3.1 Effects of Altitude and Atmospheric Conditions

The effects of altitude and atmospheric conditions on aerodynamic drag are very closely coupled and must be treated together. This was not understood very well by ballisticians until about the beginning of the 20th century. Many firing tests took place in Europe in the latter half of the 19th century, especially in England, Germany, France and Italy, in an effort to understand aerodynamic drag and develop theoretical models for drag. Ballisticians found it difficult to compare measured data when the firing tests were made at locations having different altitudes and different atmospheric conditions.

Ballisticians gradually came to realize that drag measurements made in different locations, or even at the same location under different atmospheric conditions, could not be compared unless the measurements were somehow referenced to a set of standard altitude and atmospheric conditions. This led to the adoption of a standard set of altitude and atmospheric conditions to which measurements could be referenced. At the same time, analytical methods were developed to convert data measured at nonstandard altitude and atmospheric conditions to their standard values. Data from different locations and/or different atmospheric conditions could then be compared.

In the United States, standard altitude and standard atmospheric conditions were adopted by the U.S. Army Ballistic Research Laboratory at the Aberdeen Proving Ground in Maryland at about the beginning of the 20th century. These conditions, called the Standard Metro conditions, are used for ballistics computations. The Standard Metro conditions are:

Altitude: Sea Level  
Barometric Pressure: 750 mm Hg = 29.53 inches Hg  
Temperature: 59°F = 15°C  
Relative Humidity: 78 percent  

(Hg denotes the chemical element mercury)

The values of air density and speed of sound corresponding to these conditions are:

Air Density: Speed of Sound:

0.0751265 lb/ft3 = 1.2030 kg/m3 1120.27 fps = 341.46 m/s

Also, the acceleration due to gravity used for ballistics computations is:

Acceleration due to Gravity: 32.174 fps = 9.80665 m/s

The drag function G1 is referenced to these standard conditions, and ballistic coefficients are therefore referenced to the same conditions. Of course, these standard conditions are used for reference only; it would be a very rare event if anyone were to shoot a gun under these standard conditions. So, the question and the problem is how to calculate real world trajectories at different altitudes and under different atmospheric conditions.

The historical approach to this problem has been to first extend the Standard Metro atmospheric conditions to altitudes higher than sea level, that is, to create a “Standard Metro atmosphere” versus altitude. Table 3.1-1 shows the Standard Metro atmospheric conditions versus altitude up to an altitude of 15,000 feet above sea level, which is sufficient for hunting and target shooting on the North American continent. The next step is to treat the differences between actual atmospheric conditions at any altitude point and the standard atmospheric conditions at that altitude as small variations from the standard conditions. This approach has been successful for several reasons. The main reason is that air density decreases dramatically with altitude, while it changes much less dramatically with small differences between actual atmospheric conditions and standard conditions at any given altitude.

Furthermore, the small change in air density caused by a small difference between actual air temperature and standard air temperature at any altitude point tends to be offset by the change in air density caused by a small difference between actual barometric pressure and standard barometric pressure at that altitude point. This is because a higher-than-normal temperature (a warm, balmy day) tends to be accompanied by a higher-than-normal barometric pressure of the atmosphere. That is, high temperature tends to decrease air density, while high pressure tends to increase air density.

The air density ratio column in Table 3.1-1, which is the ratio of standard air density at altitude to the standard air density at sea level, shows that the air density decreases rapidly as altitude increases. Air density is a direct

Table 3.1-1 Standard Metro Atmospheric Parameters versus Altitude

Altitude   Air Density Ratio   Temperature   Baro Pressure   Speed of Sound  
(Feet)   (At Altitude /   (Deg F)   (mm Hg)   (in Hg)   Ratio (At Altitude /  
    At Sea Level)               At Sea Level)  
                       
Sea Level   1.0000   59.0   750.0   29.53   1.0000  
1000   0.9702   55.4   722.7   28.45   0.9873  
2000   0.9414   51.9   696.3   27.41   0.9744  
3000   0.9133   48.3   670.9   26.41   0.9614  
4000   0.8862   44.7   646.4   25.45   0.9483  
5000   0.8598   41.2   622.7   24.52   0.9350  
6000   0.8342   37.6   599.8   23.62   0.9216  
7000   0.8094   34.1   577.8   22.75   0.9080  
8000   0.7853   30.5   556.6   21.91   0.8943  
9000   0.7619   26.9   536.1   21.11   0.8805  
10000   0.7392   23.4   516.3   20.33   0.8666  
11000   0.7172   19.8   497.3   19.58   0.8525  
12000   0.6959   16.2   478.9   18.85   0.8383  
13000   0.6752   12.7   461.1   18.16   0.8239  
14000   0.6551   9.1   444.0   17.48   0.8094  
15000   0.6356   5.5   427.6   16.83   0.7948  

multiplier in the equation for the drag force on a bullet, and because of this, the drag force also decreases rapidly as altitude increases. This decrease in air density with altitude has by far the largest effect on a bullet trajectory, compared to the actual atmospheric conditions and the speed of sound versus altitude. As mentioned in the preceding paragraph, the differences between actual temperature and standard temperature, and between actual barometric pressure and standard barometric pressure, have small effects on a bullet trajectory compared to the effect of decreasing air density, and these effects usually tend to offset each other due to weather patterns. The speed of sound ratio column in Table 3.1-1, which lists the ratio of the standard speed of sound at altitude to the standard speed of sound at sea level, shows that the speed of sound also decreases quite rapidly with altitude. However, the speed of sound is not a direct multiplier in the equation for drag force. In fact, it enters the equation in such a way that its effect on the drag force is much smaller than the effect of the decrease in air density. The true speed of sound does vary slightly from the standard value because of actual weather conditions, but the effect of the variation is considerably smaller than the small effect of the standard speed of sound.

Humidity also has a small effect on a bullet’s trajectory, and at all altitudes. Humidity affects the air density, tending to decrease the air density a small amount, depending on the relative humidity in the atmosphere and the vapor pressure of water at the temperature of the atmosphere. The effect of humidity is generally worst at locations near sea level on very hot days, but even under these conditions, the effect is small. For example, for a location near sea level on a 90°F day with barometric pressure the same for both situations, absolutely dry air (zero relative humidity) is not quite 0.02 percent MORE dense than air saturated with water vapor (fog, meaning 100 percent relative humidity). This seems strange; wet air feels “heavier” than dry air. But it is true because a water molecule weighs less than a nitrogen molecule, which it displaces if the pressure and temperature remain the same. This tiny change in air density is not completely negligible for long-range shooting. For example, under these same atmospheric conditions, the drop at 1000 yards for Sierra’s .308” diameter 168 grain MatchKing bullet fired at 2700 fps muzzle velocity will be about 2.4 inches more for absolutely dry air than for saturated wet air.

A word about barometric pressure. In this country, the National Weather Service and local weather bureaus report sea level-referenced barometric pressures regardless of location. For example, if you were in New York City (at sea level) on a balmy day, the barometric pressure might be reported near 30 inches of mercury (in Hg). If you were in Denver, CO, (5200 ft altitude) on a balmy day, the barometric pressure might also be reported near 30 in Hg. Now, the true barometric pressure at the altitude of Denver should be about 25 in Hg, not 30 in Hg. Our weather bureaus report sea level-referenced barometric pressures so that citizens can compare the weather in Denver with the weather in New York, or Los Angeles, or Fairbanks, AK, or Salt Lake City, or anywhere else in this nation. Also, the barometer instruments that we can purchase in stores are designed to read out sea level-referenced barometric pressures. Now, of course, the trajectory of a bullet at any location depends on the true atmospheric pressure at that location, not at sea level. Sierra’s Infinity program takes this into account. It is important to realize that Infinity is designed so that the user must enter the altitude of the shooting location and the sea level-referenced barometric pressure at that location, as well as the temperature and the relative humidity (if known). These parameters can be obtained from TV, a local weather station, or portable instruments. Then, Infinity will calculate the true barometric pressure at the firing point from atmospheric variation laws coded into the program.

A great advantage of the standard atmospheric conditions is that, based only on altitude, bullet ballistics can be calculated for locations where the true atmospheric conditions are unknown or unpredictable, and the resulting trajectories will be accurate enough for most practical purposes.

To illustrate this, let’s consider an example. Suppose that a hunter living near St. Louis, MO, has a Model 70 Winchester rifle in 300 Winchester Magnum that he uses to hunt mule deer and elk in western Colorado at an altitude near 8500 feet above sea level. His gun is telescope sighted. He loads Sierra’s .308" dia 200 grain Spitzer Boat Tail (SBT) GameKing bullet at 2800 fps muzzle velocity for hunting. He sights his gun in at a target range near St. Louis that is located at an altitude near 500 feet above sea level. The question is, if he sights his rifle in at the target range near St. Louis, where will his gun shoot in western Colorado where he intends to hunt? Sierra’s Infinity program will be used to answer this question.

Suppose he sights his gun in on a late summer day in St. Louis when the temperature at the target range is 92°F, and a local weather report lists the barometric pressure at 30.25 in Hg and the relative humidity at 90 percent. For the 300 Winchester Magnum, he uses a zero range of 300 yards. When in Colorado he will use a laser rangefinder, and he will limit his shots at mule deer or elk to no more than 500 yards.

After he finishes sighting his gun in, he returns to his home and performs the following calculations on his personal computer using the Infinity program. He calculates three trajectories for the 200 grain SBT GameKing bullet in the 300 Winchester Magnum cartridge and carefully examines the bullet path parameter from the output data. [Bullet path is the trajectory variable that locates the bullet relative to the shooter’s line of sight through the gun sights as the bullet travels downrange. It is most important because it tells the shooter how high or low his bullet will strike the target, or how much he has to hold over or hold under a target at any downrange location.] The first trajectory is a reference trajectory for the environmental conditions at the target range near St. Louis. Then, he uses the “Trajectory Variations” feature inInfinity to calculate a trajectory in his hunting location, first based on standard atmospheric conditions only, and then based on atmospheric conditions that he predicts based on his previous experiences in the hunting area.

So, for the .300 Winchester Magnum cartridge, he selects the Sierra .308" dia 200 grain Spitzer Boat Tail GameKing bullet from the “Load Bullet” library in Infinity, and selects the “Normal Trajectory” mode of operation of the program. In the “Trajectory Parameters” list, he sets the muzzle velocity at 2800 fps, maximum range at 500 yards, range increment at 50 yards, zero range at 300 yards, the elevation angle at 0 degrees, and the sight height at 1.75 inches because his telescope sight has a large objective bell. In the “Environmental Parameters” list he sets the conditions for the target range near St. Louis, that is, barometric pressure at 30.25 in Hg, temperature at 92°F, altitude at 500 ft, humidity at 90 percent, and the wind speeds to 0 mph. He then commands Infinity to calculate the reference trajectory for the St. Louis environs. The bullet path numbers versus range are listed in Table 3.1-2. It is evident that between the muzzle and the zero range the bullet rises a little more than 5 inches maximum above the line of sight, but at 500 yards. the bullet is nearly 30 inches low.

The next trajectory calculation is made using the “Trajectory Variations” capability in Infinity. The hunter selects the “Environmental Parameters” option in that mode and makes the following changes to calculate the trajectory variations based on standard atmospheric conditions at the hunting location. The standard conditions are barometric pressure at 29.53, temperature at 59, and humidity at 78. He sets the altitude at 8500 ft. [Recall that Infinity automatically adjusts the standard atmospheric conditions at sea level to the values appropriate for 8500 ft altitude.] He again commands a calculation, and Infinity outputs the bullet path differences shown in the third column of Table 3.1-2. It can be seen that the 300 Winchester Magnum always will shoot high compared to the reference trajectory at St. Louis, but the hunter really needs to make no sighting correction unless possibly when the game animal is close to 500 yards away.

The third trajectory for the 300 Winchester Magnum is calculated again by using the “Trajectory Variations” capability. From previous experiences in western Colorado, the hunter believes that the weather will be fair with low humidity, but cold. So he adjusts the barometric pressure to 29.90, the temperature to 20, and the humidity to 20, leaving the altitude at 8500 ft. After the calculation is commanded, Infinity outputs the bullet path differences in the fourth column of Table 3.1-2. Note that these bullet path differences are relative to the bullet path values in the second column of Table 3.1-2 for the reference trajectory at St. Louis, and not to the numbers in the third column. It can be seen that the trajectory calculated for the non-standard atmospheric conditions is very close to the trajectory calculated with only standard atmospheric conditions at the hunting location.

The data in Table 3.1-2 support two observations. The first is that this 300 Winchester Magnum cartridge has a trajectory that is quite flat. The reference bullet path at 500 ft above sea level stays between a little over 5 inches above the line of sight and does not fall more than 5 inches below the line of sight until the range exceeds a little more than 350 yards. At 8500 ft above sea level the bullet path stays within this band until about 365 yards. This is excellent performance, as expected for this very popular magnum cartridge for western hunting.

The second observation is that calculating a trajectory for the hunting location based on standard atmospheric conditions gives an accurate representation of the trajectory for actual atmospheric conditions, as pointed out above. Comparing columns three and four in Table 3.1-2 shows that the bullet path changes based on the predicted actual atmospheric conditions are very close to those based on standard conditions. The largest difference between the bullet paths is at 500 yards, and it is just 0.3 inches. This observation holds true for the vast majority of cartridges and atmospheric conditions at all shooting locations. We recommend that when the actual atmospheric conditions are unknown or unpredictable at any shooting location, standard atmospheric conditions be used for the altitude of the location. The altitude of any location usually can be estimated from a topographical map, a local weather station, or an atlas of North America.

Table 3.1-2 Reference Bullet Path and Changes for the 300 Winchester Magnum Cartridge loaded with Sierra’s 30 caliber 200 grain SBT Bullet at 2800 fps.

Range   Reference Bullet   Bullet Path Changes (2)   Bullet Path Changes (3)  
    Path (1)   at Hunting Location   at Hunting Location  
(yards)   (inches)   (inches)   (inches)  
               
0   -1.75   0.0   0.0  
50   1.72   0.0   0.0  
100   4.01   0.02   0.02  
150   5.06   0.07   0.06  
200   4.80   0.17   0.15  
250   3.14   0.34   0.30  
300   0.0   0.60   0.54  
350   - 4.71   0.98   0.89  
400   - 11.10   1.52   1.37  
450   - 19.28   2.23   2.03  
500   - 29.35   3.18   2.88  

 

(1) Reference trajectory from sighting the rifle in near St. Louis, 500 ft altitude and actual atmospheric conditions at the shooting range (see text).
(2)
  
From trajectory calculated for the hunting location, 8500 ft altitude and standard atmospheric conditions (see text).
(3)
  
From trajectory calculated for the hunting location, 8500 ft altitude and predicted atmospheric conditions (see text).
Note that a positive bullet path change in columns 3 and 4 means that the gun will shoot higher than the reference trajectory.