Using Calculus to determine the mass of a comet

--------- #1292 - Using Calculus to Measure the Mass of a Comet

- NASA spacecraft caught up with Comet Hartley-2 on November 4, 2010 and took a picture of the icy dirtball that was 1.4 miles in length. Scaling the picture and using Calculus to determine the volume, the mass of the comet was put at 290,000,000 metric tons. This review shows the math involved.

- News, will the Comet Elenin fulfill the 2012 prophecy of the end of the Earth to occur December 21, 2012. Alarmist want to make the most of this modest-sized icy dirtball that will pass within 22,000,000 miles of our planet. The comet is 3 to 5 kilometers wide. Astronomers measure the mass of one object by its gravitational effect on another. But, these comets are far too small and too distant to have any measurable influence on Earth. Except, for alarmist news headlines.


- Attachment - comet

- Comet Hartley-2 flew by the Earth at its closest 11,000,000 miles away on October 20,2010. That is between us and the Sun which is 93,000,000 miles away. The Earth-Sun distance is called an Astronomical Unit.

- NASA sent a space craft to catch up with the comet. On November 4, 2010 the space craft took pictures from a distance of only 435 miles. The comet was dumbbell shaped and 1.4 miles in length. It was composed of water ice, carbon dioxide ice, and silicate dust. It was rotating a complete cycle every 18 hours. This comet was first discovered in 1986. The picture showed a pitted surface free of large craters but evidence of active ejection of gas plumes. The scale of the picture was 1 millimeter equaled 25 meters.

- The comet was 2,000 meters long. We can use Calculus to calculate the volume of the comet by measuring the radius at each point along its length then summing up (integrating) an infinite number of areas of circles with each radius over the entire length.

---------------------- Length ------------------- Radius
------------------------ 0 ---------------------------- 0 meters
------------------------ 200 ------------------------- 420 meters
------------------------ 400 ------------------------- 500 meters
------------------------ 600 ------------------------- 410 meters
------------------------ 800 ------------------------- 290 meters
------------------------ 1000 ------------------------ 210 meters
------------------------ 1200 ------------------------ 220 meters
------------------------ 1400 ------------------------ 290 meters
------------------------ 1600 ------------------------ 340 meters
------------------------ 1800 ------------------------ 310 meters
------------------------ 2000 ------------------------ 0 meters

- This dumbbell shape can be approximated by a function with Volume as a function of “x” length. The plot of the function is the y axis radius versus the x axis length, from 0 to 2 kilometers.

---------- Volume = pi * radius^2 * length

----------- y(radius) as a function of length x = -1.22x^4 +5.04x^3-6.78x^2+3.14x +0.03)

----------- Volume = ( -1.22x^4 +5.04x^3-6.78x^2+3.14x +0.03)^2 * length

----------- Volume = Integral 0 to 2 * pi * ( -1.22x^4 +5.04x^3-6.78x^2+3.14x +0.03)^2 * dx

----------- Volume = Integral 0 to 2 * pi * ( -1.22x^4 +5.04x^3-6.78x^2+3.14x +0.03) * ( -1.22x^4 +5.04x^3-6.78x^2+3.14x +0.03) * dx

----------- Volume = Integral 0 to 2 * pi * ( 1.49x^8 - 12.3x^7 + 41.94x^6-76x^5 + 77.55x^4 - 42.28x^3 + 9.46x^2 + 0.18x + 0.0009) * dx

- The integral of a function of the form: Integral x^n * dx = x^(n+1) / (n+1)

Example : Integral 1.49x^8 = 1.49 / 9 * 2^9 = 0.17* (512) = 87

Applying this integration to each term of the above function:

---------- Volume - pi* (0.17x^9 - 1.54x^8 + 5.99x^7 - 12.67x^6 + 15.51x^5 - 10.57x^4 + 3.15x^3 +0.09x^2 + a constant ) from x equal 0 to 2
　
--------- Volume = (3.14) ( 0.1655*2^9 - 1.5375*2^8 +5.9913*2^7 - 12.6667*2^6 + 15.51*2^5 - 10.57*2^3 + 0.09*2^2 +0.0009*2 + a constant

---------- Volume (3.14) * (0.157)

---------- Volume = 0.49 kilometers^3

--------- Volume = 4.9 *10^14 centimeters^3

- To get the mass of the comet we need to know the density in grams / cubic centimeter. Taking the average density from other comets:

---------------------- Halley’s Comet ----------------- 0.6 gm / cm^3
---------------------- Templ-1 Comet ----------------- 0.62 gm / cm^3
---------------------- Borrells Comet ----------------- 0.3 gm / cm^3
---------------------- Wild Comet ----------------- 0.6 gm / cm^3

- A good estimate would be Hartly-2 density is 0.6 gm / cm^3.

------------------------- mass = density * volume

------------------------- mass = (0.6 gm / cm^3) * (4.9*10^14 cm^3)

------------------------ mass = 2.9*10^14 grams

------------------------- 1000 kilograms = 1 metric ton

------------------------ Hartlley-2 mass = 290,000,000 metric tons

- To double check this calculation another way is to estimate the dumbbell volume as 2 spheres connected by a cylinder. Using the scale of the picture where 1 millimeter = 25 meters. The two spheres are 1000 meter and 750 meter in diameter.

------------------- 40 mm * 25 m/mm = 1000 meters

------------------- 30 mm * 25 m/mm = 750 meters

- The overall length is 2000 meters. Subtracting the diameters of the 2 spheres, 2000- 1750 gives the length of the cylinder to be 250 meters.

-------------- Volume of a sphere = 4/3 *pi*r^3

--------------- Volume = 1.33*3.14* (750 / 2)^3 = 2.2 * 10^8 m^3

--------------- Volume = 1.33*3.14* (1000 / 2)^3 = 5.2 * 10^8 m^3

----------------- Volume of a cylinder = pi * r^2 * length

-------------------Volume = 3.14 * (250)^2 * 250 = 0.49 * 10^8 m^3

--------------- Total Volume = 7.9*10^8 m^3

---------------- Total Volume = 7.9*10^14 cm^3

--------------- mass = (0.6 gm / cm^3) * (7.9*10^14 cm^3)

---------------- mass = 4.7*10^14 gm

------------ mass = 470,000,000 metric tons.

- This rough calculation is in the ballpark, but , about 60% too high. The Calculus calculation is thought to be the more accurate. The NASA mission included a probe impact on Temple-1 Comet on July 3, 2005 . The mission launch was January 12,2005. The mission was extended to meet up with Hartley-2 Comet on November 4, 2010 traveling 2,900,000,000 miles to get there. The total cost of the mission was $309 million. An expensive picture. But, a lot more than this was learned. An announcement will be made shortly, stay tuned.
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