.

Conclusion

My main focus here was a clean example to show my staff. This example demonstrated a few of things:

  • Capturing a relatively large data set from rough drawings.
  • Applying an element-by-element transformation to each element of an array.
  • Creating a program to compute a nonlinear piecewise function.
  • Plotting both the data and model for direct comparison.
Posted in Astronomy, General Mathematics | 1 Comment

Ice is Almost Out at My Cabin

Posted on 18-April-2016 by mathscinotes

Quote of the Day

You gain strength, courage and confidence by every experience in which you really stop to look fear in the face.

— Eleanor Roosevelt

I love this time of year in Minnesota. I have setup cameras around my cabin in northern Minnesota so that I can see what is going on up there. I have been watching my lake go through the entire thaw process – it is almost clear of ice (Figure 1). I plan on being up there quite a bit this summer.

The camera is facing north. The wind blew all the ice toward my cabin – so I have the last cabin with ice on its shore.

Figure 1: Remote Camera Showing the Ice Going OFF My Lake.

Figure 1: Remote Camera Showing the Ice Going Off of My Lake. My roll-in dock is in the lower right corner. The camera is mounted in a tree about 15 feet off of the ground.

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Exoplanet Revolution Period About a Dwarf Star

Posted on 15-April-2016 by mathscinotes

Quote of the Day

If you want something you've never had, you must be willing to do something you've never done.

— Thomas Jefferson

Introduction

Figure 1: Example of the Habitable Zone for 40 Eridani.

Figure 1: Example of the Habitable Zone for 40
Eridani, a nearby star smaller than our Sun
(Source).

I have been reading about the possibility  exoplanets around stars that are relatively near our solar system. I usually think about exoplanets about stars similar to the Sun. While news articles in the popular scientific press often refer our Sun as an “ordinary star”, in fact it is somewhat large compared to the general star population – about 70% of stars in our galaxy are red dwarfs, which are stars that have masses between 7.5% and 50% that of our Sun. Because red dwarfs are so numerous, some planetary astronomers are asking if life could form on exoplanets that orbit red dwarfs – the habitable zone will be closer to the star and the exoplanet's orbital period will be relatively short (e.g. Figure 1).

In this post, I look at the habitable zone around stars smaller than our Sun and the orbital radii and periods of potentially habitable exoplanets (i.e. having temperatures near that of Earth). I became curious about this subject when I read the following statement in an astronomy magazine (Source):

A planet orbiting a star like the sun would have to complete an orbit approximately once a year to be far enough away to maintain water on its surface. "If you’re orbiting around a low mass or dwarf star, you have to orbit about once a month, once every two months to receive the same amount of sunlight that we receive from the sun," Cowan said.

My plan here is to

  • Confirm the statement that an exoplanet around a low-mass star will need to have an orbital period of one or two months in order to receive enough light to have an Earth-like temperature.
  • Determine the nominal orbital radius for an exoplanet in a red dwarf star's habitable zone.

Background

Definitions

Equilibrium Temperature (AKA Effective Temperature)
The planetary equilibrium temperature is a theoretical temperature of a planet if it is assumed to be a black body being heated only by its parent star. (Source)
Luminosity
Luminosity is the total amount of energy emitted by a star, galaxy, or other astronomical object per unit time. It is related to the brightness, which is the luminosity of an object in a given spectral region. (Source)
Exoplanet
An exoplanet or extra-solar planet is a planet that orbits a star other than the Sun. (Source)
Red Dwarf
A red dwarf is a small and relatively cool star on the main sequence, either late K or M spectral type. Red dwarfs range in mass from a low of 0.075 solar masses (MSun) to about 0.50 MSun and have a surface temperature of less than 4,000 K (Source).

Key Equations

Orbital Period Equation

Equation 1 allows us to compute the orbital period of an exoplanet given the mass of the star it orbits and orbital radius. Equation 1 can be derived using Newton's and Kepler's laws (Appendix A).

Eq. 1 \displaystyle {{P}_{{\text{Planet}}}}=\frac{{2\cdot \pi \cdot R_{\text{Planet}}^{{\frac{3}{2}}}}}{{\sqrt{{G\cdot {{M}_{\text{Star}}}}}}}

where

  • MStar is the mass of the star about which the planet is revolving.
  • RPlanet is the orbital radius of the exoplanet.
  • G is universal gravitational constant.
  • PPlanet is the orbital period of the exoplanet.

Planet Equilibrium Temperature Equation

An exoplanet's average temperature (TEffective) is temperature at which the amount of energy absorbed equals the amount of energy radiated away. Equation 2 provides an approximate value for this temperature. Appendix B contains a derivation of this equation.

Eq. 2 \displaystyle {{T}_{\text{Effective}}}=\sqrt[4]{{\frac{{L_\text{Star}\cdot \left( {1-a} \right)}}{{16\cdot \pi \cdot \sigma \cdot \varepsilon \cdot {{R_\text{Planet}}^{2}}}}}}

where

Many analyses omit the emissivity term because it usually is close to 1.

Star Luminosity Versus Mass

There is an empirical relationship between the mass of a star and its luminosity – it is very rough and has errors for low mass stars. Equation 3 shows curve fit for the data stellar luminosity versus mass data shown in Appendix C. For my work here, I will assume that this relationship is close enough.

Eq. 3 \displaystyle {{L}_{\text{Star}}}={{L}_{\text{Sun}}}\cdot {{\left( {\frac{{{{M}_{\text{Star}}}}}{{{{M}_{\text{Sun}}}}}} \right)}^{{3.5}}}

where

  • LSuntar is the luminosity of the Sun.
  • LStar is the luminosity of the exoplanet's star.
  • MSun is mass of the Sun.

Analysis

Approach

My analysis approach is to:

  • Determine the luminosity of a typical red dwarf star using Equation 3.
  • I will then solve Equation 2 for the orbital radius of an exoplanet orbiting this red dwarf with the same effective temperature as the Earth.
  • I will then use Equation 1 to determine the orbital period of this planet.

Exoplanet Period

In Figure 2, I look at the orbital period for an exoplanet with the same equilibrium temperature as the Earth (-21 °C) about a red dwarf star with a mass of 0.3· MSun. I should note that the actual mean temperature of the Earth is ~16 °C. The temperature rise above the equilibrium temperature is caused by the greenhouse effect.

I wanted to test out Equation 3, so I used a nearby star (40 Eridani) as a test case. It was reasonably close (see green highlight).

Figure 2: Exoplanet Period Calculation.

Figure 2: Exoplanet Period Calculation.

My calculation shows that the planet would have an orbital period of about 2 months, which agrees with what I read in the magazine article. The orbital radius is only about 0.2 AU.

Using the framework established for Figure 2, I can plot the range of orbital radii and periods for exoplanets with Earth-like temperatures about a range of dwarf star masses (Figure 3).

Figure M: Orbital Radii and Periods for Exoplanets.

Figure 3: Orbital Radii and Periods for Dwarf-Star Exoplanets.

Conclusion

I have been able to confirm the statement that habitable exoplanets orbiting a red dwarf star will have orbital periods of ~2 months and radii of ~0.2 AU. Exoplanets with short orbital periods are easier to detect than long-period exoplanets, so we should have a good chance of finding these worlds.

The galaxy is full of red dwarf stars, with a number of these stars in our neighborhood (i.e. within 5 parsecs). I am sure that there  numerous challenges remain for life to form on these worlds, but there are lots of potential life-sustaining candidates. Some estimates claim as many as 60 billion habitable red dwarf planets are in the Milky Way.

The Wikipedia has a good article on this topic.

Appendix A: Formula for the Revolution Period of a Planet

Figure 4 shows how to derive Equation 2.  I also show all the various units and constants that I use in all of my other calculations for this post.

Figure M: Exoplanet Period Calculation.

Figure 4: Exoplanet Period Calculation.

Astronomical Unit Gravitational Constant Stefan-Boltzmann constant Solar Luminosity Earth Orbital Radius Solar Mass Bond Albedo Emissivity

Appendix B: Formula for the Effective Temperature of a Planet

Figure 5 shows how to derive Equation 2.

Figure M: Derviation of Formula for Effective Temperature of a Planet.

Figure 5: Derivation of Formula for Effective Temperature of a Planet.

Earth Equilibrium Temperature

Appendix C: Star Luminosity vs Mass.

Figure 6 shows a plot of stellar luminosity versus stellar mass. I will use the rough curve curve fit shown in Figure 5 as Equation 3.

Figure M: Star Mass-Luminosity Diagram.

Figure 6: Star Mass-Luminosity Diagram (Source).

Posted in Astronomy | 3 Comments