16-in Battleship Gun Ballistic Coefficient

Posted on 5-October-2013 by mathscinotes

Quote of the Day

Criticism may not be agreeable, but it is necessary. It fulfills the same function as pain in the human body. it calls attention to an unhealthy state of things.

— Winston Churchill

Introduction

Figure 1: Mk 13 HC (High Capacity) 16-inch Shells.

Figure 1: Mk 13 HC (High Capacity) 16-inch Shells.

I received the following question on one of my earlier blog posts:

Naval guns claim a range of up to 16 miles, and apparently do so with an initial velocity of only approx. 3000 fps [, which is similar to rifle bullet]. how is this possible[?]

You can view my answer on that blog post, but I started to think about alternative, but equivalent, ways to answer this question. From a numerical standpoint, one way to answer his question is by comparing the ballistic coefficient of a 16-inch projectile (Figure 1) to that of a standard firearm projectile. A projectile with a large ballistic coefficient is less affected by drag than a projectile with a smaller ballistic coefficient. We can use the the ballistic coefficient to compare the effect of drag on different projectiles. A 16-inch projectile goes so much farther than a rifle bullet because the drag on the 16-inch projectile is relatively small compared to its momentum. Ultimately, this is because mass increases by the cube of the projectile dimensions and drag increases by the square of the projectile dimensions. This means that larger projectiles tend to have higher ballistic coefficients and drag has less effect.

Let's put some numbers together ...

Background

The Wikipedia defines the ballistic coefficient as follows:

Ballistic Coefficient
The Ballistic Coefficient (BC) of a body is a measure of its ability to overcome air resistance in flight. It is inversely proportional to the negative acceleration — a high number indicates a low negative acceleration.

A projectile with a small deceleration due to atmospheric drag has a large BC. Projectiles with a large BC are less affected by drag and have performance closer to their performance in a vacuum.

While the Wikipedia definition is accurate as far as it goes, it does not allow you to compute the BC of a projectile. Equation 1 shows you how to compute the BC of any projectile.

Eq. 1 \displaystyle B{{C}_{\text{Projectile}}}\triangleq \frac{{{a}_{\text{ReferenceProjectile}}}}{{{a}_{\text{Projectile}}}}

where

  • aReferenceProjectile is the acceleration of a reference projectile (eg. G7).
  • aProjectile is the acceleration of the projectile we are interested in.

Analysis

I will compute the BC of the US Navy's 16-in, high-capacity (1900 pound) shell that was fired by the Iowa-class battleships using a couple of different methods.

  • Estimate the deceleration of the battleship projectile and compute Equation 1.
  • Apply a geometry-based formula.

Both methods give similar answers.

Method Based on Deceleration Estimate

To estimate the 16-inch projectile's deceleration. There is data available for this. Figure 2 shows a snippet from a US Navy range table for the 16-inch projectile (Source).

Figure 1: Snippet from 16-inch Projectile Range Table (1900 lb).

Figure 2: Snippet from 16-inch Projectile Range Table (1900 lb).

For my BC estimate calculation (Figure 3), I will use only the forward velocity component because that component is experiencing the bulk of the drag.

Figure 2: Ballistic Coefficient Calculation Using Deceleration Data.

Figure 3: Ballistic Coefficient Calculation Using Deceleration Data.

How to Compute Acceleration

Ballistic Coefficient Calculation Using Formula

Figure 4 shows the ballistic coefficient calculation using a geometric formula that compares the 16-inch projectile to a 30-06 projectile (Ball, M2, 152 grain, 0.308 in diameter).

Figure 4: Computing the BC of a 16-inch Projectile Using a Formula.

Figure 4: Computing the BC of a 16-inch Projectile Using a Formula.

Conclusion

Here is a quote that supports my analysis from "Modern Practical Ballistics" on page 115 (ISBN 0-96212776-3-7).

Modern long-range guns like the US Navy's 16-inch guns are able to attain ranges in excess of 25 miles using an initial elevation angle near 45 degrees; their shells reach an altitude of approximately seven miles. Although their muzzle velocities are generally only about 2600 to 2800 fps, their enormous size and weight -- in excess of 2000 pounds -- gives them ballistic coefficients of around 15, which are 30 to 50 times as great as small arms projectiles!

Posted in Ballistics | 5 Comments

Parameter Determination for Pejsa Velocity Model

Posted on 3-October-2013 by mathscinotes

Quote of the Day

Character, in the long run, is the decisive factor in the life of an individual and of nations alike.

— Theodore Roosevelt

Introduction

Figure 1: Picture of Pejsa Book Cover.

Figure 1: Picture of Pejsa Book Cover (Source).

I have had several people ask me questions about the Pejsa ballistic model (previous post) and I thought it would be useful to include some additional posts on the topic. In this post, I will discuss how the formula and parameters were determined for the velocity versus range formula for the range of velocities from 1400 feet per second to 4000 feet per second – sorry about the use of US customary units. Pejsa's formula were setup specifically to used ranges in yard, velocities in feet per second, and projectile drops in inches.

I do recommend that folks read through Pejsa's book (Figure 1) for themselves. Unfortunately, it is not an easy read and the formulas derived are not as general as I would like. The main issue with the formulas are that they are quite specific to US customary units – e.g. the fact that one yard equals three feet is used in final formulas. Also, some of the approximations assume a specific range of ballistic coefficients. However, the formulas are very useful because they provide accurate answers to common ballistic situations using simple algebraic formulas.

Background

For background, see this post. Pejsa provides different formulas for the variation of projectile velocities with respect to range based on the projectile's velocity. In this post, I will only address his formula for projectile velocities greater than 1400 feet per second. In later posts, I will work through the lower velocity ranges.

Analysis

Derivation

Figure 2 shows how one can derive Pejsa's velocity versus range formula for the velocity range of 1400 ft/s to 4000 ft/s.

Figure 1: Derivation of Pejsa Velocity Versus Range Equation.

Figure 2: Derivation of Pejsa Velocity Versus Range Equation.

Drag Coefficient

Parameter Determination

Figure 3 shows how to determine the single constant factor, K1, in the Pejsa equation.

Figure 2: Determine K1 Using Empirical Data.

Figure 3: Determine K1 Using Empirical Data.

Verification

One question involved how to generate a plot of the rate of velocity change with respect to distance -- this is a graph that appears in Hatcher's Notebook, a commonly cited ballistics reference. Figure 4 shows how I derived an expression for this curve. I will use modern data from Berger Bullets to make my comparison. I used Berger as a data reference rather than Hatcher because the Berger web site gives me raw numbers instead of a graph (i.e. quicker to work with).

Figure 3: Derive Rate of Velocity Change with Range.

Figure 4: Derive Rate of Velocity Change with Range.

Graph of Results

Figure 5 shows how I setup my graph.

Figure 4: Graph Setup.

Figure 5: Graph Setup.

Figure 6 shows a comparison of Pejsa's projectile formula (blue line) with the Berger web simulator (orange line).  As Pejsa states, his  formula provides good accuracy until the projectile velocity approaches 1400 feet per second, which occurs at a range of 1858 feet. Figure 6 also compares the rate of change of velocity with respect to distance and the data from Hatcher's Notebook for a similar projectile. Again, the results are similar until the projectile velocity nears 1400 feet per second.

Figure 6: Graph of Empirical and Theoretical Results.

Figure 6: Graph of Empirical and Theoretical Results.

Conclusion

As Pejsa states, the agreement of the velocity versus distance is pretty good for velocities above 1400 ft/s.

Appendix A: Hatcher's Deceleration per Foot Graph.

Figure 7 shows the projectile rate of deceleration per foot of travel from Hatcher's Notebook.

Figure M: Deceleration Graph from Hatcher's Notebook.

Figure 7: Deceleration Graph from Hatcher's Notebook.

Posted in Ballistics | 4 Comments

Image of a Submarine At Periscope Depth

Posted on 28-September-2013 by mathscinotes

If you have ever wondered what a submarine looks like at periscope depth, here is a photo of the USS Key West (SSN-722). Source is the Wikipedia.
800px-Periscope_Depth

Posted in Naval History, Personal | Comments Off on Image of a Submarine At Periscope Depth

My Favorite Animation Video

Posted on 28-September-2013 by mathscinotes

I love animation and I just stumbled upon an excerpt from a television show that I saw as a boy -- "An Adventure in Art." I found this video fascinating. As an added treat, one of the artists featured is Josua Meador, who drew the famous "id monster" in the movie Forbidden Planet.

http://www.youtube.com/watch?v=OAl9P4T12Rs&w=560&h=315

Posted in Personal | Comments Off on My Favorite Animation Video

Physical Interpretation of a Model Parameter

Posted on 26-September-2013 by mathscinotes

Introduction

I frequently get very specific questions on my posts. Normally, I simply reply directly to the question. One recent question required an answer that I thought might be interesting to a broader audience.

Background

Here is the question that I received on this post. This post contains my review of work by Pejsa that presents simple algebraic expressions for projectiles experiencing aerodynamic drag.

Has anyone provided a physical explanation for Pejsa’s “retardation coefficient”? I suspect I know what it actually represents, but I’m not sure.

I frequently need to include parameters in my system models, and I frequently end up staring at these parameters and asking "what do they represent physically?" This post will show the thought process that I go through to provide some physical meaning to these numbers.

Analysis

Pejsa derived the following expression for how a projectile's velocity varies with range (Equation 1).

Eq. 1 \displaystyle V={{V}_{0}}\cdot {{\left( 1-\frac{n\cdot r}{{{F}_{0}}} \right)}^{\frac{1}{n}}}

where

  • r is the projectile's range.
  • V is the projectile's velocity at range r
  • F0 is the retardation coefficient at the time of projectile launch.
  • n is a modeling parameter that is function of the velocity of the projectile.
  • V0 is the projectile's initial velocity.

The reader's question is focused on interpreting the meaning of F0. Figure 1 shows my derivation of a simple expression that shows me that F0 is analogous to the time constant of an RC circuit. F0 represents that distance at which the projectile's velocity has dropped to 1/e of its initial value.

Figure 1: Derivation of Simplified Expression for Velocity as a Function of Distance.

Figure 1: Derivation of Simplified Expression for Velocity as a Function of Distance.


Pejsa used a slightly different approach to interpreting F0, which I show in Figure 2.
Figure 2: Derivation of Simplified Expression for Velocity as a Function of Distance.

Figure 2: Derivation of Simplified Expression for Velocity as a Function of Distance.


Here is a quote from Pejsa's "Modern Practical Ballistics" that describes his interpretation.

As an example, for a bullet to have a retardation coefficient F equal to 2700 [feet] means that for 27 feet (1 percent of 2700), the bullet loses 1 percent (or retains 99 percent) of its remaining speed to air drag. It can be said that that the speed of the bullet "decays exponentially" at a rate of 1 percent for every 27 feet.

This statement is completely consistent with what I derived in Figure 1. In fact, a completely analogous statement can be made for RC circuits.

Conclusion

This is just a quick post to help a reader develop some physical insight into the meaning of a number. Engineers are always looking for insight into numbers. I am reminded of my favorite quote on computing and insight by Richard Hamming.

The purpose of computing is insight not numbers.

For me, I do not really understand a concept until I can describe how it behaves without having to resort to mathematics. A physical interpretation of the modeling parameters helps with developing intuition.

Posted in Ballistics, General Mathematics | 7 Comments

Measuring the Distance to the Moon and Photon Counting

Posted on 19-September-2013 by mathscinotes

Introduction

While looking up some information on the Moon, I ran into an interesting set of web pages that describes an experiment to measure the distance to the Moon with centimeter-level accuracy. This experiment sends a stream of laser pulses and determines the delay between when the pulses were sent and when a tiny fraction of them return to Earth. The project is called APOLLO, which stands for Apache Point Observatory Lunar Laser-ranging Operation.

On one of their pages, I encountered the following statement:

... the APD [Avalanche Photo-Diode] array enables simultaneous measurement of multiple photons returning from the moon. Throughput estimates for APOLLO predict a mean photon return rate as high as 5 photons per pulse.

The photons are reflecting off of a retro-reflector left on the Moon by the Apollo space project. I started to become curious about the small number of photons that were being returned with each pulse. I wondered if I could understand that number. This is another Fermi-type calculation. Let's dig in ...

Background

The APOLLO folks are trying to measure the distance between the Earth and Moon VERY accurately. Here is their approach:

  • Send a stream of pulses toward one of the various reflectors left on the moon by robots or people
  • Detect the returning photons and determine the time delay between the transmission and reception of the pulses.

While this sounds simple, achieving the required level of accuracy requires a tremendous effort. They are trying to model all sorts of subtle effects, like:

My analysis will simply look at the their photon budget to see if I understand (1) how many photons are being launched, and (2) where their photons are being lost in the measurement process.

Analysis

Number of Photons Transmitted Per Pulse

Figure 1 shows my calculations for the number of photons per pulse being launched toward the Moon.

Figure 1: Number of Photons Launched Toward the Moon.

Figure 1: Number of Photons Launched Toward the Moon.

Transmit Pulse Characteristics

As I thought, an enormous number of photons are being launched toward the Moon.

Reflected Photon Count

Figure 2 shows my calculations for the number of photons per pulse that will be reflected back toward Earth. I guessed at the percentage of photons that were lost because of absorption when the transmit pulse passed out of the Earth's atmosphere.

Figure 2: Count of the Photons Reflected from The Lunar Reflector.

Figure 2: Count of the Photons Reflected from The Lunar Reflector.

Retro-Reflector Characteristics

Only a very tiny fraction of the photons reflect back toward the Earth.

Received Photon Count

Figure 3 shows my calculations for the number of photons received at the detector. I had to make some guesses here for parameters like:

  • The percentage of the photons absorbed by passing through the atmosphere again (kAT= 48%)
  • The percentage of photons lost because of imperfect sensor alignment (kA=20%)
  • The percentage of photons lost because photons get lost in the receiver (kDE= 40%)

These are all guesses. However, my experience says that they are not unreasonable.

Figure 3: Calculation of the Number of Photons Received Per Pulse Transmitted.

Figure 3: Calculation of the Number of Photons Received Per Pulse Transmitted.

Earth Spot Size Air Transmission Efficiency Five Photon Reference

Given all the losses, just a handful of photons are available for measuring the time delay.

Conclusion

Making reasonable assumptions, I got a number very close to 5 photons received per pulse. I think I understand what they are doing.

Posted in Astronomy | Comments Off on Measuring the Distance to the Moon and Photon Counting

Calculating the Number of Observable Life-Supporting Planets

Posted on 11-September-2013 by mathscinotes

Introduction

I like thinking about the possibility of habitable exoplanets. There are many interesting questions that people can ask about exoplanets. Here are a few questions that are interesting to think about:

  1. How many intelligent civilizations are present in our galaxy?
  2. How likely is it that a particular exoplanet could support life?
  3. How many exoplanets are out there with detectable signs of life?

Question 1 is addressed by the Drake equation, which the Wikipedia covers nicely. I wrote about Question 2 in this post. I will discuss in this post an interesting article in Astrobiology magazine that addresses Question 3. The spirit of this work follows a similar path to that of Drake.

Background

The best background information I could find is presented by Sara Seager, whose research is the subject of the Astrobiology article.

Analysis

From the given number of planets examined, the article presents an equation that allows one to estimate the number of planets around these stars that are:

  • rocky
  • lie in the habitable zone
  • support life
  • the life is generating detectable biosignature gases.

Equation 1 was presented in the article.

Eq. 1 \displaystyle N={{N}_{X}}\cdot {{F}_{Q}}\cdot {{F}_{HZ}}\cdot {{F}_{O}}\cdot {{F}_{L}}\cdot {{F}_{S}}

where

  • N is the number of planets that have amounts of biosignature gases that can be detected from Earth.
  • NX is the number of stars examined.
  • FQ is the fraction of stars that are quiet (i.e. provide a stable source of energy for a planet). In the Astrobiology article, Sara said that the fraction of stars that are quiet is 15%.
  • FHZ is the fraction of stars with rocky planets in the habitable zone.
  • FO is the fraction of rocky planets in the habitable zone that are observable.
  • FL is the fraction of rocky planets in the habitable zone that are observable and support life.
  • FS is the fraction of rocky planets in the habitable zone that are observable, support life, and the life generates observable biosignature gases.

Equation 1 is only an estimate because many of its terms (other than NX and FQ) are unknown. However, we are beginning to get estimates for all the fractions. Sara Seager has estimated the fractions present in Equation 1 and these numbers project that two inhabited planets will be found in the next decade. She also published an interesting article that discusses candidate biosignature gases -- oxygen and ozone figure prominently.

Let's try an example calculation to see the kind of parametric values that we would need to have to find 2 habitable planets in the next ten years. My quick analysis assumes that we are looking close to the Earth (i.e. within 100 light-years). Figure 1 shows my speculations.

Figure 1: My Wild Speculations on the Parameters for Equation 1.

Figure 1: My Wild Speculations on the Parameters for Equation 1.

Star Counting Article on Seager's work

Conclusion

I hope Sara Seager is correct about the odds being good on finding two inhabited planets in the next decade. I think this is the most interesting area in astronomy at the moment.

Posted in Astronomy | 4 Comments

Morning in Northern Minnesota

Posted on 10-September-2013 by mathscinotes

Fall is coming ... soon the leaves will start to change color. This is my favorite time of year here.

MorningAtTheCabin

The photographer is my neighbor, Joel Teigland.

Posted in Cabin, Personal | Comments Off on Morning in Northern Minnesota

A Management Generation Gap

Posted on 7-September-2013 by mathscinotes

A management mentor of mine (let's call him Gary) contacted me the other day. This contact brought back many memories. Most of what I know about managing people I learned from him. Gary is near retirement now and he is the best example I know of an "old school" engineering manager. Today, engineering teams are composed of both men and women. Back in Gary's day, women engineers were rare but starting to become more common. During my first days under Gary, I worked as a two-person team with a very good female engineer -- let's call her Sue.

Back in the 1980s, we were supporting US Navy contracts on three coasts: East, West, and Gulf. Since I live in Minnesota, I spent much of my time flying across the country. The work Gary assigned to me was absolutely miserable! I would get calls late in the evening and he would tell me things like, "there is a problem in Seattle -- get your ass out there first thing tomorrow morning." This happened all the time. I put 200,000 air miles on my first year -- most of the trips were 2 to 3 days. I was always traveling at night and going out to some dock in a nasty part of a large city. I would load electronic equipment onto a research vessel and then get my tail out to some ocean test site. Sue, on the other hand, always had the office duty. This entailed writing reports, giving presentations, and other office functions. Both Sue and I noticed the difference in how we were treated.

One day, Sue had reached her limit and she started to yell at Gary, "You treat Mark like dirt and me like a princess -- this has got to stop." Of course, I am no more than 10 feet away and I am thinking to myself, "You go girl! -- let him have it." Sue was fully capable of doing everything I did and I was darn tired -- I have a family with two kids myself. She beat him up for at least ten minutes and then walked away. Gary had little to say while she was berating to him. Sue clearly had made the point that she was not getting the kind of experience that I was getting and her career would suffer because of it.

After Sue left the area, Gary came over to me and said, "I am the father of daughters -- I could not live with myself if one of the women was hurt while working." To be fair to Gary, the job did have its hazards (I will relate those tales some other time). Gary stood up, looked at me with a big grin and said, "fortunately, I have no guilt about how I treat you" and he walked away. He never did change.

Today, things are very different. I work hard to ensure that every engineer receives equal treatment. I will agree with Gary on one thing. Being a parent has made a difference in how I manage -- particularly for young folks. I now understand the importance of mentoring and providing a role model. Gary was my mentor -- I hope I do as good a job with my young engineers as he did with me.

Posted in Management | Comments Off on A Management Generation Gap

Counting Nearby Stars

Posted on 31-August-2013 by mathscinotes

In my previous post, I casually stated that there are 61 stars in the region within 5 parsecs of the Sun -- at least according to the Gliese Catalog of Nearby Stars. I did not state where you could get that information for yourself. This short post describes how to figure this out for yourself. Here is my approach:

  • Get a copy of the Gliese Catalog here.
  • Look in the readme file here to understand the document format.
  • Many of these stars have common names that you can find .
  • Analyze the text file for the nearby stars in Excel like I did here.

My quick and dirty analysis shows that there are 61 stars within 5 parsecs of the Sun. I put together an Excel table but it is my second favorite software tool -- Mathcad is my favorite. I did add a couple of columns to the original Gliese columns: (1) a column of common names, and (2) the distances to the stars in light-years. The Gliese catalog just gives the milliarcseconds of parallax and I needed to convert the milliarcseconds to light-years using the following formula.

\displaystyle d[\text{light-years }\!\!]\!\!\text{ =}\frac{{{d}_{AU}}[\text{km }\!\!]\!\!\text{ }}{\frac{\pi \left[ \text{radians} \right]}{180{}^\circ }\cdot \frac{\theta [\text{milliarcseconds }\!\!]\!\!\text{ }\cdot \text{0}\text{.001}\left[ \frac{\text{arcseconds}}{\text{milliarcseconds}} \right]}{3600\left[ \frac{\text{arcseconds}}{\text{degree}} \right]}\cdot {{d}_{LightYear}}[\text{km }\!\!]\!\!\text{ }}

Here is the table I put together. I should note that new stars have been discovered since the Gliese table was put together (example). This is just a rough order of magnitude calculation (Fermi problem) and the results are good enough for what I am doing here.

Gliese Reference

Common Name

Distance (Light-Years)

Sun

Sun

0.00

GL551C

Proxima

4.24

GL559A

αCentauriA

4.37

GL559B

αCentauriB

4.37

GL699

Barnard's

6.00

GL406

Wolf359

7.82

GL411

Lalande21185

8.23

GL65A

BLCeti

8.59

GL65B

UVCeti

8.59

GL244A

SiriusA

8.60

GL244B

SiriusB

8.60

GL729

Ross154

9.59

GL905

Ross248

10.36

GL144

εEridani

10.70

GL447

Ross128

10.86

GL866A

Luyten789-6A

11.11

GL15A

Groombridge34A

11.30

GL15B

Groombridge34B

11.30

GL845A

εIndiA

11.32

GL820A

61CygniA

11.33

GL820B

61CygniB

11.33

GL725A

Struve2398A

11.43

GL725B

Struve2398B

11.43

GL71

tauCeti

11.44

GL280A

ProcyonA

11.44

GL280B

ProcyonB

11.44

GL887

Lacaille9352

11.50

GJ1111

DXCancri

11.86

GL54.1

YZCeti

12.23

GL273

Luyten's

12.37

GL825

Lacaille8760

12.65

GL191

Kapteyn's

12.66

GL860A

Kruger60A

12.98

GL860B

Kruger60B

12.98

GL628

Wolf1061

13.37

GL234A

Ross614A

13.51

GL234B

Ross614B

13.51

GJ1061

L372-58

14.04

GL473A

Wolf424A

14.09

GL473B

Wolf424B

14.09

GL35

VanMaanen's

14.16

NN3522

Gliese3522

14.60

GL83.1

TZAri

14.61

NN3618

Luyten143-23

14.68

GL1

Gliese1

14.75

NN3622

Gliese3622

14.80

GL674

Gliese674

14.89

GL440

Gliese440

14.97

GL832

Gliese832

15.21

GL380

Groombridge1618

15.34

GJ1002

Gleise1002

15.37

GL687

Gliese687

15.38

GJ1245A

Gliese1245A

15.43

GJ1245B

Gliese1245B

15.43

GL682

Gliese682

15.46

GL876

Gliese876

15.48

GL166A

40EridaniA

15.79

GL166B

40EridaniB

15.79

GL166C

40EridaniC

15.79

GL388

ADLeo

16.04

GL768

Altair

16.27
Posted in Astronomy | Comments Off on Counting Nearby Stars