How does the surface gravity on Mars vary between the equator and its poles?

Question

A day on Mars is similar in length to the Earth's day. On Earth, due to rotation, the surface gravity varies from 9.78 m/s² (0.997 g) on the equator to 9.832 m/s² (1.003 g) on the poles. I wonder how the gravity varies on Mars where the day is of similar length. We hear of different values, such as 3.69, 3.711 or 3.72076 m/s². All values around 0.38 g. But I wonder what is the exact value of the equatorial gravity on Mars (including centrifugal forces), what it's on the midlatitudes and on the poles.

The Phoenix probe landed near the north pole; do we know whether it has a gravimeter and measured the surface gravity on its landing place? I know the Curiosity rover which landed close to the equator measured a gravity of around 3.71 m/s² from which I conclude the 3.711 value is for the equator (0.378 g).

When I calculate the gravity of Mars with (G x m)/r² it doesn't include the centrifugal force of Mars' rotation.

Answer 1

The following is excerpted from from this answer to the Space Exploration SE question "Do Curiosity's reported measurements of Mars' surface gravity (~3.717 m/s^2) include centrifugal effects?".

It essentially answers my question. I would appreciate if someone gave the info about the gravimeter on the Phoenix lander too.

So the surface gravity on Mars is:

• 3.70703 m/s² (0.378 g) on the equator
• 3.71683 m/s² (0.379 g) on the midlatitudes
• 3.73493 m/s² (0.381 g) on the poles

So here's a complete tally using

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

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

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

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

            magnitudes shown only (sign indicates generally "up" or "down")
                  at the equator        at the pole
              h = 0     h = -4500 m        h = 0    
GM          -3.71317     -3.72303        -3.75729
J2          -0.01092     -0.01088        +0.02236
centri      +0.01706     +0.01697         0.0

vector sum  -3.70703     -3.71683        -3.73493

Do Curiosity's reported measurements of Mars' surface gravity (~3.717 m/s^2) include centrifugal effects?

Yes they do! At 5.2 degrees south latitude and -4500 meters altitude (bottom of Gale Crater) the acceleration is -3.7168 m/s taking into account GM, J2, and centrifugal effects.

So as Cheap Trick tells us and Meatloaf reiterated more elegantly in Roadie:

Everything works if you let it!

Here's some Python for double-checking my math:

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

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 ''

aaacs = accelerations(np.array([Req, 0, 0]))
print "Req: ", Req
for thing in aaacs:
    print thing, np.sqrt((thing**2).sum())
print "total: ", sum(aaacs), np.sqrt((sum(aaacs)**2).sum())

print ''
lat = rads * -5.4
aaacs = accelerations(np.array([R*np.cos(lat), 0, R*np.sin(lat)]))
print "R: ", R
for thing in aaacs:
    print thing, np.sqrt((thing**2).sum())
print "total: ", sum(aaacs), np.sqrt((sum(aaacs)**2).sum())

print ''

aaacs = accelerations(np.array([0.0001, 0, Rpo]))  # avoid divide-by-zero (lazy!)
print "Rpo: ", Rpo
for thing in aaacs:
    print thing, np.sqrt((thing**2).sum())
print "total: ", sum(aaacs), np.sqrt((sum(aaacs)**2).sum())