Interplanetary Mission: Viking-2 Reconstruction Plan

Viking-2 (Viking program to Mars, NASA, 1975)

Figure 1. Booklet The Viking Mars Mission created by Martin Marietta (1972).  check this

Launching Site

The launching site for the Viking-2 was in Cape Canaveral, Florida, USA. With the latitude of 28.4740° N, 80.5772° W. Cape Canaveral was chosen for rocket launches to take advantage of the Earth's rotation. The linear velocity of the Earth's surface is greatest towards the equator that is available, the relatively southerly location of the cape allows rockets to take advantage of this by launching eastward, in the same direction as the Earth's rotation. It is also highly desirable to have the downrange area sparsely populated, in case of accidents, an ocean is ideal for this. Moreover, modern rockets now consist of many stages, when one stage is no longer needed (runs out of fuel), it can be discarded from the rocket into the ocean. The east coast of Florida has logistical advantages over potential competing sites.

Figure 2. Location of Cape Canaveral

Figure 3. Launch Complex 41, Cape Canaveral

Launch Vehicle

The launch vehicle for the Viking-2 was the Titan IIIE/Centaur. With the specifications of the rocket of Payload: 15,400 kg (33,900 lb). Thrust: 10,586.80 kN (2,380,007 lbf). Gross mass: 632,970 kg (1,395,450 lb). Height: 48.00 m (157.00 ft). Diameter: 3.05 m (10.00 ft). Apogee: 185 km (114 mi).

Figure 4. Titan III-Centaur stages

And for each stage of the launch vehicle, the specifications are

Stage 0. 2 x Titan UA1205. Gross Mass: 226,233 kg. Empty Mass: 33,798 kg (74,511 lb). Thrust (vac): 5,849.411 kN. Isp: 263 sec. Burn time: 115 sec. Isp(sl): 238 sec. Diameter: 3.05 m. Span: 3.05 m. Length: 25.91 m. Propellants: Solid. No Engines: 1. Engine: UA1205.

Stage 1. 1 x Titan 3A-1. Gross Mass: 116,573 kg. Empty Mass: 5,443 kg. Thrust (vac): 2,339.760 kN. Isp: 302 sec. Burn time: 147 sec. Isp(sl): 250 sec. Diameter: 3.05 m. Span: 3.05 m. Length: 22.28 m. Propellants: N₂O₄/Aerozine-50. No Engines: 2. Engine: LR-87-11.

Stage 2. 1 x Titan 3A-2. Gross Mass: 29,188 kg. Empty Mass: 2,653 kg. Thrust (vac): 453.714 kN. Isp: 316 sec. Burn time: 205 sec. Isp(sl): 145 sec. Diameter: 3.05 m. Span: 3.05 m. Length: 7.90 m. Propellants: N₂O₄/Aerozine-50. No Engines: 1. Engine: LR-91-11.

Stage 3. 1 x Centaur D/E. Gross Mass: 16,258 kg. Empty Mass: 2,631 kg. Thrust (vac): 131.222 kN. Isp: 444 sec. Burn time: 470 sec. Isp(sl): 0.0000 sec. Diameter: 3.05 m. Span: 3.05 m. Length: 9.60 m. Propellants: Lox/LH2. No Engines: 2. Engine: RL-10A-3.

for theoretical analysis, in our case, we choose empty mass or $m_{dry}.$ as 33,798 kg + 572 (lander) + 883 (orbiter) = 35,253 kg. As we want to find the rocket propellant mass, we can use equations below by first considering the altitude that we want to reach and find out the velocity we should have at that point. 

$$V_{cl}= \sqrt{μ_e \over r_{cl}}$$  $∆V=I_{sp} g_o \ln⁡{M_o \over (M_o-m_f (t) )}$

and by knowing the altitude (in this we choose altitude with circular trajectory) we can get the velocity we should have at that point. And also by considering the Isp used as 444 s and $g_0$ is 10 $m/s^2 .$, we can get $m_f.$ using solver, considering $M_o$ is total mass ($m_{dry} + m_f + m_{payload}.$).

Parking Orbit

Based on Potential Parking Orbits for a Periapsis Altitude in NASA Technical Paper 3256 December 1992: Aspects of Parking Orbit Selection in a Manned Mars Mission, we can choose some parking orbit to place our spacecraft. There're some choices and that can be considered.

In this case, we choose to directly go to altitude 20,000 km instead, considering we will choose the escape trajectory at the nearest distance, d, is 30,000 km.

For the theoretical analysis, the data known are:

Standard gravitational parameter:

Earth = 3.986 x 10^14 $m^3/s^2.$

Mars = 4.282 x 10^13 $m^3/s^2.$

Sun = 1.327 x 10^20 $m^3/s^2.$

Earth Orbit parameters

Eccentricity (ee) = 0.0167

Perihelion distance (rpe) = 147.1 million kilometers,

Aphelion distance (rae) = 152.1 million kilometers,

Period (Pe) = 365 day

Vae 29.2898 km/s

Vpe 30.2847 km/s

Mars Orbit parameters

Eccentricity (em) = 0.0934,

Perihelion distance (rpm) = 206.7 million kilometers,

Aphelion distance (ram) = 249.2 million kilometers,

Period (Pm) = 687.0 day

Vam 21.9702 km/s

Vpm 26.497 km/s

Escape Trajectory

For escaping the earth’s orbit, the spacecraft uses the third stage of the launch vehicle also known as Centaur. The Centaur has an isp of 444 sec. 

In our case, we will use Isp 500 (considering it will use several directly orbit). With the initial circular orbit of 20,000 km (6371 + 13,629), the d for the hyperbolic escape trajectory is chosen to be 30,000 km with the Hohmann (we consider the trajectory of escape object, later on, will have asymptote line perpendicular to axis that always goes through sun on inertial frame of reference with sun as it center). The mass of the vehicle carrying the spacecraft (Viking-2) with the fuel excluded is around 3,917 kg. The propellant mass required for the escape trajectory is 9,374.9 kg for ours reconstruct one. The original mission used 12,540 kg and had an initial orbit of 184 km above sea level or 6401 km from the center of the earth.

Figure 5. Centaur D1-T rocket stage

Time

The approximation of time of flight for the transfer orbit is half the transfer period or equal to 278 days. The original mission required 333 days cruise to mars from the earth orbit. In the original mission, the craft was launched on 9 September 1975 to earth’s orbit. The spacecraft would reach and enter Mars orbit on 7 August 1976.

Arrival

For the arrival trajectory, the spacecraft used to enter Mars orbit is the Viking-2 orbiter which has the value of Isp 294 sec.The original mission used 1445 kg of fuel. However, in the original mission, the spacecraft entered mars with an orbit of 1500 x 33,000 km and a period of 24.6 hr Mars orbit on 7 August 1976.

In our case, considering the nearest distance, d, from Mars to insertion hyperbolic trajectory the same as previous, which 30,000 km, we need reduction around 1,477 m/s to get circular orbit around 21,746 km. with propellant need 1,346.661 kg.

Calculation
For initial, known data

Table 1. GM of several bodies Involve

$μ_e$	3.986 x 10^14 m³/s²
$μ_m$	4.282 x 10^13 m³/s²
$μ_s$	1.327 x 10^20 m³/s²

Table 2. Earth Orbit Parameter

$e_{Earth}$	0.0167
$r_{pe}$	147.1 million km
$r_{ae}$	152.1 million km
$V_{ae}$	29.2898 km/s
$V_{pe}$	30.2847 km/s
$P_e$	365 day

Table 3. Mars Orbit Parameter

$e_{Mars}$	0.0934
$r_{pm}$	206.7 million km
$r_{am}$	249.2 million km
$V_{am}$	21.9702 km/s
$V_{pm}$	26.4970 km/s
$P_m$	687.0 day

Comparison

Figure 6. Original Mission orbit transfer trajectory

Figure 7. Original mission sequence

Figure 8. Hohmann transfer illustration (our analysis)

Table 4. Comparison

	OURS	VIKING 2
Parking Orbit	20,000 km (Circular)	Initial orbit around 20,000 km
Mars Orbit	21,746.367 km (circular)	4,192 km (perigee)
Transfer Type	Hohmann – no correction	Hohmann–several correction
TOF	278.477 days	333 days
Dry Mass	833 kg	833 kg
Propellant	2113 kg	1445 kg
Total Mass	2996 kg	2328 kg
Isp	500 s	444 s

In the Viking-2 original mission, the launch vehicle used was Titan IIIE which is comprised of 4 stages. At the parking orbit, the launch vehicle had already separated and consists of only the third stage which is the Centaur that carries the Viking lander and orbiter. The Centaur was used for the transfer orbit to reach Mars from the parking orbit at Earth. The Centaur was then separated during the transfer and the Viking-2 orbiter was then used for the arrival orbit at Mars.

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Reference

Anonymous. (n.d.). Titan IIIE launch vehicle specs. Retrieved April 23, 2018, from http://www.ninfinger.org/models/vault/viking/titan_iiie(c).pdf

Desai, P. N., Braun, R. D., & Powell, R. W. (1992). Aspects of Parking Orbit Selection in a Manned Mars Mission. Retrieved from ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930006178.pdf

Mai, T. (2015, May 07). September 1975 - Viking 2 Launched. Retrieved from http://www.nasa.gov/directorates/heo/scan/images/history/September1975.html

NASA. (1975). VIKING. Retrieved April 23, 2018, from https//marsprogram.jpl.nasa.gov/mro/odyssey/newsroom/presskits/viking.pdf.

NASA Facts. (1976, August 17). Viking Mission to Mars. Retrieved from https://www.jpl.nasa.gov/news/fact_sheets/viking.pdf.

NSSDCA. (n.d.). Viking 2 Lander. Retrieved April 23, 2018, from https://nssdc.gsfc.nasa.gov/nmc/spacecraftDisplay.do?id=1975-083C

Rocket and Space Technology. (n.d.). Titan. Retrieved April 23, 2018, from http://www.braeunig.us/space/specs/titan.htm

The Viking Mars Mission - Google Arts & Culture. (n.d.). Retrieved from https://artsandculture.google.com/exhibit/RQKiJsUJbOktIw

Viking 2. (n.d.). Retrieved from http://www.jpl.nasa.gov/missions/viking-2

Wade, M. (2007). Titan IIIE. Retrieved April 23, 2018, from http://www.astronautix.com/t/titaniiie.html

Weebau.com. (2008, July 13). Viking 2. Retrieved April 23, 2018, from weebau.com/satplan/viking 2.htm
Wikipedia. (2018, April 18). Inertial Upper Stage. Retrieved from https://en.wikipedia.org/wiki/Inertial_Upper_Stage

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