<p>The following is excerpted from from <a href="https://space.stackexchange.com/a/34181/12102">this answer</a> to the Space Exploration SE question "Do Curiosity's reported measurements of Mars' surface gravity (~3.717 m/s^2) include centrifugal effects?".</p>

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

<p>So the surface gravity on Mars is:</p>

<ul>
<li>3.70703 m/s² (0.378 g) on the equator</li>
<li>3.71683 m/s² (0.379 g) on the midlatitudes</li>
<li>3.73493 m/s² (0.381 g) on the poles</li>
</ul>

<blockquote>
  <p>So here's a complete tally using</p>
  
  <p><span class="math-container">$$a_G = -GM \frac{\mathbf{r}}{r^3}$$</span></p>
  
  <p>From <a href="https://en.wikipedia.org/wiki/Geopotential_model#The_deviations_of_Earth&#39;s_gravitational_field_from_that_of_a_homogeneous_sphere" rel="nofollow noreferrer">Geopotential_model; The_deviations of Earth's  gravitational field from that of a homogeneous sphere</a>:</p>
  
  <p><span class="math-container">$$a_{J2x} = J2 \frac{\mathbf{x}}{r^7}(6z^2 - 1.5(x^2+y^2))$$</span>
  <span class="math-container">$$a_{J2y} = J2 \frac{\mathbf{y}}{r^7}(6z^2 - 1.5(x^2+y^2))$$</span>
  <span class="math-container">$$a_{J2z} = J2 \frac{\mathbf{z}}{r^7}(3z^2 - 4.5(x^2+y^2))$$</span></p>
  
  <p><span class="math-container">$$a_C = \mathbf{r_{xy}} \omega^2 $$</span></p>
</blockquote>

<pre><code>            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
</code></pre>

<blockquote>
  <p>Do Curiosity's reported measurements of Mars' surface gravity (~3.717 m/s^2) include centrifugal effects?</p>
  
  <p><strong>Yes they do!</strong> At 5.2 degrees south latitude and -4500 meters altitude (bottom of Gale Crater) the acceleration is <strong>-3.7168 m/s</strong> taking into account GM, J2, and centrifugal effects. </p>
  
  <p>So as Cheap Trick <a href="https://en.wikipedia.org/wiki/Everything_Works_If_You_Let_It" rel="nofollow noreferrer">tells us</a> and Meatloaf reiterated more elegantly in <a href="https://en.wikipedia.org/wiki/Roadie_(1980_film)" rel="nofollow noreferrer">Roadie</a>: </p>
  
  <p><strong><em>Everything works if you let it!</em></strong></p>
  
  <p>Here's some Python for double-checking my math:</p>
</blockquote>

<pre><code>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())
</code></pre>

CC BY-SA 4.0
2020-01-28T13:31:54.277Z
I've made some cosmetic edits, but this is not a complete answer yet. Future readers wondering "How does the surface gravity on Mars vary between the equator and its poles?" and "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?" won't be able to find the answer here, at least not clearly. Do you think you can add a short section at the top to directly answer the question as asked? Thanks!
uhoh
CC BY-SA 4.0
2020-01-28T14:20:48.837Z
@uhoh If someone answers me whether the Phoenix lander has a gravimeter and if it has one what gravity it did measure, I will add it to our answer.
user30007
CC BY-SA 4.0
2020-01-28T10:56:06.277Z
2020-01-28T14:24:13.017Z
2020-01-28T14:24:13.017Z
user30007
<p>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. </p>

<p>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).</p>

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

CC BY-SA 4.0
2020-01-28T09:31:16.340Z
Are you asking how to calculate it?Or what it has been measured as?
ProfRob
CC BY-SA 4.0
2020-01-28T09:59:23.800Z
@RobJeffries Both, I'd like to know how you calculate centrifugal forces into (G x m)/r² and if any probe measured Mars' gravity on the equator/midlatitudes/poles (on Mars or from its orbit, but including centrifugal forces).
user30007
CC BY-SA 4.0
2020-01-28T10:18:26.143Z
There are some different but related questions in Space Exploration SE; feel free to write an answer to your own question based on information there if there's enough: [How much is the difference of gravity attraction between the highest and lowest point on Mars?](https://space.stackexchange.com/q/34174/12102) and [Do Curiosity's reported measurements of Mars' surface gravity (~3.717 m/s^2) include centrifugal effects?](https://space.stackexchange.com/q/33961/12102) and [Did the Mars rovers actually confirm the gravity of Mars?](https://space.stackexchange.com/q/14947/12102)
uhoh
CC BY-SA 4.0
2020-01-28T10:31:37.163Z
@uhoh Thank you! May I copy your answer from "Do Curiosity's reported measurements...?" into here?
user30007
CC BY-SA 4.0
2020-01-28T10:38:51.643Z
@user30007 Sure, this is Stack Exchange so everything is for everybody to use! When I quote a section from another answer I use the block quote feature with `>` and if it already contains a block quote I just add a 2nd one (i.e. `>>`). It's always good to add a bit in your own words to address anything specific to your new question.
uhoh
CC BY-SA 4.0
2020-01-28T10:49:05.480Z
@user30007 Also, don't forget to give proper attribution to copied material, as mentioned in https://astronomy.stackexchange.com/help/referencing
PM 2Ring
CC BY-SA 4.0
2020-01-28T11:00:07.627Z
Sorry, I don't understand how to link within the stack exchange. Feel free to edit my answer.
user30007
<gravity><rotation><mars>
How does the surface gravity on Mars vary between the equator and its poles?
user30007
# How does the surface gravity on Mars vary between the equator and its poles?

(Asked by: username_0 on 2020-01-28)

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.

**Tags:** gravity, rotation, mars
> **Comments:**
> **username_1**: Are you asking how to calculate it?Or what it has been measured as?  
> **username_0**: @RobJeffries Both, I'd like to know how you calculate centrifugal forces into (G x m)/r² and if any probe measured Mars' gravity on the equator/midlatitudes/poles (on Mars or from its orbit, but including centrifugal forces).  
> **username_2**: There are some different but related questions in Space Exploration SE; feel free to write an answer to your own question based on information there if there's enough: [How much is the difference of gravity attraction between the highest and lowest point on Mars?](https://space.stackexchange.com/q/34174/12102) and [Do Curiosity's reported measurements of Mars' surface gravity (~3.717 m/s^2) include centrifugal effects?](https://space.stackexchange.com/q/33961/12102) and [Did the Mars rovers actually confirm the gravity of Mars?](https://space.stackexchange.com/q/14947/12102)  
> **username_0**: @uhoh Thank you! May I copy your answer from "Do Curiosity's reported measurements...?" into here?  
> **username_2**: @user30007 Sure, this is Stack Exchange so everything is for everybody to use! When I quote a section from another answer I use the block quote feature with `>` and if it already contains a block quote I just add a 2nd one (i.e. `>>`). It's always good to add a bit in your own words to address anything specific to your new question.  
> **username_3**: @user30007 Also, don't forget to give proper attribution to copied material, as mentioned in https://astronomy.stackexchange.com/help/referencing  
> **username_0**: Sorry, I don't understand how to link within the stack exchange. Feel free to edit my answer.  

---

## 1 answers

### Answer by username_0 on 2020-01-28. Score: 2 | Accepted answer

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())

> **Comments:**
> **username_2**: I've made some cosmetic edits, but this is not a complete answer yet. Future readers wondering "How does the surface gravity on Mars vary between the equator and its poles?" and "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?" won't be able to find the answer here, at least not clearly. Do you think you can add a short section at the top to directly answer the question as asked? Thanks!  
> **username_0**: @uhoh If someone answers me whether the Phoenix lander has a gravimeter and if it has one what gravity it did measure, I will add it to our answer.  

---

