5.1  Effects of Altitude and Atmospheric Conditions

 Drag depends on the density of the air and on the speed of sound, and these in turn depend on altitude, temperature, barometric pressure, and relative humidity at the firing point. Because altitude and atmospheric conditions vary from location to location and from day to day, a standard altitude and standard atmospheric conditions have been adopted for the purpose of computing ballistics tables. For the ballistics tables in Sierra's atmospheric conditions are the ones used for many years by the U.S. Army Ballistic Research Laboratories at the Aberdeen Proving Ground in Maryland. These standard conditions are:

   Altitude:    Sea level
   Barometric pressure:  750mm Hg-29.53 inches Hg
   Temperature:    59 degrees F
   Relative humidity:   78 percent

(The symbol Hg denotes the chemical element mercury.)

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

   Air density:    .0751 lb/ft 3
   Speed of sound:   1120.27 fps

The values of ballistic coefficient in the table in Section 1.0 also correspond to these standard conditions, as does the drag function G 1 . The standard barometric pressure and temperature are nominal, or average, values for the Aberdeen location.

 Two questions now arise. First, the ballistics tables provide us with trajectory data for these sea level standard atmospheric conditions, but what will happen to our bullet trajectories if we shoot at a location above sea level or with other atmospheric conditions? The second question is, how can the ballistics tables be used for higher altitudes and other atmospheric conditions?

 To answer the first question, we need to understand that changes in altitude and atmospheric conditions affect bullet drag in two different ways. One way is by a change in air density, and the other is by a change in the speed of sound. Air density tends to decrease if altitude increases, or barometric pressure drops below the sea level standard pressure value, or air temperature rises above the sea level standard temperature value, or a combination of high temperature and high humidity occurs. When air density decreases, bullet drag decreases because the air is thinner. Opposite changes in these altitude or weather conditions tend to increase air density and, therefore, drag.

 The effect on drag of a change in the speed of sound is more complex, because the effect depends on bullet velocity relative to the speed of sound, as well as on altitude and atmospheric conditions. Generally, as we go up in altitude, the speed of sound decreases, and this tends to further decrease drag.

 Consequently, a general answer to the first question posed above is that at higher altitudes bullets tend to shoot flatter and buck the wind better than they do at altitudes near seal level, provided of course that the bullets are fired with the same muzzle velocities at each altitude. This is not always true, because the muzzle velocity developed by a gun depends on the temperature of the powder in each round. Higher powder temperatures cause higher muzzle velocities, and vice versa. Thus, if a shooter develops a load in pleasant weather at a low altitude location, and then shoots the same load at a higher altitude, his trajectory should be flatter due to lower bullet drag. However, the cooler temperature at altitude will cause a lower muzzle velocity, offsetting the improvement due to lower drag. (See Section 5.6 .)

 The answer to the second question is that there is a simple procedure which allows us to use the sea level standard atmosphere ballistics tables for other shooting conditions. The procedure is to calculate an adjusted (or equivalent) ballistic coefficient for the new conditions, and then use ballistics from the tables for a "fictitious" bullet with this new ballistic coefficient. It must be emphasized that this procedure is only approximately correct, because the true bullet drag under other conditions is only approximated by the sea level standard drag ( G 1 ) and the equivalent ballistic coefficient. Nevertheless, the approximation is very good for hunting situations, and quite reasonable for target shooting as well.

 The calculation of the equivalent ballistic coefficient is carried out in three distinct steps. The first is to adjust the ballistic coefficient for a change in altitude which causes predictable changes in the standard atmospheric conditions. Stated another way, standard atmospheric conditions are calculated for all altitudes, not just sea level. The first adjustment to the sea level standard ballistic coefficient adjusts for the change in air density which results for the change in altitude.

  Table 5.1-1 shows how the standard temperature T std and the standard barometric pressure P std change with altitude over the range from sea level to 15,000 ft. The table also lists an altitude adjustment factor F A which is used in the ballistic coefficient calculation (more about this below).

 The second step in calculating the equivalent ballistic coefficient is to adjust for variations between the true atmospheric temperature and pressure at the firing location and the standard temperature and pressure for that location's altitude, which can be found from Table 5.1-1 .

 The final step in the calculation is to adjust for the true relative humidity at the firing point versus the standard 78 percent relative humidity at sea level.

 The procedure for calculation of the equivalent ballistic coefficient works as follows. Let us call C o the value of ballistic coefficient for the sea level standard atmosphere. C o for any Sierra bullet can be found in the table in Section 1.0 . then, for any altitude of the firing point and any set of atmospheric conditions, the equivalent ballistic coefficient is calculated from the equation

C eq = C o [F A (1.0+F T -F P )F RH ] (5.1-1)

 Step 1 is to find the altitude adjustment factor F A , which can be found from Table 5.1-1 since the firing location altitude is known. (Interpolation must be used for altitudes between the thousand foot levels listed in the table.) The standard temperature and standard pressure at the firing location also can be obtained from Table 5.1-1 .

 Step 2 in the procedure is to adjust for the nonstandard temperature and pressure. This adjustment is the factor (1.0 + F T - F P ) in equation (5.1-1). The temperature adjustment factor F T is given by

where T is the true temperature at the firing point and T std is the standard atmospheric temperature at that altitude, both in degrees Fahrenheit. The pressure adjustment factor F P is given by

where P is the true barometric pressure at the firing location and P std is the standard pressure for that altitude. Both pressure values may be in either millimeters or inches of Hg , as long as both are in the same units.

 Step 3 of the procedure is to adjust for relative humidity both in the sea level standard atmospheric conditions and the true conditions at the firing location. The relative humidity correction factor F RH is given by the

where P is the true barometric pressure at the firing point, RH is firing point relative humidity expressed as a decimal fraction (e.g., 45 percent = 0.45), and V pw is the vapor pressure of water at the temperature of the firing point location. Table 5.1-2 lists the vapor pressure of water versus the atmospheric temperature at the firing location. Both the true atmospheric pressure P and vapor pressure V pw can be known in either millimeters or inches of Hg , but both must be in the same units.

 Equation (5.1-4) shows that the relative humidity correction factor is the product of two terms. The first term, 0.9950, adjusts the sea level ballistic coefficient from the standard 78 percent relative humidity to a dry air (zero humidity) condition. The second factor within the brackets in equation (5.1-4) adjusts the ballistic coefficient for the relative humidity at the firing point.

 Because the molecular weight of water is lower than the molecular weight of dry air (18 versus 29), for any specified atmospheric temperature and pressure the density of humid air is lower than the density of day air. Consequently, there would be more drag on a bullet in dry air than in humid air, provided temperature and pressure remained the same in both cases. Therefore the equivalent ballistic coefficient increases if we go from dry air to humid air, and vice versa.
 This is exactly the effect we see in the two terms in equation (5.1-4). To adjust the ballistic coefficient for the relative humidity at the firing location, we first "back out" the 78 percent standard relative humidity at sea level, and then we adjust for the relative humidity at the firing point. The first term in the equation, 0.9950, is a reduction of one-half of one percent caused by going from humid air to dry air at sea level. The second term in brackets is an increase caused by going from dry air to humid air at the firing location, if the relative humidity is greater than zero.

 An example will illustrate how this procedure is used. To begin with, suppose that your shooting range has an altitude of about 1500 ft above sea level. You are shooting on a balmy day when the temperature is 75 degrees Fahrenheit, the barometer reads a pressure of 29.20 inches of Hg, and the relative humidity is a comfortable 30 percent. You are going to target your hunting load, a .30-60 using Sierra's 165 gr hollow point boat tail bullet with a sea level standard ballistic coefficient of 0.375 from the table in Section 1.0 . The question is, how do you expect your loads to perform relative to the information in our ballistics tables for this bullet?

 To answer this question, we calculate the equivalent ballistic coefficient. In step 1, we get the standard temperature and pressure and the altitude adjustment factor from Table 5.1-1 . Since the range altitude (1500 ft) is halfway between 1000 and 2000 ft, we interpolate from the values at 1000 and 2000 ft in the table to get:

     T std = 53.7 degrees F
    P std = 27.93 inches Hg
    F A = 1.047

In step 2 we compute the factors F T and F P from equations (5.1-2) and (5.1-3):

 It can be noted here that since the temperature at the range is higher than the standard value, which tends to decrease the air density, the drag tends to be lessened, and so F T tends to increase the effective ballistic coefficient. The opposite is true of the pressure, which is higher than the standard, tending to increase air density and drag.

 The humidity adjustment factor is calculated in step 3 as follows. Because the relative humidity at the range is 30 percent

RH = 0.30

From Table 5.1-2 the vapor pressure of water at 75 degrees F is

VP W = 0.88 inches of Hg

Then, equation (5.1-4) gives

Finally, the equivalent ballistic coefficient for your bullet at your shooting range on that particular day is calculated from equation (5.1-1):

 So, you can expect the performance of your loads on this day to be the same as those of a bullet with about a 4 percent higher ballistic coefficient. You can use the sea level standard ballistic tables for a bullet with a ballistic coefficient of 0.390 rather than 0.375.

 Now, suppose that after targeting your rifle, you are planning a hunt in the Rockies where the altitude will be 8000 to 9000 ft. How will you expect your loads to perform there? The same procedure will help you find out. Since you do not know the weather conditions in advance, it is best to assume that they will be about standard for that altitude range. The first step is to find F A for an altitude of, say, 8500 ft from Table 5.1-1 . Using interpolation between the F A values at 8000 and 9000 ft, F A turns out to be 1.293.

 In step 2, since we have assumed standard temperature and pressure will prevail, the factors F T and F P are both equal to zero.

 So far, we haven't said anything about humidity, and this is difficult to predict in your hunting area. Fair weather in the Rockies generally means low relative humidity, but the opposite can be true for bad weather. So, let's look at the extreme conditions that might happen. First, consider that the relative humidity will be zero while you are hunting. If we put RH = 0.000 in equation (5.1-4), then F RH = 0.9950 for this extreme.

 Next, consider that the relative humidity might be 100 percent. From Table 5.1-2 the standard temperature and pressure at 8500 feet are 28.7 degrees F and 21.51 inches Hg. From Table 5.1-2 the vapor pressure of water at 28.7 degrees F is 0.15 inches Hg. Substituting these values into equation (5.104) gives

for 100 percent relative humidity.

 So, if temperature and pressure are near standard in your hunting area, the relative humidity adjustment to ballistic coefficient will be between a half and a quarter percent. This is negligible compared to the altitude adjustment, which improves your ballistic coefficient nearly 30 percent.

 This example calculation suggests some observations which are generally true. First, changes in altitude can have a very significant effect of bullet trajectories. The values of F A in Table 5.1-1 show that the equivalent ballistic coefficient of a bullet can be 30 to 40 percent higher in the mountainous regions of North America than at sea level. On the other hand, the factors F T and F P usually cause change of only a few percent in the ballistic coefficient value, because temperature and pressure are not much different from their standard values. Furthermore, the effects of temperature and pressure variations from standard values often offset each other, as in our example above. This is because high temperatures often go with high barometric pressures, and vice versa, at least in North America.

 Note that the humidity adjustment to the ballistic coefficient is also quite small in our example (that is, F RH is very nearly unity). This is the usual case, unless both the temperature and the relative humidity at the firing location are high.

 A question which naturally arises is, when can we forget about the humidity adjustment to ballistic coefficient? In other words, when can we set F RH = 1.0 in equation (5.1-1) without causing a significant error in C eq ? At Sierra we believe that the humidity adjustment is negligible if it would cause a change in C eq not greater than a half percent. That is, if F RH is in the range of 0.9950 to 1.0050, then we can use a value of 1.0 without significant error. This is simply because the accuracy to which we can measure C o is about that amount.

  Table 5.1-3 has been prepared to offer some guidance in this regard. The table lists temperature values for altitudes to 15,000 ft and relative humidity up to 100 percent, which are limits for the plus-or-minus half percent adjustments in ballistic coefficient. To use Table 5.1-3 , you need to know altitude, temperature, and relative humidity at your shooting location. If you look in the table for your altitude and relative humidity, if your range temperature is less than the value shown, you can safely neglect the humidity adjustment. But if your temperature is higher than the value shown, you should calculate the adjustment factor F RH .

 To illustrate, look back at the example previously described. Your target range had an altitude of 1500 ft. The temperature was 75 degrees F and the relative humidity was 30 percent. Looking at Table 5.1-3 for 40 percent relative humidity, we can neglect the humidity adjustment if the temperature is below about 98 degrees F. Because the relative humidity is 30 percent rather than 40, the temperature limit would be even higher. So, for a 75 degree F range temperature you could safely neglect the humidity correction. Indeed, when we calculated F RH , we found that it caused only a quarter percent change in C o , while the altitude adjustment alone contributed more than a four percent change.

 It is clear that the three-step method of adjusting the ballistic coefficient is very convenient, because the first step is the most important and easiest, and the other steps often are not needed at all.

 If you have Version III of Sierra's Exterior Ballistics Program for your personal computer, you can input the altitude of the firing point and all atmospheric conditions from a standard weather report.  Version III calculates the true barometric pressure from the altitude and local barometric pressure in  the weather report, and then automatically performs all the ballistic coefficient adjustments described above.  As a default, the Program calculates standard atmospheric conditions at the firing point altitude, unless you input actual conditions.  The Program also corrects both the ballistic coefficient and the drag function for changes in speed of sound with altitude.  It then calculates the trajectory of the bullet using the fully adjusted ballistic coefficient as well as the corrected drag function. If you are shooting a Sierra bullet, for which Version III has ballistic coefficient values which change in different bullet velocity ranges, the Program applies the ballistic coefficient adjustments to each ballistic coefficient value.