How much is the difference of gravity attraction between the highest and lowest point on Mars?

Question

Mars has a very high mountain, Olympus Mons (25 km), while Hellas Planitia is a 9 km deep crater.

How much would the weight of a person weighing 100kg on Earth, change between the highest peak of the Olympus Mons and the Hellas Planitia crater's deepest point?

Answer 1

tl;dr: Relative to Earth, the weight of an object on Mars would be about 37.36% and 38.09% on Olympus Mons and Hellas Planitia. The difference between those two is about 2.0%. That compares to only a 0.7% range on Earth.

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The three main terms for the phenomenon of weight on a typical rocky planet's surface are:

1. gravitational monopole (GM) force
2. gravitational quadrupole (J2) force
3. centrifugal effects

I've copied the math from this answer.

$$a_G = -GM \frac{\mathbf{r}}{r^3}$$

From Geopotential_model; The_deviations of Earth's gravitational field from that of a homogeneous sphere:

$$a_{J2x} = J2 \frac{\mathbf{x}}{r^7}(6z^2 - 1.5(x^2+y^2))$$ $$a_{J2y} = J2 \frac{\mathbf{y}}{r^7}(6z^2 - 1.5(x^2+y^2))$$ $$a_{J2z} = J2 \frac{\mathbf{z}}{r^7}(3z^2 - 4.5(x^2+y^2))$$

$$a_C = \mathbf{r_{xy}} \omega^2 $$

Convert (x, y, z) to latitude and longitude

Reference ellipsoid of Mars:

$$a \approx 3396200$$

$$b \approx 3376200$$

From this answer:

According to Wikipedia's Geographic_coordinate_conversion#From_geodetic_to_ECEF_coordinates

The 3D cartesian coordinates $X, Y, Z$ in Earth-centered, Earth-fixed coordinates assuming an ellipsoidal shape is given by:

$$X = \left(N(\phi) + h \right) \cos\phi \cos\lambda $$

$$Y = \left(N(\phi) + h \right) \cos\phi \sin\lambda $$

$$Z = \left(\frac{b^2}{a^2} N(\phi) + h \right) \sin\phi $$

where $\phi, \lambda, h$ are latitude, longitude, and altitude, and $a, b$ are the equatorial and polar radii of the ellipsoid used, and

$$N(\phi) = \frac{a^2}{\sqrt{a^2\cos^2\phi + b^2 \sin^2\phi}}. $$

• Olympus Mons: 18.65°N, 226.2°E, 21,287m: X=-2,242,521, Y=-2,338,480, Z= 1,080,757 meters

• Hellas Planitia: 42.4°S 70.5°E, -7,152m: X=837,649, Y=2,365,449, Z=2,264,417 meters

Using Python below (similar to the script in that answer):

            magnitudes shown only (sign indicates generally "up" or "down")
                      OM            HP         
GM     (m/s^2)     -3.67139     -3.74879  
J2     (m/s^2)     -0.00984     -0.01172 
centri (m/s^2)     +0.01628     +0.01261  

vector sum (m/s^2) -3.66344     -3.73562

So for a 100 kg mass:

                       OM            HP         
weight (N)           366.34        373.56 
Earth relative(%)    37.357        38.093

where "relative" is relative to Standard Earth gravity of 9.80665 m/s^2.

def accelerations(rr):

    x,   y,   z   = rr
    xsq, ysq, zsq = rr**2

    rsq   = (rr**2).sum()
    rabs  = np.sqrt(rsq)
    nr    = rr / rabs
    rxy   = np.sqrt(xsq + ysq)
    rrxy  = rr * np.array([1.0, 1.0, 0.0])
    nxy   = rrxy/rxy
    rm3   = rsq**-1.5
    rm7   = rsq**-3.5

    acc0  = -GM_mars * rr * rm3

    # https://en.wikipedia.org/wiki/Geopotential_model#The_deviations_of_Earth.27s_gravitational_field_from_that_of_a_homogeneous_sphere
    acc2x = x * rm7 * (6*zsq - 1.5*(xsq + ysq))
    acc2y = y * rm7 * (6*zsq - 1.5*(xsq + ysq))
    acc2z = z * rm7 * (3*zsq - 4.5*(xsq + ysq))

    acc2  = J2_mars * np.hstack((acc2x, acc2y, acc2z))

    accc = nxy * omega**2 * rxy

    return acc0, acc2, accc

def get_xyz(latdegs, londegs, alt):
    lat, lon = rads*latdegs, rads*londegs
    clat, slat = np.cos(lat), np.sin(lat)
    clon, slon = np.cos(lon), np.sin(lon)

    N = a**2 / np.sqrt((a*clat)**2 + (b*slat)**2)
    X = (N+alt) * clat * clon
    Y = (N+alt) * clat * slon
    Z = ((b/a)**2 * N + alt) * slat

    return np.array((X, Y, Z))
    

import numpy as np

halfpi, pi, twopi = [f*np.pi for f in [0.5, 1, 2]]
degs, rads        = 180./pi, pi/180.

R_mars  = 3396200.0
GM_mars = 4.282837E+13   #  m^3/s^2  https://en.wikipedia.org/wiki/Standard_gravitational_parameter
J2_mars = GM_mars * R_mars**2 * 1960.45E-06     # https://nssdc.gsfc.nasa.gov/planetary/factsheet/marsfact.html

Req = 3396.2 * 1000. # meters https://en.wikipedia.org/wiki/Mars
R   = Req - 4500.    # https://en.wikipedia.org/wiki/Gale_(crater)
Rpo = 3376.2 * 1000. # meters https://en.wikipedia.org/wiki/Mars

sidereal_day  = 1.025957 # https://en.wikipedia.org/wiki/Mars
T             = sidereal_day * 24 * 3600.
omega         = twopi/T
print "omega:   ", omega
print ''

a, b = Req, Rpo

OM_xyz = get_xyz(18.65, 226.2, 21287)
HP_xyz = get_xyz(42.4,  70.5, -7152)


print "OM: ", OM_xyz
print "HP: ", HP_xyz

OM_acs = accelerations(OM_xyz)
HP_acs = accelerations(HP_xyz)

print ""

print "OM: "
for thing in OM_acs:
    print thing, np.sqrt((thing**2).sum())
print "vector sum: ", np.sqrt((sum(OM_acs)**2).sum())

print ""

print "HP: "
for thing in HP_acs:
    print thing, np.sqrt((thing**2).sum())
print "vector sum: ", np.sqrt((sum(HP_acs)**2).sum())